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Tying Knots With Light

Soulskill posted more than 5 years ago | from the knot-easy dept.

Math 125

thedreadedwiccan points out a summary of a recently released physics paper about tying knots with light. A pair of researchers showed that a relatively new solution to Maxwell's equations allows light to be twisted into stable loops. They are designing experiments to test the theory now, and it could have a big impact on fusion technology. The paper's abstract is available at Nature, though a subscription is required to see the rest. Quoting: "In special situations, however, the loops might be stable, such as if light travels through plasma instead of through free space. One of the problems that has plagued experimental nuclear fusion reactors is that the plasma at the heart of them moves faster and faster and tends to escape. That motion can be controlled with magnetic fields, but current methods to generate those fields still don't do the job. If Irvine and Bouwmeester's discovery could be used to generate fields that would send the plasma in closed, non-expanding loops and help contain it, 'that would be extremely spectacular,' Bouwmeester says."

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125 comments

first roast (-1, Offtopic)

Anonymous Coward | more than 5 years ago | (#24997985)

spaced ghost

Subscription required?? (5, Insightful)

linhares (1241614) | more than 5 years ago | (#24998327)

These science publishers are as evil or worse than the RIAA/MPAA with this paywall BS. To paraphrase, science is too important to be left to those that can pay 40 bucks per paper. I can't understand why Google, who wants to "organize the world's information", has not done anything to prevent the world's most valuable information from being inaccessible.

Re:Subscription required?? (2, Insightful)

boto (145530) | more than 5 years ago | (#24999043)

Google has already done it: the researchers just need to make their papers publicly available *anywhere* on the Web, and you'll find the articles on Google Search and Google Scholar Search.

Google can't do much else if the authors aren't interested in making their works openly acessible.

Re:Subscription required?? (1, Interesting)

Anonymous Coward | more than 5 years ago | (#24999085)

Well, they could zero the pagerank on sites that show different stuff to googlebot vs ordinary mortals.

Re:Subscription required?? (4, Insightful)

causality (777677) | more than 5 years ago | (#25000407)

Well, they could zero the pagerank on sites that show different stuff to googlebot vs ordinary mortals.

If THAT'S all it is, then set your user agent to "Googlebot/2.1 (+http://www.google.com/bot.html)" and say fuck 'em. But you're right, Google should actively resist this sort of double standard because it's a detriment to the usefulness of the search engine. It doesn't matter how many great results you get with a search engine if you can't actually access the information in those results.

You know, I still don't understand why there is even such a thing as a user agent string. That is, I can see why i.e. Microsoft would want such a thing but I do not see any way that it's in the interests of users. If we really want standards and we really want openness, having no way for a Web server to determine what the browser is can only advance this goal. Then the only concern is whether that browser is standards-compliant.

Re:Subscription required?? (1)

GigaplexNZ (1233886) | more than 5 years ago | (#25002233)

You know, I still don't understand why there is even such a thing as a user agent string. That is, I can see why i.e. Microsoft would want such a thing but I do not see any way that it's in the interests of users. If we really want standards and we really want openness, having no way for a Web server to determine what the browser is can only advance this goal. Then the only concern is whether that browser is standards-compliant.

At the time of writing, this only achieves 3, Insightful? Come on mods, be a little more generous...

Re:Subscription required?? (1)

Fanro (130986) | more than 5 years ago | (#25002517)

If THAT'S all it is, then set your user agent to "Googlebot/2.1 (+http://www.google.com/bot.html)" and say fuck 'em.

That trick has long since stopped working. All these subscription sites with fake google result have switched to detecting the googlebot by IP.

Re:Subscription required?? (1)

ChameleonDave (1041178) | more than 5 years ago | (#25002539)

If we really want standards and we really want openness, having no way for a Web server to determine what the browser is can only advance this goal. Then the only concern is whether that browser is standards-compliant.

Not true. The only way to provide valid XHTML is to serve it as text/html to IE and application/xhtml+xml to modern browsers.

Re:Subscription required?? (2, Interesting)

ceoyoyo (59147) | more than 5 years ago | (#24999777)

Most journals make you transfer copyright to them. Making your paper available is then illegal.

It's changing, faster and faster. More journals are opening their archives after one or two years.

Of course, you can always go to a library and get a paper for free. Even the local library in the town of 800 people I grew up in had a borrowing agreement with more than one university library.

Re:Subscription required?? (1)

boto (145530) | more than 5 years ago | (#25000289)

Most journals make you transfer copyright to them. Making your paper available is then illegal.

That's pure evil. Why do people keep submitting material to them? Journals that do that should lose their credibility.

It's changing, faster and faster.

I hope you are right.

Re:Subscription required?? (2, Interesting)

causality (777677) | more than 5 years ago | (#25000453)

Most journals make you transfer copyright to them. Making your paper available is then illegal.

That's pure evil. Why do people keep submitting material to them? Journals that do that should lose their credibility.

It's changing, faster and faster.

I hope you are right.

How about this: if you received any (one penny or more) public grants or public funds to perform your research, then that research must be available to the public free of charge. If you are wealthy and want to entirely fund your own research (for example), then you may do whatever you like with the results. The part that I consider bullshit is the idea that tax dollars are taken from me by force or threat of force under a confiscatory tax system and then I am denied access to what this money is purchasing.

Re:Subscription required?? (0)

Anonymous Coward | more than 5 years ago | (#25000651)

How about having grants that pay the entire costs of publishing? I see no reason why a researcher should be required to pay out of pocket to publish research funded by the taxpayer. And no, putting a PDF online is not publishing.

Re:Subscription required?? (1)

ceoyoyo (59147) | more than 5 years ago | (#25001413)

In the US publicly funded research does have to be freely available after some period of time.

Just like anything, the issue is a little more complex than "why don't you just publish it yourself on the Internet?" For an entrenched industry the journal publishers actually seem to be responding to reality fairly well. For a comparison point, see the RIAA.

You are not denied access. ANY library should be able to get you a copy, for free (or possibly the cost of a library card), of any article you like. It might take up to a week. The $40 or whatever stupid fee the journal charges on their web site is purely for the convenience of being able to get the PDF without moving from your couch or picking up a phone.

How would you suggest publication costs be handled? There are some, even on the web. Don't forget peer review. Security. Archives. Should the government go into the journal publishing business? Would that not require taking more tax dollars from you by force?

Re:Subscription required?? (1)

Usquebaugh (230216) | more than 5 years ago | (#25003315)

Costs?

