Video Infoblog: An Introduction to Optical Vortices and Topological Fluids of Light
An Introduction to Optical Vortices and Topological Fluids of Light
-
https://youtu.be/JAobQq8aEWk?si=EXxCtuuemnVKXWvi
Hydrodynamic whirlpools have fascinated scientists for centuries, seeking to understand their individual structure, stability, and the ways in which they interact with one another. Who hasn’t marveled at tornadoes or watched as soap bubbles get sucked into the vortex of a bathtub drain? To reduce ideas to their essence, such fluid vortices are often considered in a two-dimensional setting where they amount to current swirling around a singularity. These, in turn, bear a striking resemblance to cross-sections of optical vortices that can be created with lasers, but with the propagation axis now treated as time. The vortex center is a then a dark spot about which the phase of light rotates like a barber shop sign. Such engineered light can therefore be interpreted as a two-dimensional, compressible fluid, and the vortices it harbors exhibit all sorts of odd and potentially useful behavior. For instance, optical vortices can attract, repel, scatter, and even annihilate one another. Even more intriguing, these two-dimensional topological objects have a lot in common with the macroscopic quantum states of Bose-Einstein condensates and fractional quantum Hall systems. Pairs can even be used in Bell tests to demonstrate lack of local realism. This motivates a serious consideration of optical vortices as quantum objects that might be harnessed in emerging quantum information technologies. With these deeper issues in mind, our colloquium lecture is intended to serve as an introduction to optical vortices and their classical few-body dynamics. We tag-team an experimentalist and a theorist to provide a fuller perspective of what makes this form of light so interesting.
Transcript
0:02
awesome well thanks everybody for coming to this last seminar of the semester we are really really i
0:10
mean personally i am very very excited to hear this talk we have um former alum uh mark siemens here and
0:17
mark graduated with the bs in engineering physics from mines in 2003
0:23
and then uh moved up highway 93 to gila that's at the university of colorado
0:28
boulder where he did his phd work in the group of henry captain and margaret bernard after
0:34
graduating from that group mark moved down uh the hall uh still in gila to join steve kunds
0:40
group as an nrc postdoc in the fall 2010 he moved down highway 36 to the university
0:45
of denver where he is now a professor in his research he studies the physics of optical vortices
0:52
and coherent electron dynamics and semiconductors he's also the sps advisor and loves doing hands-on
0:58
physics demonstrations for k-12 students mark is a tech foundation science and engineering research grantee
1:04
an nsf career awardee and a university of denver top principal investigator so
1:11
that's awesome we are super excited that you're here we also have here one of our own mark lusk um and i'm
1:17
about to read some things that i i actually didn't know and i i doubt a lot of people here did
1:22
uh so mark received his bs in electrical engineering from the u.s naval academy in 1982 uh in a navy
1:30
nuclear engineering certificate in 1984 and subsequently served as a navy nuclear
1:36
engineering officer aboard the uss theodore roosevelt cbn71
1:41
after fulfilling his service obligation mark obtained a master's in electrical engineering from colorado state
1:46
university and a phd in applied mechanics from california institute of technology
1:52
in 1992. he then joined the electrical and computing engineering department of iowa state university as an assistant
1:58
professor with an experimental research program focused on millimeter wave electronics
2:03
mark subsequently joined the colorado school of mines as an assistant professor of engineering in 1994
2:08
with his research program focused on computational condensed matter theory so pretty broad uh swing
2:14
there he received uh also an nsf career award in 1995 was promoted to associate professor of
2:19
engineering in 1998 received an r d 100 award in the year 2000 and started
2:26
the csm high performance computing center in 2007 now in its 13th year so dr lusk became a
2:32
professor of engineering in 2004 but eventually shifted to a professor of physics thank goodness
2:38
uh in 2006 where he has been for 14 years now his research uh investigates light
2:45
matter interactions using quantum electrodynamics with a current focus on topological quantum
2:50
optics so a lot of stuff that i didn't know but um that being said thank you so much we
2:57
really really appreciate your willingness to give this colloquium and uh with that i'll hand it over to you
3:03
right thanks and hi everybody it's really fun for me to be back at mines in this way um
3:11
mines has a very special place in my heart as i think it does for all alums and so it's really fun to be here
3:16
um i'm in sort of a unique situation so i'm actually not in denver i'm up in the mountains in salida where my wife's
3:23
family is from we're not quite in a log cabin but the internet is a little bit slower than
3:30
mark lusks in my comfort level and so i'm actually going to turn off my video in a second and mute myself here and talk to you
3:37
through my phone uh so we're going to do that here
3:45
over here here okay okay turn off your mute microphone too yeah
3:52
there we go okay all right so this is uh it's gonna be a different
3:58
colloquium than i've ever seen and probably you've ever seen in that there's two of us tag teaming it
4:04
um and so i'm the experimental side and so uh you get to see both sides because
4:10
mark lusk is the theory guru and so he'll talk to you about some cool quantum mechanics applied to optics and
4:16
i'll tell you about how you actually do this stuff in the lab so there's kind of three stages to our talk the first stage
4:23
is i'm just going to introduce optical vortices a lot of work has been done in the last 25 30 years on them and uh
4:31
normally in the context of orbital angular momentum and lager gaussian mode that's that that figure in the lower left and then
4:37
i'll move to what we're currently working on now which is thinking about putting more than one vortex into a laser beam
4:42
and seeing how those vertices move like do they influence each other as they propagate so that gif there is showing you two
4:49
opposite charge vortices and you see z increasing the propagation distance is increasing and
4:54
eventually they recombine and there's no vortices in the beam anymore so we're trying to understand the
5:00
physics of that process and out of that we realized we need to have a much better basic understanding
5:06
of the theory of this and so that's where mark lusk is going to come in and tell you about the the theory that he developed we
5:14
we i helped him understand some of the basic optics but he really took the lead for sure on the theory part and developed some brand new ideas
5:21
for understanding these vortices that make it possible to understand this with fluids okay also i'm going to be saying next a
5:29
lot because he's controlling the slides so next we should be on slide two now um okay so let me tell you about optical
5:35
vortices and laser beams and before i tell you about optical vortices and laser beams let me tell you about just laser beams so a typical laser mode
5:43
has flat phase fronts and what differentiates a laser from say the sun or any other star or a light
5:48
bulb right is that we have this coherence and usually we think of that as being a flat face front across the beam so you
5:55
can represent that in the top image here with the three planes each of these planes is a flat phase front so all the way
6:01
across there across one of those planes is that the phase is constant and then if you go back to the next plane
6:06
that would say be say one wavelength uh away and you'd have another plane of constant in same phase
6:13
you can also represent it and i'm gonna do this a lot in in our talk here is to show this in one amplitude and
6:20
phase combined diagram and so here you're seeing the brightness is showing you the amplitude
6:26
and the color is showing you not the color of the laser beam but it's showing you the phase so all the way over on the right there you can see that phase scale
6:32
from 0 to 2 pi so the fact that this is red just tells you that this has zero phase and the phase is flat it's constant
6:38
