Research Journal

Monday, July 30 2012

    Jeff made two more interferometers with the same 170 micron apertures: a 9-pinhole 2mm radius and a 6-pinhole 1.75mm radius.  The 6-pinhole interferometer works as expected.  I'm looking for the best distance to get clear patterns.  Since these patterns are unique rather than shifted it isn't important to find the center of the patterns, although they've occurred clearly in the overlapping outer rings of the Airy patterns.  The intensity plot shows that I can expect 6-lobed snowflake patterns for l=3,4 and this seems to be the result, but the eye isn't the best judge of an exact match.  It also suggests what I might see if certain holes are blocked.  I need to block them completely without accidentally damaging them.  I think, due to the easy to interpret patterns (compared to 5 pinholes), the 6 hole MPI might be better suited to the eventual goal of profiling a speckled wavefront. 

    It's been a little difficult producing pure modes with the open cavity laser, and the mode sometimes changes while I'm moving the CCD back and forth.  Sometimes I admit mixed modes by mistake.  It has stopped lasing several times.  I spent most of yesterday adjusting the laser to get higher order modes, but there is a tradeoff.  When many modes are present before filtering, the chance of the mode changing in mid measurement are greater but higher order HG modes can easily be produced.  When fewer modes are present initially, it is very difficult to reach l=4 with the astigmatic mode converter, but modes often disappear rather than change when there is instability.

    The 9 pinhole MPI has been a surprise.  I didn't know what to expect except that the interference patterns might resemble a Bessel function.  I'd need to take hundreds of measurements to know with certainty, but after measuring l=1 through 4 with it several times the patterns appear to have a clear center, and to be consistent after reforming the measured modes.  I speculated earlier that calibrating the MPI with known modes at a certain distance would work, and so far it seems to give an adequate guide from which new modes can be read.

Wednesday, July 25 2012

    Things are happening.  Jeff from the shop drilled a 1.75mm radius 5 pinhole interferometer that successfully forms the interference patterns that are needed to determine the OAM of the input LG beam.  The apertures were 170 microns.  The problem is how to calibrate it since the patterns are periodic for 5 pinholes.  I am thinking about calibrating it with known OAM beams and leaving everything locked into place, or finding the center of the pattern from the known positions of the pinholes.  A lucky accident occurred when I was trying unsuccessfully to achieve the interference pattern for an LG 2 beam.  It wasn't appearing, but while troubleshooting I realized that a bump to the laser had changed the mode to something that was not producing an LG beam.  So the correct interference pattern won't just appear arbitrarily.  Next, I will trouble the shop to drill a 6 hole and a 9 hole interferometer.  The 9 hole MPI will allow us to measure OAM beyond what was included in the original papers.  The 6 hole MPI has the special characteristic that the patterns are totally distinct, rather than shifted.   My model for the 6 hole MPI is displaying correctly as well, so I can make predictions for what will be seen.  

Thursday, July 19 2012

    The long awaited Optical Vortex Party arrived.  We spent the day at the City College of New York listening to introductory lectures and student research presentations.  Kiko Galvez and Giovanni Milione delivered the overview lectures.  Giovanni started off, with emphasis on light as a wave.  Although the presentation was general, he jumped right into the area that interests me most about vortices, optical communication.  I'm hoping to build up my knowledge about that, and how vortex beams behave in optical fibers this fall.  Giovanni said I was welcome to join their lab.  I will.  This will be helpful since I must skip the undergraduate optics class.  Dr. Galvez focused on light as a particle.  He showed different methods of representing its polarization state, the Poincare sphere and a type of cross sectional graph. 

    The City College students gave presentations.  The most interesting to me were the ones that took data from the two-lens mode sorter that converts OAM into transverse momentum, and another that described the apparatus.  The students from Colgate and one from City College brought posters describing their summer research projects.  The student from City College made Bessel beams using a Bessel diffraction grating.  Somehow in all the talk at the LTC about methods of creating Bessel beams, I'd never seen this method.  Another brave soul from Colgate took on the challenge of quantum computing.  When I joined the APS and had to choose two of their journals for free access, I selected the publication on quantum computing because many of my classmates want to work in that area.  I've heard it's fraught with pitfalls.  Melia, Marissa, and I also gave presentations on the work we've done so far.

    We didn't get a full lab tour because one lab was locked, but what we saw was still exciting.  After reading about them dozens of times, I finally saw an SLM in action.  The program that runs the SLM simply outputs to the device as though it were a secondary computer monitor.  It could be changed almost instantly, and easily produce more exotic superpositions of vortex beams than can be achieved with an astigmatic mode converter.

    We ended with dinner at Loi, a Greek restaurant on W. 70th street.  The scallops and octopus were among the best I've eaten.  I thank Dr. Noe for taking us there.

Tuesday, July 17 2012

    Dr. Noe suggested I use the MPI to measure the OAM of higher modes.  This covers material alluded to in Berkhout and Beijersbergen's work, but not included in the paper.  The measurements they included stopped at l=2.  Marty and I had discussed using a 5 pinhole MPI.  The number of OAM states that can be observed is governed by the number of pinholes.  For odd number N pinholes, you can observe N different OAM states.  So if we wanted to measure up to l=+/- 5, produced by the open cavity laser, 11 pinholes are necessary.  It's known that for pinhole numbers greater than 5, not including 6, the diffraction pattern has a unique center.