Peer review is currently free.

Bandwidth costs are minimal, Read tenths of a cent.

Security, Archiving is bet not left to private firms with no ongoing interest in the information.

Personally, I think there should be a number of world wide repositories for all papers. The cost of each country/corporation hosting one would be minimal.

What should be charged for is the peer review. You can get any paper for free but if you want a critique of it you have to pay. But no paper can include peer review information. All papers must come from public sources.

Very quickly the reviewers would organize and there would be no more publishers of papers but rather hosting of reviewers.

Re:Subscription required?? (1)

ceoyoyo (59147) | more than 5 years ago | (#25001359)

Because up until quite recently that was a reasonable system. It is in your interest for the journal you submit to to remain solvent.

The Internet has changed things, and the journal industry is trying to figure out how to deal with it. Like any big industry, they're pretty slow at it. They ARE figuring it out though. Science releases freely any article older than a year, and there are open access journals springing up all over the place. One of the problems that has yet to be worked out is that the open access journals often charge the paper author for publication, which is even worse for underfunded labs.

Another factor is that most of the major journals have been busy digitizing their archives. That's a big undertaking (some of them have archives going back hundreds of years). It's expensive. The last five years of journal subscriptions can be seen as subsidizing that process.

Re:Subscription required?? (0, Flamebait)

Anonymous Coward | more than 5 years ago | (#24999095)

Dude, you'll also need a PhD to understand the science so make sure you get that free PhD. tuition from Google. Look at it this way, at least you're saved the embarrassment of reading a paper you will not understand.

Re:Subscription required?? (1)

Prof.Phreak (584152) | more than 5 years ago | (#25000461)

And the stated goal of these journals: to make research -more- available.

Re:Subscription required?? (1)

Usquebaugh (230216) | more than 5 years ago | (#25003325)

And the unstated goal of any organization is to remain alive at all costs.

Re:Subscription required?? (0)

Anonymous Coward | more than 5 years ago | (#25002071)

welcome to academia, now pay up chumps.
We got PloS.org now thats a start i guess...

Done in 1984 - Flux Capacitor (4, Funny)

Anonymous Coward | more than 5 years ago | (#24997989)

Anybody who is anybody saw the flux capacitor in what 1984 - this is old work. the flux capacitor had loops and curves etc.

Re:Done in 1984 - Flux Capacitor (1, Funny)

Anonymous Coward | more than 5 years ago | (#24998667)

At least one of them has a fucked up foreign-sounding last name like Bouwmeester, so I predict this will be a success! Seriously, when's the last time a scientific advancement was made by Cooper & Smith?

Re:Done in 1984 - Flux Capacitor (1)

lanc (762334) | more than 5 years ago | (#25000369)

that doesn't mean that all of the strangely named ones are successful.

Re:Done in 1984 - Flux Capacitor (0)

Anonymous Coward | more than 5 years ago | (#24999009)

Don't you mean November 5th 1955? The day Doc fell on the porcelain, man. Sheeesh.

Re:Done in 1984 - Flux Capacitor (0)

Anonymous Coward | more than 5 years ago | (#25002149)

Cool, somebody is possibly figuring out how to make light go into loops. Maybe photon orbits? And can they be modulated?

Now all we need are two micro-singularities. Anybody at CERN making progress on that yet?

Then it's just a matter of resonance and figuring out to make the right soliton within the light loops that can make a nice Kerr-like pairing or something so we can stretch out the effective event horizon...

Now we just need to figure out who the hell that Titor guy is, since we'll need a volunteer once we actually get the crazy experiment ready.

Real technical vocabulary (5, Funny)

Nuclear Elephant (700938) | more than 5 years ago | (#24998015)

If Irvine and Bouwmeester's discovery could be used to generate fields that would send the plasma in closed, non-expanding loops and help contain it, 'that would be extremely spectacular,' Bouwmeester says."

Bouwmeester continued by saying that light is, "way cool" and the ability to tie knots with it would be, "totally freaking awesome".

Re:Real technical vocabulary (0, Redundant)

Anonymous Coward | more than 5 years ago | (#24998121)

that would be extremely spectacular,
way cool and totally freaking awesome.

Welcome to the University of California, Santa Barbara, California.

Re:Real technical vocabulary (0)

Anonymous Coward | more than 5 years ago | (#24998363)

Let's not be hasty here, that kind of talk gets you tapped for the vice presidency!

Re:Real technical vocabulary (1)

jd (1658) | more than 5 years ago | (#25001077)

I always wondered what the "Vice" in "Vice President" meant.

Re:Real technical vocabulary (-1, Offtopic)

Alsee (515537) | more than 5 years ago | (#24998487)

Totally Gnu/arly dude!

-

Light tied in circles? (1)

philpalm (952191) | more than 5 years ago | (#24998019)

Or maybe the scientists are running around in circles? The goal is to figure out a way to bottle up plasma (for fusion energy harvesting). Since the magnetic bottle has not proven to be viable.

Light sabers? (5, Funny)

slapyslapslap (995769) | more than 5 years ago | (#24998055)

Please tell me this is getting me closer to owning a light saber. PLEASE!!!

Re:Light sabers? (1)

Jerf (17166) | more than 5 years ago | (#24998177)

Yup, it is. But only if it's perfectly spherical and never touches anything.

(Disclaimer: No, I just made that up, but something quite like it is probably true. Might have to be toroidal.)

Re:Light sabers? (0)

Anonymous Coward | more than 5 years ago | (#24998179)

this gets you closer to owning a light saber.

Re:Light sabers? (0)

Anonymous Coward | more than 5 years ago | (#24998183)

This is getting you closer to owning a light saber. I'm like the Santa Claus of Slashdot.

Re:Light sabers? (0)

MartinSchou (1360093) | more than 5 years ago | (#24998269)

In Soviet Empire light saber pwns you!

Re:Light sabers? (1)

DittoBox (978894) | more than 5 years ago | (#24998617)

Shit! I hate it when my Schwartz gets twisted!

The summary misses the key point (4, Insightful)

Kupfernigk (1190345) | more than 5 years ago | (#24998093)

The light knots are secondary, the key point is solutions to the equations in which the electric and magnetic fields form closed loops. Otherwise the submission makes no sense, because the plasma in fusion experiments consists of matter, not photons.

Even so, why do I think this is not actually going to work? Because for the last fifty years, fusion power has been constantly just twenty years in the future, that's why. The authors don't claim a solution to fusion containment, they are talking about possible new ways of trapping photons or creating condensates.