across the beam so next we should see oh sorry what does phase mean for laser light just to make this very concrete for you
6:44
let's say we have linear polarization then we know that that the electric field component of the electromagnetic
6:50
wave is oscillating up and down and so the phase tells you at some instant in time uh the orientation and magnitude of the
6:57
field in that in the course of that oscillation all right next we should see slide three so i'm just showing you a picture of a
7:03
wave here so the idea is that you know all along the crest of that wave this is coherence all along there that
7:10
wave is in phase if you go back you can start to see the the next wave crest forming that would be one wavelength
7:16
and you've got another place of coherence so all the way across there there's coherence and we have a similar thing
7:22
with our waves here next on slide four so now we see what happens when you introduce a vortex
7:27
so uh let's start with the image on the right that says l equals minus one you'll notice that the phase front now
7:34
is no longer a flat disc it's actually a helix uh so the phase is described by e to the i
7:40
l in this case minus one times phi being the azimuthal angle and the polar coordinates
7:46
and now our representation on the bottom with the the brightness being the
7:51
amplitude and the color being the phase you see it goes red yellow green blue and then back to red
7:57
in a clockwise direction so if you click next then you'll see that arrow going in a clockwise direction so by the right-hand rule we'll call
8:03
that a negative topological charge it wraps once so we'll call that a charge of minus one the other the
8:09
figure right next to that with l equals plus two this is now a double helix for the phase structure
8:14
and you look at the the phase diagram on the bottom next you'll see those two red arrows so
8:20
this wraps two pi twice and it does so in the counterclockwise direction so we call this a topological charge of plus two
8:27
next you'll see just you know the water analogy this is like a vortex or the swirling going down your tub drain next
8:34
on slide i think it's six i can't see the background because it's white angular momentum of light so i just want to make this concrete for
8:40
you to try to help understand the connection between you know what kind of angular momentum this is because this
8:45
this vortex carries real angular momentum so the analogy is to classical mechanics in classical mechanics we have
8:51
both spin and orbital angular momentum and spin angular momentum is like a ball spinning or the earth spinning on
8:58
its axis right and in light this has to do with the circular polarization so uh
9:03
every photon has either a plus one or a minus 1 h bar of angular momentum
9:10
that corresponds to left and right circular polarizations respectively and next we see for the orbital
9:16
component in classical mechanics we think about the earth you know rotating around the sun
9:21
orbiting the sun and we see that's what the orbital angular momentum or this vortex component
9:26
is related to and here the orbital angular momentum the angular momentum contribution of the vortex is l times h bar so l could be any integer
9:34
number it could be 1 2 negative 3 anything and so already you see this is a kind of
9:40
an exciting playground for dealing with angular momentum and a lot of people have explored this for
9:45
applications in communications or particle rotation but so that's that's one exciting aspect
9:51
of these so next on free space modes with oam
9:56
um a little decision tree here if you're a theorist you're going to talk about vessel modes because vessel modes are
10:03
great they're non-diffracting they you know they don't change with propagation um they have some challenges in that they
10:10
require infinite energy and infinite spatial extent so in the lab we prefer to use lager gaussian modes which are just
10:16
higher order gaussian modes you've probably heard of gaussian either gaussian statistics you know a gaussian shape a standard curve
10:23
a normal curve normal distribution um and so this is that's the shape of a normal laser beam
10:28
that's normally what you know dan and jeff and chip are trying to get in their lab but in this case we're looking at a
10:34
higher order version that has a vortex in it so these lager gaussian modes they have
10:40
two quantum numbers that describe them one is the azimuthal mode l that script l which is exactly the
10:45
same as what we were just talking about that's going to be the vortex winding number and they have a radial mode p which i'll
10:51
show you intuitively what that does in just a minute so down at the bottom here we've got the whole gnarly expression for one of these
10:57
air gaussian modes so it's not too bad i'll just walk you through bit by bit so the first part there
11:02
that grayish part is the just the normalization so that's to make sure that as we change l or p so we go to higher
11:08
order modes we still have the same power in the mode the next term is is a familiar should be
11:14
familiar that's just the normal gaussian the next turn is what gives us the doughnut hole um
11:19
and if you'll notice that the mode in the top right there we have a dark spot in the middle so the reason
11:24
is that we have to have a dark spot because this is a there's a phase singularity right in the center so in the center of the beam the phase
11:30
is undefined because we're wrapping some integer of two pi as we go around the beam so you can imagine wrapping
11:36
closer and closer and closer to the middle until you are just at the exact center point the phase is completely undefined so the
11:42
only way for this to be a physically realizable mode is for the amplitude to be exactly zero
11:47
and um you could imagine lots of shapes that could give you uh something zero in the middle
11:53
and it turns out that this r to the power of absolute value of l is what gives you the the eigen mode of free space for these
12:00
lager gaussians the next term they're circled in blue is a lager polynomial that's usually
12:05
neglected because for radial modes p equals zero which is what people mostly use it's just equal to one
12:11
um but there could be some interesting physics in that term that we're we think about from time to time
12:16
and then the last term there is the spiral phase e to the i l phi so that's what we started talking about this whole vortex
12:21
idea and that's where that angular momentum component comes in next slide where we see the so there
12:27
should be a diagram here showing you different l values and p values um and what we see so this is
12:33
just showing you some some examples and again the color is not what we see in the lab right the color is just showing you
12:39
um the the phase and as you go from red red yellow green blue and then back to red that's a two pi phase wrap
12:44
so p equals zero l equals zero at the the middle on the top there that's a normal gaussian mode and then as you move down this table
12:51
you're going to higher radial mode index and you'll see that each time you do that you add an additional ring
12:56
around the outside that's pi out of phase with the ring next to it and then as you go to higher orbital angular momentum or higher l values
13:03
you add an additional two pi phase wrap as you go around the beam you'll also notice that as you go from l
13:09
equals zero to one to two uh and as you increase p as well the modes get larger and this is
13:15
i mentioned earlier people were really excited about this additional uh kind of phase space of having
13:20
additional angular momentum that you could put into the beam in terms of communications one
13:26
reason why this isn't really so exciting is that to go to those higher order modes you need to have a larger and larger
13:32
fiber basically and basically what you're doing is spatial multiplexing so you could also just run other fibers or use other
13:37
mode sets um so those larger beams require a larger mode uh just a quick note for the those of
13:44
you who are working with lasers you may be familiar with hermit gaussian modes which are represented in cartesian coordinates
13:50
just x y instead of in polar coordinates and you can convert back and forth between these uh so an l equals
13:56
one p equals zero like our gaussian mode can be written in terms of a one zero and a zero one mode with the
14:02
correct phase between them with a pi over two phase shift okay next slide
14:07
