Usually, light beams shine in a straight line, with the possible exception of light being bent by gravity. But scientists are now investigating how to make light beams into looped and knotted configurations. The possibility for these structured light beams arises from some curious solutions to Maxwell’s equations, which describe the fundamentals of electricity and magnetism.

Physicists William Irvine of New York University and Dirk Bouwmeester of the University of California, Santa Barbara, and Leiden University in The Netherlands have published a study in *Nature Physics* on their analysis of knotted light. The scientists looked at the physical properties of knotted light and discovered that it can be experimentally generated using circularly polarized laser beams.

In some little-known solutions to Maxwell’s equations, all the electric and magnetic field lines form circles that are all linked to each other. These loops of field lines can be used to construct the donut shape of a torus. In such a scenario, each circle wraps around the torus once, and no two circles cross each other. Smaller tori could then be nested within larger ones, filling three-dimensional space with circles of light beams.

These solutions differ from any known existing form of light because of the intricate knotted structure, the physicists told *PhysOrg.com*. As they explained, this structure is based on the Hopf fibration, which was introduced in 1931 to mathematically investigate the structure of spheres in four and higher dimensions, a topic that seemingly has nothing to do with light.

After investigating knotted light’s properties, the physicists determined that they could use laser fields to create the structures. Starting with a single-pulsed beam of circularly polarized light, and tightly focusing the beam, it should be possible to create various shapes of looped light beams. By using holographic techniques and a spatial light modulator, the shape and profile of the looped light could also be controlled. These same techniques have recently been used to produce Airy beams, which are light beams that don’t spread out as they propagate.

Currently the physicists are preparing for an experimental realization of the new solutions either using electromagnetic radiation in the optical regime, i.e. light, or in the microwave regime. The main challenge will be to deal with ultra-short pulses of radiation in order to create a broad spectrum of frequencies as needed for the construction of the light knots.

Because knotted light beams have both beamlike properties and unique unexplored properties, the physicists predict that creating the beams could have applications in several areas. These could include applications in plasma confinement, atomic particle trapping, manipulating cold atomic ensembles, and generating soliton-like solutions in nonlinear media.

__More information:__ Irvine, William T. M. and Bouwmeester, Dirk. “Linked and knotted beams of light.” *Nature Physics,* Vol. 4, September 2008, pp. 716-720.

*Copyright 2008 PhysOrg.com.
All rights reserved. This material may not be published, broadcast, rewritten or redistributed in whole or part without the express written permission of PhysOrg.com.*

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## Falcon

## Corvidae

## CWFlink

This suggests to me that a model of particles is possible that merges this form of EM "knotting" with the warping of space to explain the stability of particles.

Question: in General Relativity, (rest) mass warps space on the hyper-macroscopic scale. Is it possible that energy warps space on the hyper-microscopic scale, allowing standing EM waves in resonant cavities of their own making?

The general objection to this has been the inability to squeeze EM energy into space smaller than the wavelength of the photon, thus the energy density of the photon in flight is certainly smaller than that which causes warping of space... but if the EM wave is "knotted" upon itself, can the energy densities be such to augment the knotting and form a stable, "particle" at or near rest?

Fascinating idea.... even if just a dream.

Comments please!

## earls

If you haven't already, check out the old-school "knot theory" and the mathematics of knots which is making a comeback, finding application in many fields including physics.

## Sean_W

## superhuman

I have been thinking about exactly the same thing during the last few days which strikes me as rather odd.

Particles as an electromagnetic wave entrapped by wrapped space, no mass just energy which by bending space entraps itself, spin as an effect of the direction of propagation, various particles represented by various modes of oscillation, strong force as some higher order effect, between knots, maybe it could even explain wave particle dualty by some sort of overlapping of such particles.

The most obvious reason to consider such a theory is the ease with which particles can be converted into energy and back again, the fact that E=mc2 and the multitude of particles which would make much more sense if they were some modes of oscillation (similar idea as in string theory only instead of strings EM in a wrapped 4d).

However especially after seeing your comment here It occurs to me that it has to be a quite common idea among physicists so there are probably good reasons why it doesn't work.

Maybe some such knotting has been missed before, we will see, would be great if it worked as it would bring more sense into the mess of quantum physics.

## phystic

does anyone remember a certain article released last year?

http://www.physor...529.html

"the neutron has a negative charge both in its inner core and its outer edge, with a positive charge sandwiched in between to make the particle electrically neutral. "

Does anyone see any relevance perhaps? ;-)

Just wondering, seems like some very interesting theories in the comments thus far.