Re:The summary misses the key point (0)

Anonymous Coward | more than 5 years ago | (#24998113)

The magnetic field already forms closed loops.

Re:The summary misses the key point (4, Informative)

quanminoan (812306) | more than 5 years ago | (#24998189)

Exactly - the magnetic confinement for a fusion torus is already completely closed. With a torus, as I understand, there are issues with plasma stability that limit the performance of the devices. However, there is no need for this light looping when you can just alter the magnetic field. Stellarators use a sort of 'helical' magnetic field twisting around a toroid to create a much more stable environment. See: http://www.physics.ucla.edu/icnsp/Html/spong/w7x_with_coils.JPG [ucla.edu] .

Re:The summary misses the key point (3, Interesting)

sedm1143 (1253596) | more than 5 years ago | (#24998291)

Yes, but plasma consists of charged particles which can be trapped by electro-magnetic fields. Light (in the wave picture at least) is simply an electro-magnetic field, so if you can tie light in loops theoretically you can also trap the plasma too. Now I agree that applications are a long way off - this is a theoretical paper so presumably no one has (intentionally) done it yet. If this proves interesting someone would have to build/modify an existing experiment to create and detect the phenomenon, and then there's a long way from there to a practical device, assuming it actually proves technically feasible.

Re:The summary misses the key point (1)

John Hasler (414242) | more than 5 years ago | (#24998403)

> Yes, but plasma consists of charged particles which can be trapped by electro-magnetic
> fields. Light (in the wave picture at least) is simply an electro-magnetic field, so if
> you can tie light in loops theoretically you can also trap the plasma too.

Also, plasma affects the propagation of light in such a way that it may help stabilize the light loop.

Ball lightning?

Re:The summary misses the key point (2, Interesting)

jd (1658) | more than 5 years ago | (#25001149)

I read the summary as more of a better feedback system. The earlier you can detect abnormalities, the earlier you can correct them. If the loops are only stable when the plasma is correctly configured, then your feedback becomes almost instantaneous from the time the plasma begins to destabilize, rather than being a rather slow interpretation of data from sensors that will only spot a problem once it passes the error threshold for that sensor. It would be like using the interference pattern from a tuning fork, rather than trying to copy the sound - the feedback loop becomes a part of the system.

Re:The summary misses the key point (0)

Anonymous Coward | more than 5 years ago | (#25002069)

First app, suckers, let me see you patent that.
  http://flickr.com/photos/94803303@N00/

Re:The summary misses the key point (2, Informative)

JaredOfEuropa (526365) | more than 5 years ago | (#24998691)

Because for the last fifty years, fusion power has been constantly just twenty years in the future, that's why.

No.

The ITER guys [iter.org] state that it will take until the 2050s until the first production fusion powerplant comes online.

Re:The summary misses the key point (1)

Instine (963303) | more than 5 years ago | (#24998695)

"It is impossible, therefore, that any arguments from experience can prove this resemblance of the past to the future,..." Hume [marxists.org]

Re:The summary misses the key point (1)

Karganeth (1017580) | more than 5 years ago | (#25001889)

Even so, why do I think this is not actually going to work? Because for the last fifty years, fusion power has been constantly just twenty years in the future, that's why.

We have certainly made a lot of progress towards sustainable fusion power in the last 50 years. There's no need to dismiss advancements just because we haven't reached the final stage of sustainable fusion power.

oooh lights (0, Flamebait)

Ryogo (1303193) | more than 5 years ago | (#24998165)

wow, loops.... watch them untie damn it. but no really, thats a bad ass discovery, if only we could find an application for it

Re:oooh lights (1)

The End Of Days (1243248) | more than 5 years ago | (#24998185)

Well get on down to your lab and get to work. You won't be coming up with any advanced uses of new physics while you're screwing around on Slashdot.

Your sig (1)

alfs boner (963844) | more than 5 years ago | (#25000237)

"I really want it" does not mean the same as "I need it" or "I deserve it"

That's a very nice crypto-shill for the RIAA.

Mods, please downmod every single post by this fucker until he changes his signature, thanks.

Re:oooh lights (0)

Anonymous Coward | more than 5 years ago | (#24998323)

... so, the article says that you can make stable loops for photons, if they travel through plasma, and that producing stable EM fields would help contain plasma. Probably a long shot made for publicity, but how can you not see the connection here?

The Real Question is.... (3, Interesting)

cychem1 (942136) | more than 5 years ago | (#24998211)

The real question is was a silver hammer necessary?

It's just cool (though maybe unrealistic)! (5, Informative)

gardyloo (512791) | more than 5 years ago | (#24998219)

The (slashdot) summary really does miss some of the key points, and emphasize the "fusion containment" aspect, which I doubt anyone takes seriously as a use of this. One of the points that I think is key is the whole subject of homotopy groups (which I've really just learned about).

Maxwell's equations (and the wave equation, the Helmholtz equation in momentum space, etc.) have a family of solutions characterized by various parameter values. When you first start learning physics, you typically only allow real-valued wavevectors, which leads to only propagating waves and so on. Later on, you start to realize (as did George Green around 150 years ago, and Newton realized experimentally) that allowing for complex wavenumbers is more appealing mathematically (because it allows for more complete solutions), and actually leads to physically realizable solutions that propagating waves just don't give you. The effect of passing from real to complex wavenumbers is, on the face of it, crazy, but easily understandable once the analysis is carried out, and simple to visualize on an Argand diagram.

However, homotopy groups (if I understand it correctly) say that there may be other solutions to such equations (in nonlinear/dispersive media) which one can't get to from just simple replacements of real with complex numbers, and so forth---these divisions are the "families" of solutions. There just isn't a simple projection from one family of solutions to another, and the solutions of from one may bear no resemblance to the solutions from other famililes. This means that there may, in sufficiently complicated systems, be physically realizable behaviors which a system may fall in to, which aren't describable by the "usual" solutions of the equations. Of course, Maxwell's equations work wonderfully in all situations I've ever heard of (no concession to the "Electric Universe" wackos!), so perhaps nature, for some reason, won't allow other families of solutions to make themselves known on any scale I know of.

Re:It's just cool (though maybe unrealistic)! (1)

mrops (927562) | more than 5 years ago | (#24998307)

Whooooosh....

Re:It's just cool (though maybe unrealistic)! (0)

Anonymous Coward | more than 5 years ago | (#24998313)

Yeah, what he said :(

-Just another physics pwned Slashdotter.