uh generating optical vertices with photography slide nine um so here's how we actually do it in the lab there's a number of ways
14:12
of making optical vertices we uh we use holography and so what we do
14:18
is add whatever phase we want to put onto the beam let's just say it's a spiral phase like that little rainbow
14:24
uh pinwheel there and then we're gonna add to that in e to the ikx which is just give you
14:32
basically a grading term we add those two we take the absolute value and what we're left with is there on the right
14:37
the top right you see it looks like kind of a normal diffraction grating but in the middle it's got this fork this topological feature
14:44
and so we what we've done is encoded the phase that we want to put onto our beam into
14:49
this grating and through holography whatever diffracts off of this
14:55
this hologram is going to acquire the phase that we programmed onto it and so down on the bottom you see a
15:01
simple picture of how we actually do this in the lab i say simple picture this is exactly what we do it's not that complicated
15:07
so we take a laser we put in a spatial filter just to clean up the mode so that we have a nice gaussian mode
15:13
coming in and then we use a spatial light modulator so this is just a way to have computer
15:18
controlled uh real-time updatable holograms so we don't have to develop a hologram
15:23
and then if we want to switch you know pull it out and put in another one so we can do everything computer controlled that's just uh imagine taking like the
15:29
the screen from your computer and hacking off the back and sending a laser beam through it and now each pixel
15:34
instead of controlling the emission of light is going to control how much light makes it through so we can put these holograms directly on there
15:41
and control them with the computer and then we block the reflected beam that's what's coming off at the bottom there and then the diffracted beam picks up
15:48
exactly the phase that we put onto it all right next slide slide 10 um so we
15:54
were talking about phase and you know the vortex is all about the phase structure or the
15:59
yeah the phase structure is all about the vortex that you really need to have that phase there and so if you want to measure a vortex
16:05
you really need to know the phase and it's hard to do that
16:10
it's hard to measure the phase because every detector we have is really based on just measuring the intensity right our eyeballs
16:17
uh solar cells or photodiodes those are or you know ccd cameras those are all based on measuring the intensity of the
16:23
light so the way to do to to get at the phase information is to i use interference and so the first
16:29
thing you do is to just interfere the mode that you have that you get say from an oem generator
16:34
with let's say a plane wave or you know another gaussian mode and you look at the interference between those two um so that works really well but the
16:40
problem is you can't recover the full the full phase because there's a inverse cosine problem
16:46
as if you work through the math and so what we do is use uh phase shifting holography so we use four
16:52
different reference beams oh next you should see test phase uh plus a whole bunch of stuff
16:58
we use four different reference beams and you could do this next by scanning a delay on your reference
17:03
beam and but what we do instead is get rid of the separate reference beam and we put
17:08
the reference beam onto our hologram so we have a composite hologram that has two parts one is the
17:14
phase of the thing that we're trying to actually measure and then we make a second contribution to it that's just a gaussian reference speed
17:20
and then we can shift the phase as you go look at that middle column with the kind of the jail bars as you go from the
17:25
top to the next one you can see that those bars are moving over a little bit and what that's doing is shifting the phase of the gaussian the reference
17:32
component and if you look at the far right you can see those pinwheels spinning slightly and the direction of that spin is going
17:37
to be determined by the sign of the topological charge so next you see this this equation here with
17:44
kind of a picture is inverse tangent of the three pi over two interferogram minus the pi over two interferogram
17:50
divided by the zero interferogram minus the pi interferogram we do this calculation and we recover
17:56
the phase so this is not a algorithm it's not a search it's not a you know some sort of
18:04
um attempts to check or guess and check we're directly
18:09
calculating the phase in this way on a pixel by pixel basis so next you should see phase measurements modal decomposition yes
18:15
slide 12. um so here's some examples of actual measurements we did so i want to emphasize the the rainbow
18:22
colors here are actual phase measurements everything i've shown you at this point has been you know mathematica i plug in
18:27
an equation and look how beautiful it is but these are actual measurements from real laser modes uh so the right that
18:33
sorry the left two columns are hyper geometric gaussian modes they have some additional rings on them
18:38
um and then the the left the next two columns are legere gaussian modes uh so those are the ones we were talking about earlier so you see one
18:44
with l equals minus one and l equals plus four you can see they spin the right direction and they've got the right number of
18:50
two pi phase wraps the one on the right is kind of fun it's hard to tell on the top uh but it's
18:57
actually also a donut mode so it's dark in the middle so if you were to just look at this in the lab you'd say oh yeah i got a vortex there i got a dark spot
19:03
but if you phase resolve it you see that the phase is actually flat because we played games to get a mode that it's a
19:08
composite mode that has a flat phase but is dark in the middle so that emphasizes the importance of actually doing
19:14
the phase measurement and then once we have the amplitude and phase like this we can do like you would do in any uh quantum
19:21
mechanics problem and and put that into the the whatever basis you want and so um
19:26
we can put this into a legal gaussian basis for example but now we're taking actual measured data and doing that um and so for example on
19:33
the right lower right there you see for a liger gaussian mode this is on a this is the spectral power for the different
19:38
uh orbital angular momentum or vortex charge numbers you see that's on a log scale so
19:44
we have a 99.9 percent purity into the l equals minus one case uh in l equals minus one uh vortex
19:52
charge in this case which next to our knowledge is the highest purity vortex that was ever measured so
19:58
in slide 13 now so i want to transition now from talking about
20:04
orbital angular momentum like your gaussian modes to thinking about um what happens when you put more than
20:10
one vortex and we're going to talk a lot about two vortices because that's the next simplest thing to talk about next so basically the question is what
20:16
happens when two or more vortices are placed into the laser beam all right slide 14. so we're going to
20:22
think about vortex interactions we want to measure vortex pair motion so the first thing we need to do is to generate a vortex pair
20:28
and so we're going to use the same method we did before but now instead of just putting a single vortex in there we
20:34
can create some array of vertices so this example here i'm actually showing you a two by two array
20:39
just to kind of illustrate the level of control that we have so we can control the amplitude and phase
20:45
at every point on the beam which means that we can control the vortex charge plus or minus we can control it to location we can
20:51
control its tilt which is something that you'll hear about more in in a few minutes and
20:56
some relative phase information so we really have exquisite control of the initial configuration of these vortices
21:02
using this spatial light modulator approach and then if you look on the top right there you see a ccd with some arrows
21:09
what we can do is put a ccd camera on a stage and just drive it back and forth to measure the what the mode looks like
21:16
as it propagates okay next so here's our pilot study a measurement of two
21:21
uh vortex motion so this is just a little bit more detailed uh schematic of the setup
21:26
um so we use an imaging system so the slm is there in the middle then
21:31