Perhaps along the lines of nested light, relating to quarks, then to decoherence(in a "compton effect-like" style, similar to electrons). Ideas

## CWFlink

Indeed... One can imagine spin, magnetic moment, charge, energy levels, radioactive decay leading to "particle emission" upon transition of energy levels, etc. The idea allows the construction of a model for sub-particles that is analogous to the atomic model for elements. EM resonances (possibly "knotted" to achieve energy density for the imagined "warp") providing on one hand an analog to the electron shells, and on the other hand, explaining the shells themselves as resonances on a higher order.

It is a very attractive model, rich with possibilities, IF.... one assumes that EM energy can warp space on what we typically call the quantum mechanical scale.

String theory (for what very little I know) seems to be based on a similar search for a system of resonances from which a panoply of sub-atomic particles can be derived. But so far, I have not seen a conceptual framework with which to tie the "string" resonances to traditional physical models such as EM and GR.

Is it possible to evolve string theory from traditional EM (including the "knotting" solutions) crossed with the "dream" of EM energy "warping" space/time on the quantum level? And in relation to energy DENSITY, such that the phenomenon can exist only at the level of the very small (or in the center of black holes).

The "dream" would be to thus unify... No. I can't say it. ...it is just too much to hope for. But I await the "tut-tut" that explains why space-time can ONLY be warped on the astronomical scale and never by 0 rest-mass photons. :-)

## neurogalactus

Now, the question is how are those processes realized in EM realm, especially on the quantum level, and can they actually be the source of all energy transitions/interactions in Nature, including gravity and inertia?

It's not hard to postulate that mass/matter is looped/imploded EM radiation; it's much harder to actually offer any sound and testable theory.

Finally, if the Maxwell equations provide knot-theory-related solutions, maybe other solutions exist that reveal the role of resonance, harmonics, fractality and quasiperiodic symmetry on the quantum scale, and could possibly lead to many scientific and technological breakthroughs.

## superhuman

http://en.wikiped...electron

## superhuman

http://en.wikiped...electron

## zafouf

But in Penrose's twistor theory, this traveling Robinson congruence can't be regarded as the path of a spinning particle, because it doesn't Lorentz transform right.

Just as a solution of Maxwell's equations, it has to Lorentz transform right, though.

There were lots of exercises in Roger Penrose's Road to Reality book on twistor theory and these traveling Hopf fibrations. He has an article online about it, it's also called the "Robinson congruence". I put solutions to the twistor exercises online also. There's a webpage with Road to Reality solutions online.

So it has been investigated as a model for massless particles.

Of course you could add these solutions to each other, since Maxwell's equations are linear. I wonder if you get more of the same configurations or what.

I wonder how quantization works, if the spin is quantized in the theory. I don't remember what Penrose said about that.

Laura

## phystic

http://www.physor...529.html

## GoodElf

Vortex knots in light

J Leach1,3, M R Dennis2, J Courtial1 and M J Padgett1

1 Department of Physics and Astronomy, University Avenue, Kelvin Bld, Glasgow G12 8QQ, UK

2 School of Mathematics, University of Southampton, Highfield SO17 1BJ, UK

3 Author to whom any correspondence should be addressed.

Email: j.leach@physics.gla.ac.uk

Received 11 November 2004

Published 15 February 2005

Abstract. Optical vortices generically arise when optical beams are combined. Recently, we reported how several laser beams containing optical vortices could be combined to form optical vortex loops, links and knots embedded in a light beam (Leach et al 2004 Nature432 165). Here, we describe in detail the experiments in which vortex loops form these structures. The experimental construction follows a theoretical model originally proposed by Berry and Dennis, and the beams are synthesized using a programmable spatial light modulator and imaged using a CCD camera.

While it appears those of us "on the outer" can't access this current article and are unable to comment on the specifics of this particular paper you may console yourselves with this older article that was published in full on the experimental aspects of this phenomenon three years ago and more...

http://www.iop.or...055.html

This is not to diminish any of the importance of this recent experimental analysis but it was not "discovered" by these people. The real early work is shown above. These experimental results rests on earlier theory performed by the eminent Michael Berry upon whose suggestion (a lot of his suggestions are spot on) this experiment was performed. I would also suggest this is a topic also well researched by Dr. Taco Visser.

http://www.nat.vu.nl/~tvisser/

What is of note in this Nature article is the linking of this phenomenon to "Hopf fibration"

http://en.wikiped...f_bundle

and then to Chandrasekhar%u2013Kendall curl eigenstates and "ball lightning".

http://arxiv.org/.../0411153

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 Chandrasekhar%u2013Kendall 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.