    Darkroom Specialties has been in contact and they will know how small they can print circular apertures by tonight.  If they are able to, it will be easier to make MPIs to measure higher order OAM states.  Printing 13 pinholes shouldn't be more difficult than printing 5.  I also made an inquiry to a laser drilling company about a custom pinhole plate, but I suspect this will be very expensive.  Even if they could do it, I think it would be best to verify we understand the papers with the cheapest available MPI first.  If the MPI can't be printed, we can drill pinholes as small as 200 microns here with new drill bits, or 300 microns using what's in the shop.  I like this option because we can make adjustments within a day if things don't work out or we want to try something new.  Drilled pinholes probably won't be perfect, but in the Guo et al. paper the authors claim to have not been very precise in pinhole design and they were able to distinguish OAM states.  However, their patterns had some distortion, destroying the azimuthal symmetry for small wedges of the pattern.  Modeling the overlap of the Airy disks from each pinhole shows that even using the largest pinholes (300 microns) and a large MPI radius of 1mm we can achieve significant Airy disk overlap 2 meters from the MPI.  Using 100 micron pinholes this overlap can be achieved by half a meter, the beginning of the far field.

    I will continue to work on introducing tilt into the Mathematica model in order to make simulated measurements from a wavefront that is not flat and has multiple vortices passing through multiple MPIs.  Since this will be time consuming and is done by computer, it's a project I will probably carry on after the formal end of the REU.  When our MPIs are complete, I will take some tilt measurements, which Berkhout and Beijersbergen simulated but didn't physically measure.  Of the prospective uses for this research, the response of the MPI to a tilted vortex from a speckle pattern is part of the long term goal of using the MPI to characterize wavefronts.  This research area is interesting, and when I can take the time to develop a model as complete as the authors' there are all kinds of things I can look into with it.

Monday, July 16 2012

    I made more progress in putting  together a basic MPI.  I contacted Gene Lewis of Darkroom Specialties about printing out an opaque slide with transparent pinholes of around 100 microns, or even 50 if they can achieve it.  His company made a diffraction grating of some kind for Pradyoth Kukkapalli in 2010 (Note to Dr. Noe: The link to Pradyoth's webside from the past projects page erroneously goes to Annie Nam).  Mr. Lewis responded that their film recorder, used to make the slides, imprints digital images on the slide at a resolution of 2732x4000 and he's looking into whether that will achieve a pinhole size below 300 microns, the size we can produce here.  I made a couple of tools on Mathematica.  One shows the overlap of the central Airy disks on a projection screen at adjustable distances from an MPI with adjustable radius and aperture diameter.  It shows, barring any error, that the 300 micron pinholes we can make at Stony Brook should be adequate to overlap a broad area of the Airy disks within table distance.  The other tool is for making MPI patterns to send to Darkroom Specialties.

    I made a scale image from the tool, proportional to 100 micron pinholes and a 1mm ring radius.  I made another at the proportions Berkhout and Beijersbergen used when scanning across one dimension of a speckle pattern, 50 micron pinholes at 100 micron radius.  This pattern looked strange because the authors had repeatedly specified that the hole separation needs to be significant compared to the aperture diameter to avoid convolution by the diffraction of the individual holes and the aperture separation didn't look more than twice the aperture diameter.  In their paper on speckle measurement they found that pinhole separation of 10% of the average speckle size produced good results.  However, it matches the dimensions they specified.  If Mr. Lewis could get the apertures down to 50 microns we could even do something with speckle pattern profiling.  I can also consider using multiple rings of pinholes, as the authors did, and blocking the ones I'm not using to make an adjustable MPI.

    I'm thinking about how to make the Mathematica model more general, perhaps simulating a simplified speckle pattern wave and its diffraction through an MPI with multiple rings of apertures.  This is something the authors proposed, but haven't done.  Because the patterns are shifted when a vortex is off axis, and because the vorticity in speckle is known to be almost exclusively +1 and -1, comparing the known positions on the speckle pattern with the shifts in the diffraction pattern could be a method to characterize a wave front like a Shack Hartmann sensor.  It might be more feasible to simulate one vortex through one MPI with an adjustable angle to the axis of the MPI, and test those predictions with our MPI, perhaps adjusting the beam angle slightly and seeing if the pattern captures this angular discrepancy as expected.  A model of this would probably consume most of the remaining time and perhaps stretch on a little longer.  A general script for Mathematica could also let us freely adjust the hole positioning and make predictions for different types of MPI.  However, my script for showing the diffraction pattern of a circular MPI isn't displaying the odd pinhole patterns very clearly.  An odd pinhole MPI can discern the handedness of the vorticity, while an even pinhole MPI cannot.  While waiting for the slides to arrive from Mr. Lewis I may devote all the available time to making the model more useful. 