after that there's a two lens uh there's two lenses around an iris so the iris selects just the diffracted
21:37
mode that we want and then the two lenses basically image uh so that we can run our camera and start right at the
21:43
slm plane and then back off from there because we're gonna generate uh so our laser beam is coming in with some
21:50
it's a gaussian laser beam and then we're gonna put two vortices in there and see what happens okay so then on the
21:55
lower left you see two gifs running the one on the left side is two vertices with opposite
22:02
topological charge and you see that as z increases as that propagation distance increases so we're scanning the ccd
22:08
camera back you'll see that the vortices move down and start to drift towards each other
22:14
and then eventually recombine and there's no more vortices the the example next to it is for two
22:19
vertices that have the same topological charge those kind of move slightly apart maybe
22:25
and maybe a little bit vertically away from each other but there is certainly not the same recombination motion and much
22:32
much slower vortex dynamics in general so next slide so here's our results for for two
22:38
vertices and so the the top frames there are showing you the results for the plus minus
22:44
charges you can see as the propagation increases the vortices move towards each other and recombine for two positive charges they maybe
22:50
rotate a tiny bit but are pretty much stationary on the bottom i'm showing you the separation distance between those
22:57
vertices as a function of propagation distance so as z increases you're moving off to the right and
23:02
the the red dots are data for plus plus charges and you see they basically
23:08
they don't separate at all and the blue dots are the opposite charge pair and those
23:14
start pretty much constant separation and then they dive and eventually recombine and go to zero separation
23:20
on the right you can see the data kind of like a head-on view with just uh scatter plots as they are accumulated
23:27
so the the top is the plus minus charge pair and they start along the zero line um far away from
23:34
each other and then move upwards one on the left and one on the right until they recombine at the top middle
23:39
and in the bottom you see the red dots that's the plus plus charge and you see they pretty much stay in the same place
23:44
so you'll notice the the plus minus charges they seem to follow kind of a half circular sort of trajectory
23:50
and the plus plus charges seem to be pretty much stationary all right next slide so we wanted to
23:56
understand the physics of what's driving this vortex motion we could always we could decompose this in terms of
24:01
lager gaussian modes and just write out a mode basis and propagate the thing but we really wanted to understand this
24:07
from a perspective of what's driving the vortex motion itself because this is uh this is a very rich field in for
24:14
example quantum fluids looking at vertices and and super fluid helium or bose einstein condensates um there's
24:20
a lot of uh really interesting work going into understanding how those vortices move and we wanted to see
24:25
could we understand that for these these vertices so let's talk about superfluid vertices on the left here
24:30
so i've got two examples one is a plus minus pair like we talked about and one is a plus plus pair kind of like we talked about for the optics
24:37
so in superfluid you think about one vortex is driving so if you have a vortex pair
24:43
each vortex is going to move in the background field of the other or uh you know the the phase
24:49
of that spiral that spiral phase of one vortex is going to influence the other so next if we look at just at vortex 2
24:56
over there on the one on the right and we think about the background field from vortex one and we take the phase gradient of
25:02
that think about it as being kind of a wave that vortex two is going to be writing we see that phase gradient is pointing
25:07
down next we look at at the spot of vortex one and think about the phase gradient from vortex two that's also pointed down
25:14
and next now you should see white arrows on the top left diagram that's showing you what these vertices are going to do if
25:20
you have a plus minus pair in a superfluid they're going to track together they're going to move down in
25:25
this case so if it was plus minus instead of minus plus they'd move up okay so now let's turn our attention to the plus plus case
25:32
next you should see uh specializing at the vortex on the right it's the background field it experiences is a
25:38
field from the vortex on the left and so that background the phase gradient there is pointing up
25:44
if we look at the background field on the vortex on the left it's experiencing a phase gradient pointing down so those are in opposite directions
25:51
next we should see these two white arrows on the top diagram and so what happens with two plus vertices is they'll actually uh spin
25:58
around each other in a counter-clockwise direction if you have two negatively charged vertices they would spin in a
26:03
clockwise direction so i said this is vortex motion and superfluid it's also true in cyclones
26:09
and hurricanes so here's a snapshot from a weather forecaster explaining the fujiwara effect
26:16
uh in hurricanes this summer you had two hurricanes that were coming close to each other and they actually can experience this
26:22
this sort of orbiting so that's known as the fujiwara effect okay so what's happening in our laser beam
26:29
next we see uh our minus plus and plus plus situation vortex motion and laser beam
26:34
next our white arrows we see we again we saw kind of a half circular
26:40
sort of trajectory for a minus plus and the plus plus were basically stationary so if you compare these two situations
26:46
clearly things are very different um so next that makes us think maybe the hydrodynamics
26:51
explanation fails to explain optical vortex dynamics or next maybe there's more physics
26:58
so there's a really important difference between the vortex dynamics that we see
27:03
which is between the vortices that we see in a laser beam and the vortices in a superfluid or in cyclones and that difference comes
27:10
down to the amplitude structure so in the lower right here i'm showing you the
27:15
the amplitude you think of these as a fluid as a function of distance from away from the vortex center
27:21
so the blue line there you see that basically absolute value of distance away from the center
27:26
that's what it looks like for an optical vortex a plus one or minus one charge so you uh if you remember back to the
27:31
legere gaussian expression plus one charge vortex has a core um an amplitude core of r to the
27:38
one and so that's why it's a linear shape like that and in contrast the superfluid vertices
27:44
uh are basically inc they have a basically a constant density except for a little blip right in the
27:50
middle um to the size of that little blip they call the healing length and it's usually in the order of angstroms so
27:55
it's really tiny and so they don't have the same sort of amplitude modulation that we are having
28:01
that we were seeing in our optical vortex case so we thought it looks like there really is some more physics going on here
28:08
and uh so that's where we asked mark luck to help us figure out the theory here
28:14
thanks mark so the um the starting point for us is the uh it's maxwell's equations everything
28:20
that he mark has been talking about is associated with maxwell's equations and here they're embodied in a wave equation for the magnetic
28:26
vector potential a so this is the equation that we start with in trying to understand what's going on with these optical
28:32
vortices and we assume a particular type of solution for a and it looks like we're just talking about a plane wave that
28:39
where the the light is polarized in one direction but in fact this psi here is a field psi
28:45
depends on x y and z where z is the propagation direction of the beam and so psi is not just a constant in
28:52
fact if we take this a and put it into the wave equation we have a new equation a scalar equation
28:57
for psi and here i've broken down what would be the laplacian into two pieces
29:02
so this little perpendicular symbol transverse means that this is the laplacian in the xy
29:08
plane perpendicular to the direction of propagation and here's the second derivative of psi
29:13
