Banner
    Why Increasing Bandwidth Of Radio Transmissions Using Orbital Angular Momentum Is Innovative
    By Tommaso Dorigo | March 24th 2014 04:20 AM | 30 comments | Print | E-mail | Track Comments
    About Tommaso

    I am an experimental particle physicist working with the CMS experiment at CERN. In my spare time I play chess, abuse the piano, and aim my dobson...

    View Tommaso's Profile
    I received the following comment from Bo Thide', one of the authors of the paper where Fabrizio Tamburini and collaborators explain their novel method to multiply the transmission of information via EM waves (see here). I think his points are of interest to many so I decided to elect his comment to a independent posting here.

    By the way, Bo Thide' is a Swedish professor at the Uppsala department of Physics and Astronomy. For his CV see here.

    Comments welcome...

    ----
    From Bo Thide':

    A few points about MIMO vs OAM and OAM vs SAM (wave polarisation):

    1. Every EM field from any arbitrary radio or TV antenna, laser, light bulb, star or whatever object that emits radiowaves, lightwaves etc carry angular momentum density. Precisely as they all carry energy and linear momentum density (also known as Poynting vector). This follows directly from Maxwell's equations.

    2. Wave polarisation (left-hand/right-hand, or horizontal/vertical) is the spin part of the electromagnetic angular momentum. It is therefore called SAM (spin angular momentum) and is entirely different from the OAM which is the orbital part of the angular momentum. It is precisely as with the motion of the earth. It spins on its own axis at a rate of approximately 1 rev per 24 h. This is the earhts SAM. It also orbits around the sun at a rate of about 1 rev per 365 days. This is the earth's OAM. The SAM (24 h) and the OAM (365 days) have nothing to do with each other. The same is true for photons/electromagnetic fields.

    3. MIMO is based on the use of multiple antennas. It is *necessary* for MIMO to be able to increase the spectral capacity to use multiple antennas at both the transmitting and receiving end. OAM can increase the spectral capacity using only a single antenna at the transmitting and receiving ends. The fact that it is *possible* or *sufficient* to use multiple antennas to generate and detect OAM (at least approximately) does not mean that it is *necessary*.

    4. MIMO requires massive, time- and energy consuming digital postprocessing, OAM requires almost no digital postprocessing.

    Short summary of angular momentum radio:

    Transferring information wirelessy with electromagnetic (EM) fields amounts to encoding information onto physical observables carried by these fields, radiating them into the surrounding space, and detecting them remotely by an appropriate sensor connected to an information-decoding receiver. Each observable is second order in the fields and fulfills a conservation law. Of all available observables, only the linear momentum is fully exploited in present-day radio. A fundamental physical limitation of this observable, which represents the translational degrees of freedom of the fields and of the charges (typically an oscillating electric current along a linear antenna), is that it is single-mode. This means that a linear-momentum radio communication link comprising one transmitting and one receiving antenna, known as a single-input-single-output (SISO) link, can provide only one transmission channel per
    frequency (and polarization). In contrast, angular momentum, which represents the rotational degrees of freedom, is multi-mode, allowing an angular-momentum SISO link to accommodate an arbitrary number of independent transmission channels on one and the same frequency (and polarization).

    Comments

    HenryB
    I find this disappointing.  Based on the criticisms from the last post on this, I was expecting to learn more about the details of this topic.  Instead we get generalizations and no addressing of the details or criticisms.

    So I'll try to point out where there appears to be misleading statements (or at least ignoring criticisms), and hopefully others more knowledgeable here (in Science 2.0 fashion) can help fill in the gaps.
    1. Every EM field [...] carry angular momentum density. Precisely as they all carry energy and linear momentum density (also known as Poynting vector). This follows directly from Maxwell's equations.
    Those statements are technically true.  However it, in context, gives the misleading impression that the OAM represents additional intrinsic degrees of freedom. That is not the case.  QED gives only two intrinsic degrees of freedom to photons, and the orbital angular momentum 'modes' are not missing in this description.  If you want, OAM can be completely described in spacetime distributions of these intrinsic polarization modes.

    Thinking of this in terms of angular momentum density with Maxwell's equations, while 'density' is a pointwise defined notion, is it actually possible to distinguish these modes from measurements at a single point?  From the critics it sounds like the answer is no.  For example OAM sorting in "twisted light" experiments requires apparatus that covers the whole are significant portions of the beam profile.