Sunday, July 15 2012

    I've been working on two problems with the MPI and made some progress.  The first problem was to figure out what size to make the radius of the circle that the holes are arranged around, given the lower limit of hole size the shop can drill.  I assumed that diffraction through any one of the holes should produce an Airy pattern because the intensity is constant at each hole in an LG mode, since their intensities are radially symmetric.  I also assumed that the azimuthal phase change over the width of one hole was small enough to leave the pattern unaffected. 

    On Thursday night I diffracted points on the high intensity annulus of various LG modes through pinholes ranging from 75 microns to 500 microns into the CCD camera to verify that the pattern resembled the Airy pattern.  It did, but I didn't profile it and it was very faint.  I had to wait until everyone was gone so I could keep the room as dark as possible.  There was a problem with the pictures saved from the camera displaying with the wrong proportions compared to what I saw on the screen.  The only way I could find to compensate was to use windows screenshots and paste them into paint. 

    My reading, and Marty's reading, of the paper suggested that the diffraction patterns the authors made occurred in the central Airy disks overlapping from the pinholes, although they didn't specifically say this.  So I chose a radius that would overlap the central disks from each of the holes.  After discussing it with Marty, he suggested that the overlap should occur within a reasonable distance so it was easier to set up on the table.  Anything more than a meter or two, he said, would start to be awkward.  Using the larger pinholes that the shop can make will extend this distance, but not impractically so.  I had also calculated the overlap of the central disks based on the radial distance from the center to the first minimum, but much of the pattern leading up to the minimum won't be visible so we decided to reduce the radius as much as possible, careful not to overlap the holes.

    The second problem was using the known intensity pattern for a circular MPI to make a visual model.  I hadn't used Mathematica before, but since it was available through Stony Brook, I started working on it and Matlab to write a script with adjustable parameters to graph the diffraction pattern.  Mathematica turned out to be more intuitive and flexible in setting up the model.  I now have a tool that shows me the diffraction pattern where I can adjust all the elements like wavelength and MPI radius with value sliders.  I was able to reproduce many of the patterns shown in the papers at various pinhole counts and topological charge.  There was one strange thing.  The patterns didn't occur at the same distances from the interferometer when observing through different pinhole counts.  I don't think Berkhout and Beijersbergen specified the distances at which they recorded the patterns, so it may have been a varying distance that they adjusted until they found an obvious pattern.  Right now the resolution from the Mathematica intensity plot I used isn't as clear as I'd like.  I have the added problem that the readable models I'm running on my laptop at the lab overwhelm it.  I can't modify them with the laptop either, but have to go into the sinc sites while they are open or work on my full powered desktop in the Bronx to use the full version of Mathematica.

    Marty suggested that looking into non redundant masks (something like a circular MPI but without radial symmetry) might be a chance to take the project in an original direction.  Making an original model that takes this into account, as well as other factors like the pinhole size, seems possible.  Ideally, I could replace the sliders with a visualization of the MPI where the pinhole positions could be adjusted with a mouse and the predicted output displayed.  I talked to Dr. Simon about non redundant masks during the trip to the AMNH on Friday, since he's worked with them, and in response he has prepared a 30 minute presentation on them for this Wednesday's meeting.  A non redundant mask used for the soon to be launched James Webb space telescope was designed by the astronomers at the AMNH.

    Below I will list, in one place, the Berkhout and Beijersbergen sources about using a circular MPI to measure optical vortex topological charge:

2008 original paper-
2009 followup-
2010 paper about using their technique on a speckle pattern-

The following sources were cited prominently by Berkhout and Beijersbergen:
2009 paper explaining a method of reconstructing the phase of a vortex passing through an MPI using the Fourier transform of a single diffraction intensity pattern-
2009 Schoonover and Viser paper describing some general theory behind MPI-

Tuesday, July 10 2012

    I'm tentatively proceeding with the Berkhout and Beijersebergen multi-pinhole interferometer (MPI).  The authors describe it in the two papers written 2008 and 2010, shortly before they coauthored work on the two-custom-element mode sorter for single photons that we looked into last week.  The authors recommended one additional 2009 paper that describes reconstructing the phase of an optical vortex passing through an MPI using the Fourier transform of a single diffraction intensity pattern.  This work is, in some sense, a problem of geometrical optics.

    The pinholes in the intended MPI are arranged in a circular pattern.  There are other schemes that have been used for this purpose and this work also relates to a paper about two slit and two pinhole diffraction of LG beams that Dr. Noe previously pointed out as interesting.  I intend to use the circular pattern.  There may be a relationship to Melia's work in that the intensity pattern that the authors derived converges to a Bessel function of the order of the topological number of the input LG beam as the number of pinholes increases.

    Because the intensity pattern is known (in the case that the MPI is perpendicular to the beam axis), it is possible to create a model of the expected patterns which I will attempt with matlab (which I'm familiar with) or possibly mathematica (which I'm not familiar with).  I need to wait until I'm home for the weekend where I have a computer with those math programs.  I don't own a windows laptop.  The model should hopefully allow me to input the details of the spacing of my pinholes and view the expected diffraction pattern, starting with 2 pinholes reproducing the double slit diffraction pattern.