Re:It's just cool (though maybe unrealistic)! (2, Insightful)

Anonymous Coward | more than 5 years ago | (#24998389)

no concession to the "Electric Universe" wackos!

While there are undoubtedly wackos out there, it's important not to be too absolute and dogmatic about unsubstantiated explanations for physical phenomena, because wackoness is always judged relative to current models rather than relative to the full but unknowable truth.

All it takes to turn a wacko into an annoying "I told you so" is some physicist doing some lateral thinking and coming out with a new theory or an extension to a current one which just turns out to be correct. And theoretical physicists have a habit of doing that.

While the majority of wackos are inevitably going to be wrong, a few are just as inevitably going to be right. Let the scientific method decide.

Re:It's just cool (though maybe unrealistic)! (1, Insightful)

Anonymous Coward | more than 5 years ago | (#25001201)

Take, as an example the Aristotle's "Luminiferous Ether" which was viable, then crazy after MichelsonMorley, and now - depending on the next 6 months at CERN, very similar concepts may not sound so crazy any more if physicists can observe Higgs... leading to a pervasive field in the universe that creates mass (not light... which the original idea was used to explain), but nonetheless kind of sort of very much like the (a)ether ideas.

Re:It's just cool (though maybe unrealistic)! (1)

John Hasler (414242) | more than 5 years ago | (#24998449)

> The (slashdot) summary really does miss some of the key points, and emphasize the
> "fusion containment" aspect, which I doubt anyone takes seriously as a use of this.
> ...
> However, homotopy groups (if I understand it correctly) say that there may be other
> solutions to such equations (in nonlinear/dispersive media)...

Nonlinear/dispersive media such as, for example, plasmas?

Re:It's just cool (though maybe unrealistic)! (1)

Alsee (515537) | more than 5 years ago | (#24998663)

The Electric Universe....
because Creationists need someone to mock on science.

-

how the homotopy works? (1)

hypomorph (1305401) | more than 5 years ago | (#24999927)

Please correct me if I'm wrong, but I have a basic understanding of homotopy.

I guess you view a solution as a certain kind of map on R^3 that obeys Maxwell's equations and then use homotopy to deform one map into another, all the while respecting Maxwell. Then, one element of the homotopy group would correspond to one family of solutions which may all be transformed one into another via homotopy. Knowledge about the group formed (which has to come from the kind of topological space that Maxwell's equations define) must imply the existence of other families of solutions (e.g., multiplying two known solutions together in the homotopy group to get a new one).

What I'd like to know is how do Maxwell's equations define the topology on the image of the solution maps. Any help?

Re:It's just cool (though maybe unrealistic)! (2, Insightful)

Anonymous Coward | more than 5 years ago | (#25002145)

We're actually using Maxwell-HEAVISIDE equations all over the world after Oliver Heaviside rewrote Maxwell's original equations from quaternion notation into a much simpler vector notation.. throwing out some interesting stuff along the way.

Oh regarding those Electric Universe 'wackos':

You do realize that you're also calling a winner of the Nobel Prize in Physics a wacko, right?

http://nobelprize.org/nobel_prizes/physics/laureates/1970/alfven-bio.html [nobelprize.org]

And so far their successful predictions should at least be called _interesting_ and not 'wacko' for anyone who follows the scientific approach with components like theory, predictions, verification, modification etc.

http://thunderbolts.info/predictions.htm [thunderbolts.info]

While the ideas of plasma cosmology seem radical. At this point to me they don't seem any more radical than the ideas put forth by standard cosmologists of multiple universes, dark matter, multiple dimensions, black holes, neutron stars spinning from 1.4ms(!!) to thirty seconds, strange matter, dark energy, etc..

Ok, questions (3, Interesting)

Marrow (195242) | more than 5 years ago | (#24998223)

1. How do you bend light without passing it through matter or using a grav field that will crush the experiment?

2. If they can bend light, why are we using electron beams for crt's?

3. If you could build loops of light can they be modulated to store information and read it back again?

Re:Ok, questions (0)

Anonymous Coward | more than 5 years ago | (#24998293)

don't know answer to 1 & 3. for 2: it probably requires lot of power..

Re:Ok, questions (0)

Anonymous Coward | more than 5 years ago | (#24998525)

they're talking about bending light in plasma, ionized gas. If you have an idea to make a display device using this uninvented technology, well you can always spend the $800 for a patent application. As for point three, cool idea.

Re:Ok, questions (4, Informative)

mhall119 (1035984) | more than 5 years ago | (#24998529)

1. How do you bend light without passing it through matter or using a grav field that will crush the experiment?

Magnetic fields will bend light, which I believe is what this paper was based on.

2. If they can bend light, why are we using electron beams for crt's?

Because it's easier to bend a stream of electrons than a stream of photons.

3. If you could build loops of light can they be modulated to store information and read it back again?

I suppose, in theory, but it wouldn't be the most efficient means of data storage.

The reason, I think (IANAP), that this could be important to fusion reactions is that a photon loop within a plasma could heat the plasma to fusion-levels without the plasma trying to burn it's way through the outer walls of the reaction chamber. Current torus designs, I think (IANA nuclear scientist), run the plasma around the inside of a magnetic field, like cars on a racetrack, to get the energies necessary for fusion. This causes that super-hot plasma to push against the outer part of the magnetic field, which has to be extremely strong to contain it.

Re:Ok, questions (1)

Marrow (195242) | more than 5 years ago | (#24999195)

I don't think that magnetic fields can bend light.

Re:Ok, questions (1)

mhall119 (1035984) | more than 5 years ago | (#24999227)

Well, technically they'll bend space, just like gravity does, which will bend the light. However, magnetism is many orders of magnitude less effective at it.

Re:Ok, questions (1)

ceoyoyo (59147) | more than 5 years ago | (#24999879)

Magnetic fields don't bend space. At least not the way you're thinking. In string theory there's the possibility that magnetic and electric forces can be described as geometrical distortions of some of the EXTRA dimensions, but not the three (or four) we're used to.

Light in free space completely ignores magnetic and electric fields, for all intents and purposes. If you want to get technical, magnetic and electric fields, since they carry energy, do gravitate, but VERY slightly. You'd need a truly huge field (as in cosmically huge, not really big junkyard magnet huge) to have any effect worth worrying about. Also, quantum mechanics predicts that light waves interact very slightly. It's measurable, but very small, and doesn't really result in bending of light beams.