broken out along the direction of propagation and the reason that i do that is because
29:18
it's a second term that's normally quite small for beams of light that's called the paraxial approximation wherein we
29:24
we just drop this and if you look at this equation and kind of blur your your eyes you see that here it is again
29:31
that it looks like the schrodinger equation which i've drawn over here at right and the only difference is that the
29:36
schrodinger equation is in terms of time and the what's called the paraxial equation is in terms of z
29:43
and so from here on out i'm going to refer to this maxwell equation this fractional equation as a
29:49
schrodinger equation and i'm going to think of z the direction along the axis as actually time and so what this does
29:56
is it turns us into a turns everything we're doing into a consideration of two-dimensional physics
30:01
with this new type of time-like axis and it's because it's two-dimensional that all sorts of interesting topological
30:08
features can be exploited now let's talk about before we even talk about lasers
30:13
let's just talk about light and touch base with what mark siemens was saying about it having a hydrodynamic aspect so light it
30:20
turns out is a 2d compressible fluid let me explain why i say that if you take this
30:26
schrodinger field psi and decompose it into an amplitude row and it's phase fee then you could think
30:33
of a fluid density as being the square of the amplitude and we'll call that chi and the fluid velocity something that
30:40
mark already mentioned as being the gradient of the face how do i know that because if i take
30:46
this this expression and put it into the schrodinger equation the schrodinger equation magically turns into
30:52
euler's equation for a two-dimensional fluid and that's called the madelung transformation
30:58
and the right-hand side here is the gradient the transverse gradient of the pressure where the pressure is
31:05
given in terms of the density and any time you have pressure related to density it's like a spring with strain related to force
31:12
and that means that the material is compressible and so the schrodinger equation or
31:18
maxwell's equation has turned into an euler equation for a two-dimensional compressible fluid
31:23
and so we think of light as being a two-dimensional compressible fluid okay now let's talk about vortices and
31:29
i'm going to take a step back and instead of talking about compressible fluids let's make a foundation based on something
31:35
that mark siemens had just talked about which is an incompressible two-dimensional fluid and in an
31:41
incompressible fluid we think of the the bat the fluid as being a background
31:46
and we just kind of plop in a vortex a little point vortex into that fluid so when i talk about the
31:51
background um field i'm talking about an incompressible fluid say that has a vortex on top of it and one
31:59
of the things that we know has been known for a couple hundred years about incompressible fluids is that the
32:05
um vortex moves with the fluid it's called advection and so if i know
32:10
the vortex if i know the fluid velocity i also know the vortex velocity and we know from fluid dynamics that the
32:18
fluid velocity is given by the gradient of the background phase so that must also be the vortex velocity
32:24
okay so for an incompressible fluid this is an equation of motion for the vortex and what we want to do
32:30
is come up with an equation like this that tells us how those vortices move and eventually
32:35
annihilate the experiment that mark seaman showed so let's extend this a little bit into
32:42
compressible fluids we still have this decomposition of a background fluid
32:47
field and a vortex field that's our total schrodinger field we have this schrodinger equation the paraxial
32:54
equation and now we take this equation and play a little variational game
32:59
all we do is say we want to track the place where both the real and imaginary parts
33:05
of psi are equal to zero and so a little variational analysis actually gives you
33:10
an equation for the velocity of the vortex directly from the schrodinger equation and here i've
33:16
written it down and what you see is that for a compressible fluid sure you have this initial
33:21
piece which is the fluid velocity but now you have a piece that contributes to the vortex
33:26
so that the vortex doesn't any longer move just with the fluid instead the vortex can move relative to the
33:33
fluid because of a gradient in the background density of the fluid now we didn't invent that that's been
33:38
around for about 14 years in fact it's been refined several times and most recently
33:44
in i think 2018 by a fellow named groscheck that was nice enough to suggest to us that
33:49
we'd look at this and maybe it would be useful so we're very appreciative to him for helping us out in that way all right
33:56
so let's see how this works for maybe you know all we need is that equation by grocery and
34:01
um maybe we can track our vortices directly let's see so we put in a gaussian beam and
34:06
initially there's no phase so this gradient in the phase is equal to zero and if you actually put in the
34:12
expression for the the radial uh change in the density of this gaussian beam and you put a vortex on
34:19
the shoulder of the gaussian beam then the vortex will move straight up and that's what this equation predicts
34:25
so we're thinking oh great you know this this equation that gross suggested to us is beautiful it does predict exactly
34:31
what the vortex does but then we tried it for our system and it didn't work
34:37
and the reason that we thought it uh it might the difference might be is
34:43
because our vortices started as circles with circular contours but they developed into elliptical contours and got more
34:49
and more elliptical as they went along so we started thinking huh maybe this elliptical thing
34:55
is important maybe we have to include that in the dynamics and we screwed around this for a while
35:00
and eventually decided that the way to go was to think of this picture that you're seeing right now
35:05
as the projection of a three-dimensional object and so we think of this as a a true
35:12
three-dimensional vortex that is sticking up out of the plane of the
35:17
of the paper here and the projection axis that the flashlight if you will shining down
35:23
to make the shadow that you see is actually a gauge freedom we can choose any projection axis that we want
35:28
in fact that tilt of that three-dimensional axis can be described by two angles and so we have a way of describing these
35:35
elliptical vortices using two euler angles and then it raises a couple of questions for us
35:41
can we predict the vortex tilt from the schrodinger field and maybe can we figure out how this
35:47
tilt affects the dynamics of the velocity of the vortices that we see in the lab so here's the picture the red circles
35:54
the red ellipses are the the shadow what we would actually measure in the lab
35:59
the green is all in our mind with a gauge freedom in terms of what the projection axis is
36:04
and we choose the this projection axis as z prime and we call the two angles here
36:10
that describe that axis as the azimuthal angle c and a polar lean the leaning of z into
36:16
z prime as theta and so now our game plan is to be able to figure out how to
36:21
measure c and theta from the paraxial field the schrodinger field and how to predict how to use that to
36:27
figure out where the vortex goes okay um i'm thinking of my friend reuben collins who says
36:33
man theorists love to show a bunch of equations all right so sorry reuben i'm going to show a bunch of equations but
36:38
only briefly okay so it's all we have the the first step here is um to
36:44
describe the tilt these two angles in terms of the schrodinger field
36:49
and the way we do it i think is kind of fun you you take the schrodinger field it has real and imaginary parts
36:55
and you turn it into a two-dimensional vector and then you play a game in continuum mechanics i mean my phd was an
37:01
applied mechanics i got to use it for something so i take the gradient of this two-dimensional vector and that makes it
37:07
tensor and i think of f here not its inverse as being the deformation gradient
37:12
like a the the gradient that the the deformation gradient associated with the twist of an elastic continuum or associated
37:19
with navier stokes the deformation of a fluid and if you think of f as that deformation gradient you say oh any