    The work is intended to apply to LG beam cross sections of arbitrary size, but the intensity drop off will present problems in observing the diffraction pattern.  Measuring the diffraction pattern of a more focused beam involves putting very small pinholes very close together.  The authors, when measuring near the beam waist, used 50 micron pinholes as a compromise between good diffraction and a useful level of throughput.  In observing single pinhole diffraction in the lab from various distances beyond the focal point of an LG beam with a 75 micron aperture the diffraction patterns quickly became faint.  I'm hoping the camera can pick up a lot of details that my eyes can't.  I'm looking into the best method and material to make very small pinholes of a consistent size, and how to measure that size.

    On a lucky note, although it didn't seem possible to reproduce the authors' mode sorter due to the custom lenses and our lack of SLMs  to replace them, at the optical vortex party next week at City College we will visit the Alfano lab where Giovanni Milione has a pair of them that he will perhaps demonstrate.  I'm appreciative that Dr. Noe talked him into giving us a lab tour.

Monday, July 9 2012

    Dr. Noe gave me access to the He-Ne open cavity laser on Friday evening.  Using the astigmatic lens mode converter I saw LG modes for the first time, generated from HG modes up to order 4,0.  As expected, it can be difficult to get the lenses lined up.  The Miles Padgett AJP paper that has the most basic explanation of the mode converter also describes an interferometer to observe the phase structure and intensity of LG modes that I may try to build Tuesday to observe the phenomena while I learn about them.  With a day full of lectures, and a new topic Dr. Noe brought up I haven't had time to try it yet.

    Dr. Noe brought up the topic of the Talbot effect and OV.  In searching for information about the Talbot effect and diffraction in OV I came upon the topic of pinhole interferometry, explored in two papers (1, 2) by Berkhout and Beijersbergen.  I took an interest because we had recently derived the interference pattern for two slit diffraction.  This is a more complicated version of that problem.  The authors derived a formula for the interference pattern caused by diffraction through circular rings of pinholes.  This interferometer has several useful characteristics.  It can measure the OAM of an OV of arbitrary size as long as the singularity is enclosed in the ring and is proposed for use in astronomy.  This was also shown to be useful for measuring singularities in a speckle pattern.  The authors propose its use for investigating the interstellar medium by analyzing the speckle pattern caused by starlight scattering from inhomogeneities in the medium.  It is also stated that the pinhole plate does not need to be perfectly perpendicular to the source beam.  Other shapes of pinhole pattern have been explored, but one interesting feature of the circular pinhole plate is that the interference pattern, in theory, converges to a bessel function of the order of the OAM as the number of pinholes is increased.  The authors show this effect in simulation, visible through a plate with 16 pinholes. 
    We attended several lectures since the last update.  On Friday Dr. Hal Metcalf gave a second talk to familiarize us with the Rabi frequency and the Bloch vector so that we'd have some basis to understand Daniel Stack's thesis defense, given on Monday.  Hal described the energy levels of a hydrogen atom, and the effect of shining light on it at the resonant frequency,  disregarding all energy levels not at the resonant frequency.  If I recall correctly, Hal used a different approach than I had been exposed to in David Griffiths' quantum mechanics textbook.  I had been in the habit of thinking about allowable energies, without realizing the connection to modes.  I will be reviewing the hydrogen atom in August and I'll try to incorporate the point of view that Hal has been communicating with us.

    On Monday we were paid a visit by Andrew MacRae, a prospective postdoc presenting information from his thesis.  He spent some time describing, in a general sense, how quantum models of many particles weren't always of much interest because they told you about macroscopic phenomena you already knew.  We showed him some of what we'd been working on in the lab.  Although his thesis was probably lost on me, I was impressed during his lab visit at just how much he recalled about the basic things we were working on.  I found the quasi-phase space he described in his lecture interesting.  He described it as a Wigner distribution assembled by repeatedly measuring x and p separately, meaning it's not really a probability distribution.

Thursday, July 5 2012

    Dr. Noe expressed interest in the 2012 Lavery, Padgett paper Refractive elements for the measurement of the orbital angular momentum of a single photon that came up last  week.  This is a followup paper based on earlier work the authors conducted with SLMs in which they use two custom lenses to build a mode transformer.  Their transformer sorts a beam that is a superposition of LG modes with different topological numbers by transforming the OAM from the l index of the LG beam into a transverse momentum.  A third lens then separates the beams of different transverse momentum onto distinct positions on a screen.  The authors hoped to devise a simple system with fewer parts that could sort large numbers of OAM states with minimal channel crosstalk.

    The first custom optical element converts the ring pattern of the LG mode into lines, but this introduces a phase error due to the varying path length.  The second custom optical element corrects these phase errors.  The transformation itself is described in a 1974 paper detailing general types of coordinate transformations.  The optical elements were first demonstrated using an SLM before, in 2012, they were manufactured with with an ultra precision lathe.  The SLM method resulted in large intensity loss.  The manufactured elements resulted in 85% transmission efficiency that was suitable for use with single photon detection.  The formulae for the shape and phase change of the two custom elements is given, but there is  no obvious solution to replacing them without an SLM.

    Making a mode transformer with multiple elements is possible.  The authors describe previous methods achieved with multiple fork diffraction gratings, or a cascade of Mach-Zehnder interferometers.  The authors contrast their method with previously demonstrated mode transformers, noting that previous methods have a burdensome level of complexity due to the ability of holograms to detect only one mode at a time (or several modes requiring many photons), requiring large numbers of holograms and photons to sort large numbers of modes.