Re:Ok, questions (0)

Anonymous Coward | more than 5 years ago | (#24999371)

I'm no physicist, but I thought of this once, because light is electromagnetic, so I looked up how light works and what I found indicated it wasn't possible.

If I remember correctly, I came to this conclusion because light waves "twist" as they go through space, and as a result, one would have to change their electromegnet's polarity at the same speed as the frequency of the light, and on top of that you'd have to know the state of the light coming in or you'd be out of sync and bend it in the wrong direction.

In the end I decided that the best you could hope to accomplish was to disperse a concentrated light beam in random directions.

But like I said, I'm no physicist. But in theory I'm pretty sure a magnetic field should affect a beam of light. It's just a matter of in what way and if we can control it.

Re:Ok, questions (0)

Anonymous Coward | more than 5 years ago | (#25002171)

If I remember correctly, I came to this conclusion because light waves "twist" as they go through space, and as a result, one would have to change their electromegnet's polarity at the same speed as the frequency of the light, and on top of that you'd have to know the state of the light coming in or you'd be out of sync and bend it in the wrong direction.

So you'd need a predictable frequency and phase...

Like a laser? :P

Re:Ok, questions (0)

Anonymous Coward | more than 5 years ago | (#25002747)

or you can setup your, same frequency, light sources so they can act as the field themselves. Now that's some sleek elien science!

Re:Ok, questions (1)

John Hasler (414242) | more than 5 years ago | (#24999259)

> Magnetic fields will bend light, which I believe is what this paper was based on.

Not true.

Re:Ok, questions (1)

mhall119 (1035984) | more than 5 years ago | (#24999771)

I could be wrong, but I seem to remember reading about the very minuscule distortions of space-time that are produce around pulsars and other cosmological objects with very intense magnetic fields.

Re:Ok, questions (1)

drerwk (695572) | more than 5 years ago | (#25003411)

...around pulsars and other cosmological objects with very intense magnetic fields...

Given that the question you answered required not crushing the experiment, and presumably having to do with the terrestrial use of the field for fusion, the cosmological do not count...
But in complete seriocity, post a reference. I think you may be right, and it may have to do with a field so strong that the extreme vacuum energy results in non-linear propagation of the electric field. But Maxwell is linear and would not allow, if I understand correctly.

Re:Ok, questions (1)

mhall119 (1035984) | more than 5 years ago | (#25003859)

I remember is working in pretty much the same was a gravity according to relativity theory, not that the attractive properties of magnetism were distorting space-time, but the extreme amount of it did. This was nearly a decade ago, though, and I haven't been able to find anything on the internet that backs it up. Perhaps it was just an article in a physics magazine, and not something that was actually proven or observed.

Magnets do not bend light - in a vacuum (1)

drerwk (695572) | more than 5 years ago | (#24999899)

Were do you get that magnetic fields bend light? Not with Maxwell, not in a vacuum. Any reference to the contrary will be read!

Re:Ok, questions (1)

sedm1143 (1253596) | more than 5 years ago | (#25000171)

Light is composed (in the particle picture at least) of photons which are neutrally charged particles. As such they are not affected by magnetic fields (which only affect charges MOVING in said magnetic field), outside of weird scenarios like a couple of people above have mentioned.

Re:Ok, questions (2, Informative)

mako1138 (837520) | more than 5 years ago | (#25002839)

The reason, I think (IANAP), that this could be important to fusion reactions is that a photon loop within a plasma could heat the plasma to fusion-levels without the plasma trying to burn it's way through the outer walls of the reaction chamber. Current torus designs, I think (IANA nuclear scientist), run the plasma around the inside of a magnetic field, like cars on a racetrack, to get the energies necessary for fusion. This causes that super-hot plasma to push against the outer part of the magnetic field, which has to be extremely strong to contain it.

Not quite. In a tokamak, the plasma isn't accelerated around the torus to heat it. The basic method is ohmic, or resistive heating, where a current is induced in the plasma with magnetic fields. The current across the plasma resistance generates heat. This is kinda like your concept, but not exactly.

Ohmic heating is typically insufficient for reaching fusion energies. The other methods of heating rely on direct energy injection, either through RF or neutral ion beams.

Regarding containment, the magnetic field in a torus is not like a hard wall; it only presents a permeable barrier that particles are still able to diffuse across. If you turn up the magnetic field, you slow down the diffusion, but turn it up too high and you risk plasma instabilities. The key is to control the energy leakage to a point where enough energy stays in the plasma long enough to sustain the reaction.

I haven't read this new paper yet, so I can't comment on its applications to fusion.

we'll all see the light/come to our senses, soon? (-1, Offtopic)

Anonymous Coward | more than 5 years ago | (#24998249)

the 'alternatives' are not very appealing.

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Dr. Octavius? (2, Funny)

MikeUW (999162) | more than 5 years ago | (#24998281)

I'm pretty sure this was already covered in Spiderman 3 - hopefully things turn out better this time around.

Re:Dr. Octavius? (3, Funny)

Lord Fury (977501) | more than 5 years ago | (#24998325)

Doc Oc was Spiderman 2. The villain in Spiderman 3 was conforming to societal pressures, security guards outside of Hot Topic, and running out of mascara and hair spray.

This is one step closer to lightsabers, right? (1)

jmccarthy (228531) | more than 5 years ago | (#24998355)

Because I don't need flying cars. I want Jedi weaponry in my lifetime.

Don't cross the streams (4, Funny)

symbolset (646467) | more than 5 years ago | (#24998429)

That is all.

I can make knots too (0)

Anonymous Coward | more than 5 years ago | (#24998447)

Pass me some fiber optic line and I'll make you one.

Maxwell's Equations? (2, Interesting)

Snowtred (1334453) | more than 5 years ago | (#24998545)

Its been awhile since I've had anything with Vector Calculus, but doesn't a stable loop of light violate Maxwell's Equations in some way? Divergence of B = 0, Div of E = p/epsilon, Curl of E = dB/dt. Seems like a stable knot might not fit with that. Anyone more math savvy know?

Ball Lightning? (2, Interesting)

PeterJFraser (572070) | more than 5 years ago | (#24998613)

If it is possible it probably appears in nature.

Scientific papers (1)

The Creator (4611) | more than 5 years ago | (#24998715)

Why aren't they simply published on the internet, instead of some silly place that asks $18 for a pdf?

Re:Scientific papers (1)

ceoyoyo (59147) | more than 5 years ago | (#24999937)

Because somebody has to manage things such as peer review and maintaining stable, reliable, long term archives.