37:25
tensor can be decomposed into a symmetric part and a rotational part
37:30
and the symmetric part includes all the stretches and the special axes associated with the ellipse
37:36
in other words if i solve the eigen system of this symmetric uh tensor v here's the eigen system in
37:44
terms of the eigenvectors and eigenvalues it's not too hard to show how those
37:49
values relate to the two angles okay lots of math but what is it saying to you
37:55
it's saying you solve the schrodinger equation and i'll tell you given that field exactly what the tilt of the vortex must
38:01
be and that's powerful all right the next thing we decided to do was to take those tilts and see if we
38:08
could somehow figure out how to how they might be affecting the dynamics well remember an incompressible fluid
38:14
just looks like this the vortex velocity is the gradient of the background face and the compressible stuff the stuff
38:20
that groshak suggested to us had this additional term okay associated
38:25
with the gradient in the background density of the fluid if you want to think in terms of fluid
38:30
what we did is a variational analysis just like the one that they did to get this term only now we allowed for tilt and it was
38:38
fun and the result is something quite different this is the same equation
38:43
but now there's an additional tensor and this tensor is acting on that background
38:49
gradient in the density in other words this term right here includes all of the tilt stuff so we
38:56
have a tilt operator that's screwing around with the background density and now they're coupled together
39:02
all right this is like a surfer okay and the surfer can now lean their body
39:09
and if they lean their body they can change the direction that they go in so now we see that the vortex can now
39:14
affect its own velocity by tilting and that's important and that's what we're actually seeing in
39:20
the lab so these vortices are really four degree of freedom systems there's two uh two degrees of freedom associated
39:26
with position and two associated with the tilt angles like c and theta now we can turn this
39:32
back to the lab and see if it makes sense so we looked at two applications and one of
39:38
them was particularly simple it's just this gaussian where i'm looking down on the
39:43
gaussian now and i put in an elliptical vortex something that mark siemens can produce in the lab
39:48
this is still just modeling though and the paraxial field shows you the contours the the solution
39:54
to the schrodinger equation at some later time here and if we take the schrodinger field
40:00
and then go through this these this three set process we take the schrodinger field to get the
40:05
and solve for the tilt we use the tilt to get the coupling to the density and we solve to get the velocity and we
40:12
integrate it we get this green line and so the green line is predicting the track
40:17
of that vortex pretty well and so we thought oh okay this is working like we had hoped all right so now we go
40:24
to the lab and on the left is the theoretical prediction on the right for this particular set of
40:30
angles is the x and y positions of the vortex as a function of time
40:35
the dashed lines are the predictions from our theory our vortex equation that we've integrated
40:41
and the discrete points are what mark's group measured in particular
40:46
um a really great student named drew motif did this and wow you know it's easy for me to show this but it took him months to get this
40:53
stuff right and uh because it's a really difficult experiment but the standard deviation is given with the
40:59
error bars as you can see that the prediction and the experiment match up within the one standard deviation which
41:05
is really excellent so drew did this again now we tilted the vortex
41:10
so that there's an azimuthal component and you can see that the trajectory is now at an angle and once again uh drew's
41:17
measurements are exactly on that we can match up the experiment in theory within a standard deviation
41:23
easily so then drew goes crazy not goes crazy he does a great job and
41:29
he everything is compacted into this one plot but each point on this plot is a different experiment of vx and vy
41:37
and let me talk you through with through this by just looking at one point if i look at this light green circle up
41:43
here the way to read that is to say i'm on a line with theta equals 60
41:49
degrees that's the polar lean and i have a number up here in colored that is my azimuthal orientation of the
41:56
vortex the green circle in the middle is my prediction for where what the vortex velocity
42:03
should be and the black dot is what drew measured in the lab the circle around it is one standard
42:10
deviation associated with his measurements and there's two dimensions to that you know v
42:15
x and v y and all of these plots you see the idea is that the colored dot should
42:21
be near the black dot and it is in every case except wow you'd think this would be the easiest but it's
42:26
not the one where there's no tilt at all and then we're a little bit off because the vortex drifts a little bit as it goes up
42:33
okay but the bottom line is that the measurements that we have made here now pretty
42:38
extensively are consistent with our theory and so we're of the mind that we've captured the right
42:44
theory at least in the simple setting and now we can turn back to the more complicated setting of the annihilation problem that mark
42:51
siemens talked about now he talked about the top half of this where two vortices are coming together
42:57
but as long as you're doing that you may as well this is easy for a theorist to say you may as well start with the nucleation
43:03
and look at the whole circle so you can nucleate these two pairs have them move up and separate and then
43:09
eventually come back together again and we're going to follow the same approach as what we did with drew's experimental work
43:14
we solve our short integer equation to get psi we use psi to get the tilts we use the tilts to get the velocity we
43:21
integrate the velocity to get the position of the vortex done all right still in theory land here but
43:27
just to get an idea these are pictures similar to what mark showed where i'm looking at the magnitude in the face of the
43:33
uh of the schrodinger field this paraxial field and because the field is so simple it's
43:39
possible to analytically determine the position of the vortex as a function of time in fact that was done almost 30 years
43:46
ago by indebito and so we're going to use that to help check ourselves the black line here the black curve
43:54
is the indebito prediction for where the trajectory should uh what the trajectory should look like
43:59
as these two vortices come together and you can see that the vortex pair is moving as i go through
44:04
these four time slices until it finally annihilates at this last slice
44:10
and so we take that we say okay what's happening with the tilt oh my gosh it's so complicated this is
44:15
the tilt and now there's two angles one uh two sets of angles one for each vortex it looks pretty crazy
44:22
and so we're trying to make sense of this we think hey let's just plot this on a 3d plot and it looks like this and
44:29
now we can kind of make sense of what's going on down here at the lower right you have a
44:34
nucleation event and one vortex goes off to the left one vortex goes off to the right
44:40
you can think of time the beam axis as being vertical and eventually there's a nucleation and
44:46
our experimental work is going to be with this top half the nice thing about this is that you can see the vortex
44:52
tilt as a function of time and you can follow both vortices and if you don't realize that time is
44:59
pointed up you say hey man you just have one vortex it's just going around in a circle right with
45:04
this continuum of of tilts and you know it's kind of like a feynman diagram in that sense
45:09
sure it doesn't matter how you look at it it all still makes sense right so this kind of makes sense and
45:15
and uh makes us think okay we we think we know what's going on let's see if this ties in with what we actually do
45:21
in the lab all right the way we're going to do this is in pieces first first talking about just the the
45:27
theory still here's that indebted trajectory if our vortex velocity equation works
45:34
the velocity should be tangent to this at every point let's check that this is an incompressor
45:40