    Aside from custom optical elements and SLMs, custom holograms can also be designed to sort OAM states, although less efficiently and with less range of OAM.

Tuesday, July 3 2012

    I left out some of what I did on Friday with the fiber-optic cable.  During the week we had been trying to get a stable reading on the intensity of the He-Ne laser.  As described previously, the output through the single mode fiber was ~60%, but the intensity was oscillating.  After removing and then returning the coupler, using the multimode fiber, the reading stabilized.  I then replaced the multimode fiber, mistakenly, with a 1310 nm single mode fiber and the beam pattern projected on the screen clearly resembled the HG11 mode, four lobes with rectangular symmetry.  Bending the cable, which was loosely coiled, changed the projected pattern, which in some cases resembled an LG mode.  I haven't yet attempted to do this in a systematic way, nor have I found the best way to photograph the projections.  Several kinds of mode conversion can be performed in both regular and twisted optical fibers.  Marty mentioned that Jacob Chamoun of the LTC had played around with this stress induced birefringence in optical fibers using paddles but that there were too many variables to do it in a controlled way.  I am interested in output modes with orbital angular momentum, or output that comes from vortex beam inputs.

    When looking to produce particular modes as output, the picture is greatly complicated by the presence of the additional modes native to optical fibers.  Predicting the output seems difficult.  According to the Bozinovic paper I mentioned Monday, the TE and TM modes within an optical fiber have a tendency to couple with those modes that can combine to create output with angular momentum, and this TE/TM coupled output is instead linearly polarized, carrying no OAM.  This has implications for transmitting a vortex beam through an optical fiber, or simply producing one by combining other modes in a distorted fiber.  The authors made a prediction of the range of output OAMs that could be found when distorting the optical fiber based on this paper.  Measuring the OAM of the output to determine the modes might be more manageable than predicting it.  To this end I am presently reading through the papers describing known methods of measuring the OAM of light Miles Padgett listed in his own paper Measuring orbital angular momentum superpositions of light by mode transformation to see what can be conveniently installed along with the fiber-optic setup I'm using.

    Question: What can measuring the OAM of a mixture of LG modes in free space tell me about the source?  For instance, if I have combined beams of different topological charge and submit them to one of these measurement methods what will the result look like versus a beam of a single topological charge?

    To get some hands on time with the equipment, I joined Melia in profiling the central dot of the Airy pattern she produced across its central x-axis.  Another REU student, Jun, had earlier described the technique to me over lunch and shown me some intensity plots of a laser he was working on at the Schneble lab.  Jun said it was one of the basic things he learned on a previous stint in a laser lab.  Eager to learn the basics of the equipment, I was planning to try it as soon as the chance arose.  We had some difficulty locating a table to attach the photodiode to.  Dr. Noe recommended we use a pinhole so we attached a 150 micrometer one.  Failing to use the pinhole, he said, we'd find an intensity pattern that looked like a rectangle with even intensity across the middle, rather than the expected guassian.  Our output was gaussian looking, and Dr. Noe described how to overlay a gaussian curve on top of it, manually adjusting the parameters to find the best fit.  Then we could find the distances between that best fit curve and our measured curve, squaring them to remain positive, to find the error.  This is assuming the theory anticipates a guassian intensity curve, which, for the central dot in an Airy pattern, it does.

Monday, July 2 2012

    On Friday, Dr. Hal Metcalf began the first of 3 talks about quantum mechanics.  The talk focused on oscillatory modes and coupled oscillators.  One interesting point he made was that there is no way of knowing that you have found all the solutions to a differential equation.  He then showed us videos of the modes of several one and two-dimensional oscillating systems.  He concluded his talk by showing us a system of two coupled pendula of the same length, and from that the same resonant frequency.  When one of the pendula is left still and the other set to oscillate in a normal mode, the moving pendulum slowly excites the stationary one until it becomes stationary and the initially stationary pendulum is fully oscillating.  This then occurs in the opposite direction.  I read that the combination of two different normal modes does not result in a normal mode.  In this system the center of mass appears to be oscillating back and forth, and the center of this oscillation appears to be undergoing a secondary longer-period oscillation as the pendula change back and forth between excited and stationary.  Hal asked us "how the pendulum knows" which direction to transfer its energy.

    I read two papers about vortex generation using specially made twisted optical fibers, also referred to as vortex fibers.  In the first paper by N. Bozonovic et al. the authors combined higher order modes (HE21) through a vortex fiber, and tuned the output using paddles (a polarization controller) through the expected range of OAM.  Adding the two modes of l=1 had an expected range of output between -1 and 1, which the authors were able to achieve by adjusting the paddles.  Although they noted that inducing stress in multimode fibers has previously been used to combine HE21 modes, the authors used vortex fiber because it raises the degeneracy of modes native to the fiber that normally couple with the HE21 modes to produce linearly polarized output with no OAM, thus reducing the amount of coupling.  They interfered the output beam with a reference beam to determine the phase, and also devised a method to determine the purity of the OAM states, 97%.