These are also "favorite topics" of mine over the years... Check out the references.

Cheers

## Nadayanna

## Ivars

The dissipative structure inside vortex is symmetric to de broglie wave structures outside it. If experiment allows to adress the internal structures directly, we get fractional Hall effect, for example.

The fields in Maxwell solution at these lengths are not EM, at least not in a sense they can have energy used to MOve electron as a whole, so energy SCALE, Boltzmann constant, temperature scale is different in those realms by a factor of approx 10^52. (k is 10^52 times bigger).

Energy has different quality there.

## andyd

toroidal topology? See: http://members.ch...tron.pdf

## GoodElf

Some consideration should be given to "Very Special Relativity"...

http://arxiv.org/...36v1.pdf

... Effects which the LHC probably would "miss". In such a situation I think Hawking will win his $100 bet. This "scale" is in the realm of Quantum Electrodynamics.

Cheers

## Kushelev

The Power Engineering of the Future!

http://nanoworld..../104.htm (Russian)

History (English):

Begin(English): http://www.nanowo...0505.htm

Continue (English):

http://www.nanowo...ndex.htm

Future (images): http://nanoworld.narod.ru/

## zafouf

I wonder if there's some theoretical reason why it has to move at the speed of light.

## nilbud

## GoodElf

"As described by Wheeler in the early 1960s, geometrodynamics attempts to realize three catchy slogans

* mass without mass,

* charge without charge,

* field without field."

http://en.wikiped...dynamics

Of course the near field velocity of the speed of light can be "infinite"... However you might be asking some different "deeper" question though?

## GoodElf

http://www.nature...6-s1.pdf

Regarding your last question... In the article by Roger Penrose on Twistor Theory that you referred to above is a section that I very strongly agree with... the section "2. some background ideas"... the answers are at the QED scale not at the "string scale". Recapping... Light as "rays" do not work because of packet spread in transverse modes. Simple practical optical devices such as a "Fourier Optical Processor" invert time to frequency (reciprocal time) and space to reciprocal space in a natural optical Fourier Transform in the time and space domains swapping into the reciprocal domains respectively. In the very near field though this phenomena would be subject to Berry Phase which leads to well known non-commutative properties.

http://www.mi.inf...ase.html

Regarding Fourier Optical Processors here is a simple schematic...

http://members.op...m_1.jpeg

This optical device has been used for several decades to process analog imagery directly. IMHO "Ray Theory" is not a realistic "model" in the evanescent field (but I am not an expert in this specifically).

## ZeroDelta

## dconine

Sounds a lot like modified ether theory to me...

Sure is a good thing there are so many geniuses around to solve this stuff.

## tigger

Perhaps we'll eventually get into the good stuff and move onto prime numbers and quantization. There's something strange going on with prime numbers... there's something strange going on with our mathematical system which quantizes (oh, *shock* (sarcasm) we end up with a quantized description of reality using a quantized mathematical system)

## zafouf

Also, the configuration has an angular momentum, how does that get quantized?

Anyway, at least in Roger Penrose's theory, this twisting configuration couldn't be regarded as the path of a massless particle. I'm not sure if he did end up being able to model massless particles as twistors somehow.

## zafouf

I'm talking about the concept of seeing the whole thing as a particle, but also curious about the "knotted light", the quantized spin for that.

## jburchel

Interesting discussion in the comments, seems a very popular topic. Much more engrossing than stories about baby polar bears dying due to our horrible impact on The Blessed Mother Earth.

## GoodElf

The suggestion that light being bent by "gravity" is not a idea that I think is plausible knowing that the relationship between electromagnetic and gravitational "interactions" are of the order of 10^40. Perhaps it has the cart before the horse. The properties of gravity may be the result of spacetime curvature due to "geometry" created by "light". It is more likely that LQG is not correct and that Roger Penrose is correct as I stated above regarding his online article "On the origins of twistor theory"