the direction of the vortex for an incompressible flow so if i rule out everything but the incompressible prediction
45:46
the vortex would move straight up and we see no that's not right but we already knew light was compressible
45:51
so then we include this grocery term that associated with the gradient and
45:56
the density and honestly this is the thing that we had hoped originally was gonna work so
46:02
when i first plotted this and mark siemens and i were looking at it were like what it doesn't work i can't believe it
46:07
but then we went through a few months of work and he pretends like it was me but we were both working on this together
46:12
every day this we finally had this term and we when we include it the velocity is now tangent to the
46:19
trajectory and in fact if we integrate the velocity we do get exactly analytically that blue line for this
46:25
simple case so we're feeling pretty good about being able to do this we go back to our
46:30
experiment and now we plot the predictions the blue curve let's just focus on that one is the
46:36
our prediction for how the vortex separation works as a function of time and it's matching up
46:42
with the experimental stuff pretty well but now we can do something completely new we can use our
46:49
our theoretical expressions for the vortex tilt and we can actually use those in
46:54
the lab to back out what the tilt of the vortices are experimentally and so we're plotting the
47:00
theoretical predictions on the horizontal and the vertical uh on the vertical it's the experimental measurements and we're
47:06
comparing those ideally to be along a 45 degree and we're pretty darn close one standard
47:11
deviation is the um the shaded areas here and honestly these are really hard
47:17
experiments easy for the theory side but hard to do experimentally and so it's it was actually surprising
47:23
to us that we got within a standard deviation for most of the data points all right let me let me pull this
47:29
together and summarize a little bit we're playing in this area of topological optics that's what that's
47:34
what this really is where we have optics in two dimensions and there's structural features and that's the topology
47:40
so we have two dimension a two plus one dimensional physics of this topological compressible fluid it's very much like a
47:47
dilute bose-einstein condensate in fact with the fetchbach resonance tuning you can change the amount of
47:53
non-linearity in a in a bose-einstein condensate and make it look very much like
47:58
one of these optical fluids the system that we're looking at has four degrees of freedom now we understand
48:05
that two tilt angles in addition to the transverse positions we have uh this what we think is a kind
48:10
of a new dynamical theory associated with these vortices and the important thing is not that just
48:16
that it explains what we saw in the lab but that we can exploit it to actually move the vortices around
48:22
many vortices and we can braid them we can um think about information transfer or
48:27
encoding of information in these things and the motion of the vortices and that's really where we're headed
48:33
we've also looked at bose einstein condensates uh and we've applied all of this even though i i didn't really talk about the
48:40
derivation everything that i showed you applies equally well to a non-linear schrodinger equation and so we're able to make predictions
48:46
for nonlinear materials and include the trap of the bose einstein condensate and predict how the vortices move
48:52
in there and with that i'm going to turn it back to mark for a couple of final comments mark all
48:58
right thanks mark so uh where do we go from here uh so we see this
49:04
as a system where we can look at the optical vortex interaction and and really look do room temperature
49:09
quantum science in what we're calling topological fluids of light or topological optics um and leading to questions about uh are
49:17
these vortices can we treat them as emergent quasi particles uh we have direct control and readout
49:22
right so compared to other systems for doing quantum science uh where you have to do things at really low temperature
49:28
the coherence of the laser beam is is easy and so we can do this out in free space uh and if you want to
49:34
know what happened to the beam you just stick in a camera and it's all at room temperature so we see this the diagram over on the left
49:41
the venn diagram it's kind of a unique triple point between condensed matter and particle and optical physics
49:47
um and we we're looking for more people to join the team so in particular if you're interested in uh looking for a phd or if
49:54
you have students looking for a phd i'd i'd love to chat with you and next uh so thanks to especially the wm keck
50:02
foundation uh so this this talk with with mark leskin me here uh brings back memories of our talk in
50:09
front of their panel and uh just a lot of fun we had when they came to do a site visit and eventually
50:15
very generously supported this work and then also thanks to a number of people in the group in particular jasmine anderson and drew
50:22
voitive took a lot of the experimental data that you saw here uh so
50:27
yeah feel free to contact me by email if you have questions or interest in uh in this topic and we'd be happy to
50:33
take questions now oh i i uh yeah i
50:39
also wanted to mention if you are interested in grad schools uh du has uh is in the top 10 percent in
50:45
terms of percentage of women in the graduate program which is unique
50:54
awesome wow that is incredible um what a
51:00
what a crazy analysis that was really really cool um so we do have a few questions if you
51:07
both would be interested in answering some of these uh so our first one is from david schmidt
51:13
who says have you looked at mixing different vortex beams of opposite but unequal charges
51:21
yeah so i'll jump in there uh so we've thought about a little bit about so let me just start with adding a
51:27
single vortex of um not the same of a non-integer charge so not plus or
51:33
minus one but two or 33 or something like that what you'll see is that if you put that vortex right
51:39
in the center of the beam it will stay as long as you've got pretty good propagation conditions and
51:45
you've centered it as best you can but as soon as you move it off center you break the symmetry and that let's say topological charge of
51:52
33 will break up into 33 unit charge vortices so then you have to
51:57
think about the the motion and the dynamics of those three questions
52:02
uh sorry of those of all of those vortices so we that that may be a next step is to
52:08
start to think about the how those things split up and how they move but um for now we've been focusing on the unit
52:14
charges because those are the ones that are stable with propagation let me let me add one thing there mark
52:20
um that in we intentionally break them up sometimes and we we can break them up to
52:25
make vortices do things uh in particular we can make them circle around each other in a braid or
52:31
we can make them scatter we can have multiple vortices that scatter off of each other and sometimes nucleate additional
52:37
vortices so turning it around that breaking up can be a good thing yep cool
52:45
so from david goldberger experimentally how do you control the vortex tilt
52:51
so we take the analysis that mark lusk showed you in terms of we got this polar tilt theta and this azimuthal tilt
53:00
c and uh then we can plug that in and get what the field should look like and then we just stamp that onto our
53:06
hologram so it's just you know we i showed you lots of examples with theta equals zero
53:12
and c doesn't matter for an untilted vortex but then we just change change our
53:17
hologram to include the tilt in there
53:26
awesome i don't know mark are you looking at the q a the questions no not me i'm just
53:34
looking at dan well you should open it up and look mark mark hassman says beat navy
53:39
just hey jay there's no response required there
53:47
that's ridiculous so i'm gonna i'm gonna say that that one was answered live
53:52
okay so so so this this is reuben and i want to ask a question but i think it's ill-posed
53:59
um usually you know these things that are governed by differential equations
54:04
usually in the first step of solving the problem you're able to figure out how many degrees of freedom the solution has in it
54:10
and it sounds like in this problem you started out with something that looked like it had two degrees of freedom