    The second paper, written by Giovanni Milione (3rd author) et al., used a simpler setup to produce and analyze a type of vortex beam called a cylindrical vector beam (CVB), also produced by Jacob Chamoun in his 2010 LTC project, starting from TEM00.  The authors name a new polarization state, called the hybrid-azimuthal-radial polarization (HARP) state, that is a superposition of TE01 and HE21.  The HARP state possesses V, H, LCP, and RCP polarizations states.  The researchers passed a vertically polarized 632.8 nm laser through a half-wave plate and then coupled it into 10 m of vortex fiber.  When the beam exited the fiber it passed through a linear polarizer to analyze the output polarization state.  The fiber was spun at 20 turns per meter and had a cutoff wavelength of 740 nm.  A vortex output was not seen with fewer than 20 turns per meter.  When the analyzer was set to 45 degrees the output appeared to be a first order LG mode, and at 0 and 90 degrees the output resembled a first order HG mode.

Thursday, June 28 2012

    We had a meeting on Wednesday with Dr. Hal Metcalf in attendance.  While Melia was explaining what she knew about Bessel beams Hal asked whether they were really non-diffracting.  The answer, as in other cases relying on the paraxial approximation, is that the beam is well behaved within the limits of the approximation.  In Hal's words the beam is only non-diffracting over a finite length.  Although I haven't read it carefully, the discussion led me to this article on the limits of the paraxial approximation in laser beams.  Hal also advised me to get a more intuitive feel for what modes are.  The study of light often seems like a field where intuition is destined to fail.

    Dr. Noe pointed me to a Miles Padgett paper wherein the authors used a specially manufactured pair of optical elements to convert the OAM of beams (made with an l-fork diffraction grating) into transverse momentum, resulting in beams of different OAM being directed to different regions of a detector.  Notably, their technique is also useful for single photons.  This paper apparently has a special connection to the LTC as Dr. Padgett spent some time on a visit to the center thinking it through.

       I spent some time looking at the Poynting vector in LG beams, covered in the book Optical Angular Momentum.  This is a calculation of personal interest because I only recently studied the Poynting vector, and I was impressed that such a thing could be calculated in a somewhat simple way.  Since it's an important underlying detail in optical vortices I intend to become comfortable with the calculation shown in the book, but unfortunately I wasn't digesting it well.  I pulled a second book from the library Progress in Optics Volume 39 editor Emil Wolf that the authors of Optical Angular Momentum pointed out as having a more detailed explanation of these basic concepts.  In my brief assessment the book has had the most basic explanations of the orbital angular momentum of light and I might recommend that a beginner taking an interest in the subject start with it (once I return it to the library).

    More importantly than those fun calculations, it seems the project may go in the direction of transmitting OV with fiber-optic cables so, like Marissa, I've done some reading about fiber optics, phase modulation, and multiplexing.  When I lived in Fukuoka, fiber-optic internet connections were cheaper, faster, and more readily available than what can be purchased in the US.  I've been amazed to learn this week that even those high quality connections could be greatly improved if a better way can be found to transmit vortex beams through existing cables. 

Wednesday, June 27 2012

    Dr. Noe talked with Giovanni Milione about what he was doing at the Alfano lab over at CCNY.  This is second hand information, but they are working on putting a vortex beam into a 2-mode fiber-optic cable, getting out a superposition of 6 states.  This is an interesting problem to work on given the news about OV data transmission yesterday, as it was mentioned in the news that an obstacle to OV communication is the difficulty in transmitting an OV through normal fiber-optic cables.  Giovanni will also be hosting an optical vortex party at CCNY tentatively scheduled for July 18th.  Kiko Galvez will come with 5 students.

    I read all the abstracts in the Journal of Optics special issue on orbital angular momentum.  Through these I found a researcher, Constantine Alexeyev, who has written several recent papers about OAM and optical fibers, including spiral optical fibers that are still on the drawing board.  He also showed up as a citation in the 2003 book Optical Angular Momentum by Allen et al. that I'm reading now.  I also read through the 1998 McGloin et al. paper 'Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam' in which the authors used a single mode fiber, deformed with weights into an ellipse, as a mode converter.  In their words "the transfer of the angular momentum to the light occurs because of a difference in phase velocity within the fiber for two orthogonal modes that comprise the input beam".  Their explanation for how this worked was that the stress changed the shape of the wire, changing the propagation constant.  

    Dr. Cohen, Marissa, and I continued trying to get a more stable reading from the 10 mW He-Ne laser that we've been using with the fiber-optic cables.  Dr. Cohen noticed that the instability was only occurring at the output of the fiber.  He also suggested we use the oscilloscope rather than the multimeter to get a better sense of the pattern of changing intensity.  I had thought the pattern was erratic, but on the oscilloscope the beam appeared to be "breathing", in Dr. Cohen's words, rising and falling  by about 20 mV at about 30 Hz.  The same occurred  when I switched to a new fiber.  We are still looking for an explanation, but the whole system is sensitive to noise, wind, and other vibrations so it could just be something about the room disturbing the cable.

