Research
Journal

### Monday, July 30 2012

### Wednesday, July 25 2012

### Thursday, July 19 2012

### Tuesday, July 17 2012

### Monday, July 16 2012

### Sunday, July 15 2012

### Tuesday, July 10 2012

### Monday, July 9 2012

### Thursday, July 5 2012

### Tuesday, July 3 2012

### Monday, July 2 2012

### Thursday, June 28 2012

### Wednesday, June 27 2012

### Tuesday, June 26 2012

### Monday, June 25 2012

### Thursday, June 21 2012

### Wednesday, June 20 2012

### Monday, June 18 2012

### Friday, June 15 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.

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.

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.

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.

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.

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- http://prl.aps.org/abstract/

2009 followup- http://iopscience.iop.org/

2010 paper about using their technique on a speckle pattern- http://www.opticsinfobase.org/

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- http://apl.aip.org/resource/1/

2009 Schoonover and Viser paper describing some general theory
behind MPI- http://pra.aps.org/abstract/

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.