Have you noticed that stuff "published on the Internet" can be unreliable?

$18 is ridiculous for one article. If you're really interested you should be able to get it free through any library.

Re:Scientific papers (1)

jd (1658) | more than 5 years ago | (#25001215)

More than a few are pre-published for free, but aren't peer-reviewed and can potentially therefore deviate substantially from what is actually published. Since an entire science journal doesn't usually cost $18, except perhaps for something truly arcane, and as most scientists can't afford to buy their own journals, relying on the University library to do it for them, the odds are extremely high that you'll find a cheaper version somewhere.

Commercially Awesome? (0)

Anonymous Coward | more than 5 years ago | (#24999037)

Great, all we need is another method to tie knots. Imagine if they made something to untie knots with light! Now that would be commercially viable! XD

tying the knot? (0)

Anonymous Coward | more than 5 years ago | (#24999365)

I like light, I'm just not sure if I'm ready for that kind of commitment.

Full Text (thanks to MIT) (1)

yfarjoun (878821) | more than 5 years ago | (#25000083)

  Unfortunately the figures, equations, and tables came out as "Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com".....if someone can suggest where I should upload the PDF, I'd do that too.

======

Letter

Nature Physics 4, 716 - 720 (2008)
Published online: 31 August 2008 | doi:10.1038/nphys1056

Linked and knotted beams of light

William T. M. Irvine1,2 & Dirk Bouwmeester2,3

Abstract

Maxwell's equations allow for curious solutions characterized by the property that all electric and magnetic field lines are closed loops with any two electric (or magnetic) field lines linked. These little-known solutions, constructed by Rañada1, are based on the Hopf fibration. Here we analyse their physical properties to investigate how they can be experimentally realized. We study their time evolution and uncover, through a decomposition into a spectrum of spherical harmonics, a remarkably simple representation. Using this representation, first, a connection is established to the Chandrasekharâ"Kendall curl eigenstates2, which are of broad importance in plasma physics and fluid dynamics. Second, we show how a new class of knotted beams of light can be derived, and third, we show that approximate knots of light may be generated using tightly focused circularly polarized laser beams. We predict theoretical extensions and potential applications, in fields ranging from fluid dynamics, topological optical solitons and particle trapping to cold atomic gases and plasma confinement.
Introduction

The concept of field lines whose tangents are the electric or magnetic field is typically used to visualize static solutions of Maxwell's equations. Propagating solutions often have simple field-line structures and so are not usually described in terms of field lines. In the present work, we study a propagating field whose defining and most striking property is the topological structure of its electric and magnetic field lines.

An intriguing configuration for field lines is to be linked and/or knotted. Two closed field lines c1(tau), c2(tau) are linked if they have non-vanishing Gauss linking integral3, 4, 5, 6,

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

whereas for a single field line c(tau) the self-linking number, L(c,c), is a measure of knottedness. The linking integral L can also be computed visually by projecting the field lines onto a plane and subsequently counting the crossings in an oriented way3. For example, the lines in Fig. 1a have linking number 1, but do not form a knot, whereas the blue and orange field lines in Fig. 4 below are knotted and linked to each other. In the case of magnetic or electric fields, averaging the linking integral over all field-line pairs together with the self-linking number over all field lines gives rise to the magnetic and electric helicities4, 5:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where B:=nablatimesA and E:=nablatimesC in free space.
Figure 1: Construction of the Hopf fibration.
Figure 1 : Construction of the Hopf fibration.

aâ"c, Left column: A torus can be constructed out of circles (fibres) in such a way that no two circles cross and each circle is linked to every other one. a,b, Each circle in such a configuration wraps once around each circumference of the torus. c, By nesting such tori into one another, the whole of three dimensional space, including the point at Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com) can be filled with linked circles. There are two 'special' fibres: the circle of unit radius that corresponds to the infinitely thin torus, and the straight line, or circle of infinite radius, that corresponds to an infinitely large torus. These two fibres will provide an economical way of characterizing the time evolution of the configuration. Right column: The Hopf map maps such circles in Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com to points on the sphere Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. Each circle is mapped to a point, each torus in Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com onto a (parallel) circle on Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. The circular (straight) special fibres are mapped to the north (south) pole and will be referred to as the n (s) fibres. In the present work, the fibres of two everywhere-orthogonal Hopf fibrations correspond to electric and magnetic field lines (see Fig. 2 for t=0).
Full size image (93 KB)

Since Kelvin proposed knotted field configurations as a model for atoms, knots and links have been studied in branches of physics as diverse as fluid dynamics7, plasma5 and polymer physics6. More recently, an approach to knotted classical fields was proposed8 and further understood and developed9, 10. Knotted vortex lines have also been considered in phases associated with the electron states of hydrogen11, with the Riemannâ"Silberstein vector of the electromagnetic field12 and in phases associated with lines of darkness in a monochromatic light field13, with the latter predictions experimentally verified14.

Here we consider a state of light whose electric field lines are all closed and any two are linked to each other as described in Figs 1 and 2. The magnetic and Poynting field lines are similarly arranged. This structure is based on the Hopf fibration defined by the Hopf map Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (see Fig. 1)15, 16. Using stereographic projections, h can in turn be expressed as a complex function in Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (for example zeta(x,y,z,0) or eta(x,y,z,0) below) whose lines of constant amplitude and phase are circles, and surfaces of constant amplitude are nested tori.
Figure 2: Time evolution of the field lines and energy density of the Hopf knot.
Figure 2 : Time evolution of the field lines and energy density of the Hopf knot.

a,b, Time evolution of the electric field lines (green) (a) and of the magnetic field lines (black) shown together with the electric field lines for reference (b; see also Supplementary Information, Videos S1â"S3). At time t=0, the Hopf fibration discussed in Fig. 1 can be clearly recognized with the n fibres aligned with the x (electric-field) and y (magnetic-field) axes, and the s fibres lying in the yâ"z (electric-field) and xâ"z (magnetic-field) planes, centred on the origin. The same sets of fibres are shown at times t=0.5 and 1. c, Time evolution of the energy density, which is initially spherical and then propagates and expands like an 'opening umbrella' along the z axis. The fibration can be seen to locally rotate about the z axis as well as expanding and deforming, with the structure remaining centred on the centre of energy of the knot. The rotation and deformation seen at these times slows down dramatically at subsequent times (see Fig. 4e), in an analogous way to the variation of the Gouy phase of a focused Gaussian beam20. The electric and magnetic field lines remain perpendicular everywhere throughout the time evolution.
Full size image (100 KB)

Electromagnetic fields derived from the Hopf fibration first appeared in ref. 17, and were extended to propagating solutions by Rañada in refs 1, 18, 19. The construction was cast in terms of differential forms, which provide a natural way to map fields between spaces of differing dimensions. The resulting electric and magnetic fields have simple expressions:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, and x,y,z,t are dimensionless multiples of a length scale a. Since both nablaeta and Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com are perpendicular to lines of constant eta, the magnetic field is tangential to lines of constant eta. A similar argument holds for the electric field and zeta. The corresponding field lines are shown in Fig. 2.