Tuesday, June 26 2012

    I went further setting up the resource page, now called Singular Optics Map, with the help of an html editor within the program Seamonkey.  Admittedly, the code looks ugly but the editing couldn't be simpler.  Newly added is a list Dr. Noe made of all former LTC singular optics projects.  What remains is to add destinations for the various topic links.  Once the destination links are completed tomorrow I can instantly add papers and resources in an organized way.  If I can figure out the address to publish to, I can connect the html editor file directly to the site without copying and pasting into terminal, and will also be able to change the link structure quickly.  Once these details are out of the way I can read more papers while conveniently sharing them.

    To my surprise, optical vortices were serendipitously in the news today.  Researchers Jian Wang et al. of the University of Southern California, LA managed to combine several channels of data encoded into beams of different OAM and a single frequency and then transmit it across 1 meter of air at a rate of 2.56Tbit/s.  Suddenly, I'm interested in optical communication and want to get my hands on this paper.  The people at my friend's tech company were very interested in this news, so I could at least use it as a conversation topic with them if I manage to understand the paper.

Monday, June 25 2012

    Today was spent getting hands-on experience with fiber-optic cables. After several hours I was finally able to focus a laser into a single mode fiber-optic cable, and do so within 20 minutes on subsequent occasions. When successfully focusing the laser into the cable the helpful point turned out to be making minute adjustments to the depth the cable was inserted into the coupler each time I found the maximum intensity with the mirror knobs. Gradually I was able to move the cable into the closest position and screw it on to the coupler. The transmission I achieved by eye was just above 60% each time.
    Dr. Cohen suggested that we use a photo-diode to help pin down the maximum transmission intensity. This turned into a good opportunity to use a lot of basic equipment. However, we weren't successful in using this setup as an aid in focusing the laser. We found that the first laser we used, when measured with the photo-diode and multimeter with a 1 kOhm resistor, was cycling between 244-262 mV every minute or so in what looked, superficially, like some kind of charge/discharge cycle. We replaced it with a second laser, but the multimeter reading jumped around roughly the same range, although this time it seemed to be erratic.
    Now that it takes minutes rather than hours to set up a cable I can play around with focusing different beams into different cables, replicating Wang Jing's interesting result of sending HG modes into a cable of the wrong wavelength and getting out LG modes, or following up on Dr. Cohen's early comment that it might be interesting to focus a vortex beam into a fiber-optic cable. Tomorrow will be spent working on the optical vortex resource page so I can mention it at the weekly REU meeting to other students interested in the topic.

Thursday, June 21 2012

    I looked into two methods to produce LG modes from HG modes. The first method, detailed in Padgett's undergrad focused paper “An experiment to observe the intensity and phase structure of Laguerre-Gaussian modes”, involves combining HG modes to produce LG modes. Mathematically, LG modes can be written as linear combinations of HG modes so it's not surprising that combining HG beams with appropriate phase differences will result in LG modes. Padgett points to another paper “Astigmatic laser mode converters and transfer of orbital angular momentum” by Beijersbergen as having an easy to understand chart of some of the combinations of HG modes that will produce an LG mode. Looking at the pictures in the chart, the results seem more intuitive.
    The second method was a project in itself, explained in the paper “A low-cost spatial light modulator for use in undergraduate labs” by D. Huang et al. from the University of Arizona, Tucson. The authors disassembled an LCD projector as a cheap way to obtain the parts necessary to make a spatial light modulator (SLM) that can produce all kinds of beam profiles on demand. In addition to LG modes, their SLM could make Bessel modes, computer generated holograms, and all kinds of apertures and gratings for beginner level diffraction experiments, at a cost of $150. Basically, the computer controlled 1024*768 LCD component is removed from the projector and placed between two polarizers. Varying the voltage to a pixel will change the orientation of the polarity of any laser light beamed through. Although the LCD component had uneven thickness that limited the quality of the phase patterns the authors produced, the apparatus they assembled from it could produce a variety of phase patterns successfully over short distances. There would be some hurdles with the electronics and programming in assembling this apparatus, especially taking into account that no two models of projector will be exactly the same. Maybe I'll build one at home some day.

    In a short lab meeting this morning I became interested in one result found by first year grad student Wang Jing at the LTC in 2001. He transmitted a green laser through a fiber-optic cable designed for a different wavelength and output some modes that looked like LG modes. I suspected that the HG beam sent into the cable somehow split and recombined at different phases to produce the output modes. The real reason may be detailed in the paper “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam” which I'll read soon. I will look into ways to determine what modes are actually being output.

Wednesday, June 20 2012

    I presented some basic information at the REU meeting about vortices, as I understand them so far. 5 minutes were alloted to each student. I started with a non-mathematical definition from Mansuripur's Classical Optics and its Applications: “an optical vortex is a phase singularity nested within the cross-sectional profile of a coherent beam of light”. In other words, there is a point of zero intensity. The beam, in such a case, will carry non-zero orbital angular momentum that can be transferred to an electric dipole.