54:15
and it turned out in the e and they have four and that's like you know more than i
54:21
would have expected so in retrospect does it make sense that there are four degrees of freedom
54:27
given you know the physics of the problem right did the question make any sense
54:34
yeah i i think so i i i get it i mean it was surprising to us we didn't
54:39
walk in thinking oh it's a four degree freedom system but on the other hand it's a continuum field
54:45
and it's our eye that's pulling out the vortex as an object of interest
54:51
right so in your when you pull it out your first thought is the vortex is
54:56
identifiable by the fact that it's a singularity the density the amplitude zero and the phase
55:02
is singular and if that's your definition of a vortex there's two degrees of freedom
55:08
but that's not enough to anticipate what the how the vortex moves
55:14
the only now we can completely determine the motion in the problems that we've done
55:20
so we think there are four degrees of freedom but isn't it like quantum numbers i mean
55:25
you you know you have a complete set when you haven't done any experiments for a long time that show you new quantum numbers so
55:33
i would i would say that uh jury's out i mean there could be a curvature effect here we don't know
55:41
it's a good question yeah um so nick matari says how does the
55:47
theoretical treatment change when considering an open system what types of unwanted interactions with the environment are
55:53
present in optical vortex experiments that sounds like nick yeah
56:03
i'm i'm thinking is that an experimental is that maybe uh i i can tell you an experimental
56:09
factor yeah yeah um so one thing that we didn't talk about really but that goes into it is
56:14
the that background field that we use to calculate you know that the gradient of the
56:19
density and the gradient of the phase um that's not just determined by the other vortex it's also determined by the beam
56:26
that the vortices are moving in and so uh when we do these recombination experiments with two
56:32
vortices what what i showed you was data where the two vertices start pretty close together
56:37
and so that the the gradients of the beam itself at least to start with are rather small um at least relative to the the the
56:44
phase gradients from the the vertices and the amplitude gradients as well um
56:49
so when you include the the density gradients of the beam that is an additional factor that has to be included that's
56:56
kind of some aspect of the environment i i would also add that there that uh
57:02
in an open system here it's less susceptible to noise and to you know the standard
57:08
effects of an open system because of the fact that light doesn't interact with itself
57:21
so i have a quick question before mark lusk has to go so where can can you point me toward
57:28
where you get this uh so you wrote psi as a product of two size one is the
57:33
vortex field scalar field and one is the background scalar field where does that actually come from do you have a you probably had
57:39
a a reference or something but i have not actually seen that type of thing done
57:44
before usually in in optics right we we say that it's a sum of two things not a product but this
57:50
is very interesting yeah um this is this is a starting point for i mean this is an initial condition
57:57
that we could program into uh our uh diffraction gradient you know
58:02
mark's diffraction gradient or we could put in as an initial condition for the shredinger equation right
58:07
so um it's it's just i can i can assert that that
58:13
is my initial condition and then i can uh follow it and i can then assert that
58:19
the um the vortex field is the same if you get really close to
58:25
the vortex and so i can always decompose it into the total field divided by the vortex
58:31
field and what's left is the background so it's not it's not a matter of um
58:38
wow this happens to work out it's more i assert that there's a product at all times and that defines the background field
58:45
okay okay because usually i would just write the background field as one plus something
58:51
right but but here um if you wanted to remove the the vortex that sum isn't going to
58:57
work i mean you have a singularity and you want to remove it you need a product the left hand side has a singularity and the
59:04
right hand side has a singularity and the singularity is embodied in that psi vortex yeah yeah that is really
59:11
fascinating to me let me say that one more way the the e to the i l phi and then potentially
59:17
tilted of the vortex is in the beam itself right so if you try to write the
59:23
that that psi as a sum with a vortex term and a background term you're not going to be able to do it
59:29
because the e to the il phi is it's it's everywhere and so you can't just subtract it off
59:34
it's everywhere on all the amplitude so you really have to uh write it as a product that's the
59:39
natural thing to do with this phase here yeah and and maybe it's worth pointing out while we're looking at this
59:46
that something that mark said at the very beginning the background field of one vortex is
59:51
includes the vortex field of the other vortex or of the other 10 vortices very so so the way that these guys talk
59:58
to each other is by influencing their background field if you were talking about this from a
1:00:03
particle physics perspective you would say that the background field mediates the interaction between two
1:00:09
particles so the the background field is bosonic you know like that
1:00:15
yeah that is so cool okay so we have one more question here i know mark you have to run so if you if you can't stick
1:00:21
around that's we'll just have to lean on mark siemens here but jonathan baralak says do these optical vortex dynamics show up
1:00:28
slash explain what is seen in speckle patterns can these vortices sort of spontaneously form in
1:00:33
pairs making weird speckle patterns for a flat beam that's all mark siemens yeah that's a
1:00:39
prescient question because that's actually where we started um so there's
1:00:45
before we went to the two vortex case we actually looked at speckle and dan came to my lab once and saw if
1:00:52
you image speckle and you get to the very near field of speckle and then change the propagation distance
1:00:57
you can actually see this really crazy it looks like it's biological or alive um these vortices moving around and they
1:01:04
do exactly what you're saying they merge together they pop apart out of nowhere and it's always the plus and minus one
1:01:11
charge pairs and they form loops and all sorts of interesting dynamics we did some
1:01:16
we tracked the vertices and did some statistics on them and found that they the statistics the velocity statistics
1:01:22
pretty much matches what you'd expect from a quantum fluid so vortices and uh helium or bose einstein condensate
1:01:28
and so that made us think hmm you know can we really understand the physics of how these
1:01:33
vortices are really interacting which is a weird thing to say right even now it feels weird to say because we're in a purely linear system
1:01:40
right so there is no interaction the photons are not interacting but if we take this vortex centric
1:01:46
perspective and look from the vortex perspective uh they're writing in this optical fluid
1:01:51
and they're the motion of each vortex or they're advecting the motion of each vortex is driven by the the presence of all the
1:01:59
other vertices in the system and we can describe their motion that way so yes absolutely
1:02:04
speckle is a great example of this
1:02:12
awesome
1:02:19
well i think that might be it doesn't look like we have any other uh questions so
1:02:26
and i think we're a little bit over five so if people want i think the students can be
1:02:33
dismissed and then um those of you who'd like to stick around and talk
1:02:39
to uh mark siemens feel feel free to stick around and mark you have to head out that's
1:02:45
i do thank you everybody i'm i'm sorry that i have to run i had a prior commitment yeah thank you so much thanks everybody
1:02:51
it's really great to share with you and mark siemens you're gonna stay here right i'm gonna i'm
1:02:57
gonna sign off on this and okay and this is gonna disappear um when i when i sign off this the
1:03:03
screen oh right yeah i can try to pull up a copy okay all right you guys thank you
1:03:12
very much see you later now thanks bye