As a first step in our investigation, we present a numerical study of the evolution of the field lines and energy density, shown in Fig. 2. The initially spherical energy density expands like an 'opening umbrella' with a preferred propagation direction (z) while preserving the Hopf structure. The propagation direction is set by the cross-product of the electric and magnetic n lines (as defined in Fig. 1). The s fibre twists around the centre of energy density. The n fibre cuts through the maximum of energy density and its tangent on the z axis undergoes a rotation with an angle analogous to the Gouy phase shift of Gaussian beam optics20.

To further characterize the physical properties of the field configuration, we compute the full set of conserved quantities that correspond to the known (conformal) symmetries of electromagnetism in free space (see Table 1). Note that all currents, when scaled by the energy density and the scale factor a, are integer multiples of one another; that the fields carry angular momentum along the propagation axis and that the momentum is a fraction of the energy, so the Hopf fields can be transformed via a Lorentz transformation to a rest frame, or to a counter-propagating frame, making them even more beamlike.
Table 1: Conserved (Noether) currents and charges of free-space electromagnetism and their value for the knots considered here. The centre column gives expressions for the energy density (corresponding to time-translation invariance), momentum density (space translations), angular momentum density (rotations), boost vector density (Lorentz boosts), special conformal current and charge densities (special conformal transformations, SCT) and dilation charge density (scale invariance). Expressions for the more familiar non-SCT currents can be found in most classical field-theory texts. The special conformal current and charge were computed following the Noether procedure for the special conformal transformations. The rightmost column gives the values of the currents and charges carried by the knots obtained by integrating the current densities over all space. The length scale a=1 and the values are rescaled by the energy density.
Table 1 - Conserved (Noether) currents and charges of free-space electromagnetism and their value for the knots considered here. The centre column gives expressions for the energy density (corresponding to time-translation invariance), momentum density (space translations), angular momentum density (rotations), boost vector density (Lorentz boosts), special conformal current and charge densities (special conformal transformations, SCT) and dilation charge density (scale invariance). Expressions for the more familiar non-SCT currents can be found in most classical field-theory texts. The special conformal current and charge were computed following the Noether procedure for the special conformal transformations. The rightmost column gives the values of the currents and charges carried by the knots obtained by integrating the current densities over all space. The length scale a=1 and the values are rescaled by the energy density.

Full table

Though the linking number is also a conserved quantity for the solution under consideration, it does not correspond to a space-time symmetry; rather, it is a topological invariant4. Indeed, the linking number is not conserved for a general free-space electromagnetic field, but only for fields that satisfy

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

In the case of the Hopf fields defined in equation (2), EdotB=0 guarantees the conservation of linking number.

For problems with spherical symmetry, a natural basis for representing electromagnetic fields is that of the vector spherical harmonics (VSPHs). Labelled by angular-momentum integers lgreater than or equal to1 and -lless than or equal tomless than or equal tol, wavevector k and polarization TE/TM (electric/magnetic field transverse to the radial direction), the vector potential Alm(k,r) for the VSPHs is21

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where L=-irtimesnabla, Ylm are spherical harmonics and fl(kr) is a linear combination of the spherical Bessel functions jl(kr), nl(kr), determined by boundary conditions. In free space Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. A general free-space vector potential A(r,t) can be expressed in the spherical harmonic basis as

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

We present the decomposition of the Hopf electromagnetic field in this basis (see Supplementary Information, Methods S1), revealing the following remarkably simple structure:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

the Hopf field is a superposition of TE and TM vector spherical harmonics corresponding to a single multipole (l=m=1), a relative phase factor i and an energy spectrum

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

Strikingly, the superposition ATE-iATM is an eigenstate of the curl operator, that is, it satisfies the eigenvalue equation

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

Such eigenstates, known as Chandrasekharâ"Kendall (CK)2 states for constant k, are part of a family of fields known as force-free fields and are of broad importance in plasma physics and fluid dynamics22, 23, 24. The Hopf fields are therefore a pulsed version of the CK curl eigenstate fields with energy spectrum omegae-omega.

To understand how such a simple superposition (equation (4)) gives rise to the remarkable field-line structure of the Hopf field, we begin by studying, in Fig. 3, the field lines of the single-frequency curl eigenstates, which have the unique property that the electric, magnetic and A-field lines have the same structure up to a rotation (Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com). The A1,1TE and iA1,1TM field lines are symmetric under rotations about the centre axis and separate into sets of nested tori, with each set centred on zeros of the field. The A1,1TE field lines follow one and the iA1,1TM lines the other circumference of each torus. A superposition will therefore have field lines that stay confined and wrap around the tori with linking (winding ratio) that depends on the field strength (Fig. 3b).
Figure 3: Field-line structure of single-frequency CK curl eigenstates.
Figure 3 : Field-line structure of single-frequency CK curl eigenstates.

For such states, the electric, magnetic and A-field lines have similar structure. a, Field lines of A1,1TE (white) and iA1,1TM (black) spherical harmonic fields at time t=0. Both sets of field lines are symmetric under rotations about the centre axis. The A1,1TE field lines make up a collection of sets of nested deformed tori with the field lines following one of the torus circumferences. The iA1,1TM are circles centred on and perpendicular to the centre axis and therefore follow the other circumference of the nested tori. The field-line direction is reversed in the centre of each set, corresponding to zeros in the field strength. A superposition of A1,1TE and iA1,1TM will therefore have field lines that wrap around the tori and stay confined to them; the ratio of the winding around each circumference will vary depending on the strength of the field at these points. b, Some of the resulting field lines on different tori, with different winding ratios.
Full size image (51 KB)

To understand the step from the single-frequency curl eigenstates to a Hopf configuration, it is necessary to take the energy spectrum into consideration. To nest all tori about the same s fibre, the spectrum must 'eliminate' all zeros of the radial function, effectively giving rise to a field without oscillations. This is achieved by the spectrum in equation (5). Interestingly, this spectrum is close to one used in research on single-cycle light 'bullets'25 based on the fields of Ziolkowski26, whose defining property is the absence of oscillations.