    One situation in which optical vortices arise is in Laguerre-Gaussian modes. Modes are solutions to the paraxial wave equation that produce equal amplitudes when traveling back and forth in the cavity of a laser between the two reflectors. In other words they are standing waves. Although my textbook (Laser Physics, Milonni) doesn't have a full derivation of LG modes, they are derived by converting the paraxial wave equation into cylindrical coordinates and using a particular ansatz that includes Laguerre polynomials. The LG modes have 2 indices p=0, 1, 2, … and l=..., -2, -1, 0, 1, 2, … . At p=l=0 the mode is the same as the lowest order Hermite-Gaussian mode, the first solution to the paraxial wave equation presented by Milonni. But, modes with the magnitude of l greater than 0 possess an orbital angular momentum of l*(hbar) per photon. The intensity of the p=0, l=1 case is described as “the doughnut” because its intensity is concentrated in a ring with low or zero intensity in the middle region. Presumably since the orbital angular momentum is l*(hbar) the plus or minus sign next to l governs the handedness of the vortex.

    Dr. Noe just sent me the title of a book, Twisted Photons, about this very topic. I will look for it in the library. I'm excited to calculate, if possible, the Poynting vectors of these new beam equations I'm learning about.

    I started a resource list of papers I've looked at or finished reading, divided by category. The link is on my main page. It's not a complete listing of last week's resources yet, but all useful papers I subsequently come across will be added as I encounter them. I will trim or sort the list when the project becomes finalized to provide useful information to other students examining my research and vortices in general.

    Dominik Schneble gave an excellent introduction to the process of producing BOE using laser cooling. His explanation of detuning the frequency of the laser so that it would only be at the resonant frequency if Doppler shifted relative to the absorbing particle answered a question I was wondering about since the first day at the LTC. The atoms then emit a photon at their ordinary resonant frequency, but lose an amount of kinetic energy that compensates for the difference in absorbed and emitted energy.

Monday, June 18 2012

    So far I've found a useful article describing the basics of an optical vortex that gives the formula for the amplitude of a simple vortex or a combination of simple vortices. The article also had visualizations of the intensity and Poynting vector for a vortex of topological charge 3, giving some idea of the circulation of energy in a single vortex.

Linear Optical Vortices by MASUD MANSURIPUR and EWAN M. WRIGHT

A second paper I read described vortex generation using the modified Michelson interferometer I mentioned previously.

Interferometric optical vortex array generator by Sunil Vyas and P. Senthilkumaran

This apparatus seems simple enough to build, since it is basically 3 Michelson interferometers. When 3 beams converge it will produce vortex lattices. Several other papers I looked at describe hitting a target with one of the beams before they are recombined and reconstructing information about the target using the changes in the vortex lattice. There are several other types of vortex interferometer. I am looking for the simplest to build, and something to analyze with it. I am not yet clear on the methods of analysis. I am presently reading an overview of vortex interferometry by Masajada to get a better understanding of this.

Interferometry with Vortices by P. Senthilkumaran, Jan Masajada, and Shunichi Satos

A paper Dr. Noe sent me detailed a microscopy technique that resolved details of a microscopic feature on a glass plate by mapping the movement of a single vortex produced by a beam as it approached and then passed the feature. This is single vortex interferometry. Producing a vortex lattice isn't the only way to conduct vortex interferometry.

New scanning technique for the optical vortex microscope by Ireneusz Augustyniak, Agnieszka Popiołek-Masajada, Jan Masajada, and Sławomir Drobczyński

Friday, June 15 2012

Week 1:

    I started at the Laser Teaching Center this week. I met Dr. John Noe and Dr. Marty Cohen, whose career at Bell Labs is something I'd like to hear more about. I also met the other REU students and summer research students.

    I've become interested in the topic of optical vortices, oft mentioned in the first few days. The idea of a beam with angular momentum surprised me somewhat. I like any topic that adds a new twist on a familiar one. After introductions Marissa Romano showed us a Michelson Interferometer in the back of the lab. I recently used one at CCNY. I was glad to see the demonstration as I later read a paper that described replacing the mirrors in the Michelson Interferometer with two additional interferometers in order to produce an optical vortex array generator, an interferometer that produces grid patterns of optical vortices on a projector screen by combining 3 or 4 beams.

    I'm also grateful to Marissa for presenting information about the basics of linear algebra. At CCNY we don't take a linear algebra class. It's included instead in the last month of our vector analysis class. Many of us study the subject independently along with our quantum physics class. Our lab spent some time discussing the topic together.

    Bruce W Shore, a visiting lecturer, gave a series of 3 undergrad level lectures about coherent manipulation of atoms using laser light. This involved many topics beyond my current knowledge of quantum physics and I look forward to revisiting his paper in the future. Fortunately he and Dr. Metcalf visited the lab while we were discussing linear algebra and raised some thought provoking points about what we'd written on the board. Notably, they talked about putting matrices into Euler's equation. There's nothing strange about this but it had never occurred to me.

    Dr. Noe made me aware of several important people working on optical vortices, and two former students at the Laser Teaching Center who did projects on vortices: Azure Hansen whose research journal on the LTC site is very impressive and a pleasure to read, and Giovanni Milione who later moved on to grad school at my university. Azure's journal was particularly useful in finding basic information about optical vortices. This week, I'd like to observe optical vortices or possibly vortex arrays using equipment in the lab. I also want to learn to visually represent them using MATLAB or a similar program. Two months is not a lot of time.