http://users.ox.a...0001/#08

Maybe gravity is an "interaction" but I think Einstein is correct and gravity is a pseudoforce and not the direct effect of "particle interactions". This does not mean that it may not be expressed as a particle interaction but what I mean is it is not "primarily" a particle interaction but a wavelet eigenstate in a cavity. These are the "essence" of that "additional material" to the paper. I suggest a careful reading of that material (as far as it goes). As stated in the abstract of this paper we have a "time evolution" which is not part of normal quantum theory (quantum theory is not explicitly expressible, except in the classical limit, as a "history"). So what we are dealing with are wave equation solutions of "measurables" not quantum statistical expectation values. Photon wavelets are a superposition of Fourier component "states" (which are individual cavity excitation solutions). The sum of the states are a sum of the respective "orbits" and compose the "particle" (as a "collapsed" state). This "particle", in the classical limit, obeys Newton's Second Law. It has been a "doctrine" of quantum mechanics that there is a point at which the quantum process is overtaken by the classical process leading to "two theories". This "doctrine" is called the Correspondence Principle. Actually... there has been recent evidence to show that such a limit is not strictly true and every time this has been attempted to be shown to be the case it has mysteriously evaded detection.

http://en.wikiped...rinciple

So instead have a look at this paper which indicates the existence of self accelerating Airy Beams.."Observation of Accelerating Airy Beams" by G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides* PRL 99, 213901 (2007)23 NOVEMBER 2007. If you can't get this one then look at the earlier work by MV Berry which can certainly be downloaded from his homepage.

"Nonspreading Wave Packets" by MV Berry and NL Balazs accepted 12 September 1978. This is the theoretical backing behind the ideas of these practical experiments. A truncated airy packet is an self accelerating wave packet. Without any additional "forces" it accelerates forever in free space in curve or "orbit". These essentially non-spreading packets confirm that a "particle" is a family of overlapping "orbits" and not to a single particle (a point source without a connection to the extended space it exists in). The "particle" is "distributed" in a cavity as a wave. A photon can be made to "loop" in it's own space and become a solution - Chandrasekhar-Kendall curl eigenstate. As Berry put it the discussion in his original paper..."the wave function corresponds to a family of orbits and not to a single particle. The unique nonspreading property is easily related to to the unique shape of the curve which is unaltered (apart from translation) as the classical motion shears the phase space. Moreover, the role played by the caustic shows dramatically how features of wave functions can be dominated by 'forms' (envelopes of families of orbits, which can accelerate, even in empty space) rather than 'things' (individual particle, which are constrained to move with constant velocity)."

What can be said is the Correspondence Principle can be applied to these squeezed states of light. A knowledge of sources leads to exact solutions. One such solution may be that indicated for an electron. The electron would be one of these "nondispersive waves". It just so happens that this property apparently leads to "mass" which is just curved spacetime. So you see that "gravity" does not curve spacetime directly but it is connected to the geometry in which light "propagates" at the quantum level.

The Fine Structure Constant is probably related to the spin angular momentum to the orbital angular momentum of the "orbits". And each successive "orbit" is an integer relationship to the previous orbit. In the Wikipedia: Correspondence Principle referenced above there is a paragraph on "Bohr Model". There you have the "classical" statements made in the first couple of paragraphs that the Fourier Transform will have periods which are related to integer multiples of 1/T and the energy levels differ by the amounts h/T and are 'equally spaced". It is possible (according to that reference) to determine the energy levels by recursion working from the outer orbits to the inner orbits. Usually this is expressed as a "law" of Angular Momentum is an integer multiple of %u045B. Some statements made there about why Bohr was a bit worried being "dismissed" by the author based on a simple understanding of the problem. I charge you all to look again.

Interestingly we have the relationship between the spin angular momentum and the orbital angular momentum being an expression of the Fine Structure Constant. See this reference...

"The Electron as Orbiting Quantum" by Ted R. King, Jr. 4/7/2001. It is a simple but effective demonstration of "cavity" processes leading to quantization through the Fine Structure Constant. The electron is treated as a sort of precessing wavepacket (a collection of waves) in a cavity.

http://en.wikiped...e_packet

The "cavity" has states that exists in the bare proton (it is an ionized atom of hydrogen) while the electron appears to be "opportunistic" in filling the states resonantly.

I would like to comment more on this but I actually do not have the original paper. I am going to make some attempt to secure a copy real soon. Until then the rest is pure speculation.

Cheers

## GoodElf

http://www.memeta...Quantum/

%u045B should read "h bar". There are "problems" with this editorial.

## spice_guru

## GoodElf

http://pdfmenot.c...0d2f.pdf

... Linked and knotted beams of light

WILLIAM T. M. IRVINE1,2* AND DIRK BOUWMEESTER2,3

Cheers