We now consider how the linking is preserved in time. A curl eigenstate has conserved helicity because its helicity integral is proportional to the norm of the state. A calculation of the helicity integral (equation (1)) for a general field in the CK basis gives

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com of equation (3) are the coefficients of the CK eigenstates. Only the second term is time dependent; the absence of ((l,m),(l,-m)) pairs therefore guarantees the conservation of helicity. The conserved part (first term) of the linking number is proportional to the difference between the amplitudes of the + and - CK states.

The understanding gained above suggests a route to generalizing these solutions: taking superpositions of curl states with different values of l and m and a similar energy spectrum. The result for l=2, m=2, is shown in Fig. 4. Not all the field lines are closed and link in the same way, as is the case for the Hopf fields; however, fibres analogous to the s and p fibres of the Hopf knot can be found. These fibres close, follow the energy density and have a fixed linking structure in the form of two intertwined trefoil knots. In addition, we note that by varying the relative strength of the TE and TM components in the single-frequency building blocks, pATE+iqATM, all possible torus knots with winding p,q can be produced at time t=0.
Figure 4: A generalization of the Hopf fields on the basis of the VSPHs, obtained using l=2,m=2 multipole fields and the spectrum of equation (5).
Figure 4 : A generalization of the Hopf fields on the basis of the VSPHs, obtained using l=2,m=2 multipole fields and the spectrum of equation|[nbsp]|(5).

aâ"c, The magnetic (red) and electric (yellow) field lines at time t=0. d, within these field lines we have numerically found a set of fibres that are analogous to the s and n fibres of the Hopf fields in that their structure is preserved by time evolution. There are two (red) lines analogous to the s fibre and four analogous to the n fibre. e, The time evolution of the s and n fibres for the Hopf field for comparison. f, The structure of the fibres analogous to the n fibre can be better understood by picking a pair and allowing it to 'relax' using a package for representing knots32, revealing doubly linked trefoil knots.
Full size image (98 KB)

We now turn to the possibility of an experimental realization using laser fields. A simple argument suggests that the relative phase factor of the CK building blocks may be fairly robust: pure TE and TM free-space VSPHs are a simple superposition of out-going and in-going VSPHs; their time-averaged Poynting vector is indeed purely azimuthal. The only way to construct a pure multipole field that propagates along the z axis in free space is by taking a superposition of TE and TM fields with a phase i; therefore, any propagating pure multipole is a Chandrasekharâ"Kendall curl eigenstate.

The fact that the Hopf fields are built of CK states with only one value of l and m suggests Laguerreâ"Gaussian beams20 as a good starting point for their production; these are pure angular-momentum eigenstates of the paraxial wave equation and are used as a basis to model laser beams. Though studied extensively, only recently has the relation between strongly focused laser beams and VSPHs received some attention27, 28, motivated in part by optical tweezing. Using the code developed in ref. 28 provided to us by the authors, we found that a strongly focused zeroth-order Gaussian beam with circular polarization converges towards a pure l=1,m=1 multipole field as the focusing angle increases toward 90Â. This suggests that an experimental implementation of the CK basis states may be simpler than we might expect. To create the full Hopf field we would start with a single- or few-cycle pulsed beam of circularly polarized light29 and focus it tightly. The pulse shape and spatial profile could be further controlled with a spatial light modulator using holographic techniques30, which have been recently used to produce, for example, pure Airy beams31.

In conclusion, we have investigated the physical properties of an exceptional solution of the charge-free Maxwell equations in which all field lines are linked once with one another. The decomposition into vector spherical harmonics has revealed the relation to eigenstates of the curl operator, led the way to new field configurations with multiple linking and given guidance on how to generate such special solutions in an experiment. Since the class of electromagnetic knots has both beamlike propagation and unique properties that have not been explored in this context, we predict a wide variety of potential applications and theoretical extensions in areas ranging from colloidal and atomic particle trapping to manipulating cold atomic ensembles and from generating soliton-like solutions in nonlinear media to helicity injection for plasma confinement.

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Acknowledgements

We gratefully acknowledge discussions with M. Srednicki, J. Hartle and K. Millett. We thank V. Vitelli, C. Simon and F. Azhar for comments on the manuscript. W.T.M.I. gratefully acknowledges support from the English Speaking Union through a Lindemann Fellowship. D.B. acknowledges support from Marie Curie EXT-CT-2006-042580.

Received 17 March 2008; Accepted 24 July 2008; Published online 31 August 2008.
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References

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      1. Center for Soft Condensed Matter Research, Department of Physics, New York University, New York 10003, USA
      2. Department of Physics, University of California, Santa Barbara, California 93106, USA
      3. Huygens Laboratory, Leiden University, PO Box 9504, 2300 RA Leiden, The Netherlands

Correspondence to: William T. M. Irvine1,2 e-mail: wi1@nyu.edu

Re:Full Text (thanks to MIT) (0)

Anonymous Coward | more than 5 years ago | (#25000225)

(Posting as AC to protect moderation...)
You can upload the PDF for free (and link to a javascript viewer) at PDFMeNot [pdfmenot.com] .

Re:Full Text (thanks to MIT) (0)

Anonymous Coward | more than 5 years ago | (#25000433)

I do believe this constitutes copyright infringement.

The real question... (1)

longacre (1090157) | more than 5 years ago | (#25000659)

When will Air Jordans have light-based shoelaces?

Maxwell's equations (1)

Doc Hoss (1062428) | more than 5 years ago | (#25001059)

For those not in the know with Maxwell's equations, here's the Wikipedia [wikipedia.org] for them.

a string?! (0)

Anonymous Coward | more than 5 years ago | (#25001347)

Light propagates so it is not a true loop, but more like the idea of a vibrating string from string theory. Does this mean nature mimics its self from small scale to large scale since the solar system is similar to an atom? Probably not, but it is a nice idea to think about.

Ball lightning? (1)

Ungrounded Lightning (62228) | more than 5 years ago | (#25001427)

Stable loops of light in plasma. I wonder if this might be related to ball lightning?

Annular Fusion, as seen in "Infinite Jest"? (0)

Anonymous Coward | more than 5 years ago | (#25003487)

The novel "Infinite Jest" mentioned something called annular fusion, which involved optics somehow. This article involves optics (in a sense) and fusion. Was David Foster Wallace about twelve years ahead of this curve?

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