    The point of the critics is that its the spatial distribution which defines the OAM modes, and thus this is not an intrinsic freedom, and thus cannot work in the far field. Therefore this can only work where the beam is confined enough to allow us to distinguish spatial correlations from the source.
    2. Wave polarisation (left-hand/right-hand, or horizontal/vertical) is the spin part of the electromagnetic angular momentum. It is therefore called SAM (spin angular momentum) and is entirely different from the OAM which is the orbital part of the angular momentum.
    Calling it "entirely different" is misleading.
    Again, OAM modes can be built completely from the "wave polarization" modes.
    We are not missing any fundamental freedoms by using spin modes as our basis. 
    3. MIMO is based on the use of multiple antennas. It is *necessary* for MIMO to be able to increase the spectral capacity to use multiple antennas at both the transmitting and receiving end. OAM can increase the spectral capacity using only a single antenna at the transmitting and receiving ends. The fact that it is *possible* or *sufficient* to use multiple antennas to generate and detect OAM (at least approximately) does not mean that it is *necessary*.
    Either this is using a bizarre definition of "antenna" or this appears to just be wrong.  From the discussion in #1, it is only by looking at the spatial distribution that these modes can even be distinguished.  Therefore correlations from multiple antenna elements spread over some distance appears absolutely *necessary* due to this being a spatial distribution and not an intrinsic freedom.  How far the antenna elements needs to be spread depends on how focussed the beam front, thus trying to prevent the "far-field" limit where no local spatial distribution can be seen.

    angular momentum, which represents the rotational degrees of freedom, is multi-mode, allowing an angular-momentum SISO link to accommodate an arbitrary number of independent transmission channels on one and the same frequency (and polarization).
    Ending with the claim that this gives extra "degrees of freedom" for "an arbitrary number of independent transmission channels" is incredibly frustrating.  This is the language that riles up the critics, because there are NOT extra degrees of freedom here.  And to repeat this without addressing any of the critics is seriously disappointing.  This is not effective outreach.
    dorigo
    Dear Edward,

    I have asked Bo Thide' to answer your comment, as I am not in the position of doing it
    meaningfully.

    Thanks,
    T.
    If Bo Thide has declined to address the critiques, would you mind at least giving some feedback yourself?

    I have an electrical engineering background, and I have to say the critics sounds very convincing. Even friends of mine with physics backgrounds don't seem to agree with the claims being made here. So I'm really really struggling to understand this "breakthrough" here.

    If you think this isn't nonsense, can you please explain to us engineers WHY? It sounds like the dispute is even at the basic level of interpretations from Maxwell's equations (and with radio we're definitely dealing with the classical regime), so I don't really understand why you don't feel comfortable giving a "meaningful" response. I really want to understand the other side of this. Any information is better than none.

    dorigo
    Please see the answer by Bo Thide' at the end of the thread,
    T.
    I think this is a great, short, concise and exciting post, about a great idea by Bo Thide! Well done. Today's radio is a waste of bandwidth because information is only communicated using the amplitude (AM) or the frequency (FM) of the radio transmission, and does not make use of the possibility of using the spin phase of the radio photons.

    Just imagine how much more data bandwidth we could achieve in the radio spectrum if we exploited the phase of the photons to carry information, not just the amplitude or frequency! It is astonishing that we are so backward. I believe this is partly because of the use of complex numbers to represent phase, both in quantum field theory and also in simple AC electrical theory.

    E.g., Weyl and Feynman used the complex component of Euler's exp(iA) = (cos A) + i(sin A), to represent the phase, and the non-complex term in the equation (cos A) to represent simple amplitude. Of course, you need both phase and amplitude to fully represent all knowledge of a wave, but you don't really need complex numbers if you just draw a picture of a particle spinning around as it propagates!

    This use of i(sin A) to represent the phase of the spin of a particle as it propagates is very convenient mathematically and as Bethe and others showed, complex numbers are helpful in the Optical Theorem (as Prof. Jacques Distler helpfully explains on his blog). But it is very tempting for professors to resort to complex math to obfuscate simple thing like the spin of particles, but using dense mathematical modelling, then claiming that the "model" is the reality... What we need to do is to recognise and applaud the successes of complex maths, without putting students off the use of spin polarization to increase the effective data bandwidth of the radio spectrum for communicating!

    This is simply wrong: "MIMO requires massive, time- and energy consuming digital postprocessing"

    If MIMO antennae are fixed and carefully positioned, then fairly simple mixing will suffice (and can be done in waveguides or analog electronics). E.g., for a point-to-point link between fixed masts.

    The reason why complex processing is typically done e.g., in MIMO WIFI units is because you need to track initially unknown and constantly changing relationships between (possibly many) end-points.

    Some articles present it as if there is just a language barrier and some engineers aren't understanding this "breakthrough" by some astrophysicists. I find this a bit insulting to engineers. And as you (and Edward above) point out, there are times where the statements go beyond misleading to being just simply wrong. I'm starting to think this has nothing to do with a language barrier; there really just isn't anything here.

    Between you and Edward, every single point has been critiqued. Can anyone here respond, supporting this article's claims? I'd like to hear a compelling explanation from the 'other side'. Anyone?

    David - here's my best shot at understanding what is being claimed...

    In an optical fibre or RF waveguide, then the use of OAM seems to make sense to me - if you can coerce signals to take helical paths, then presumably as always the corresponding angular momentum will be nicely quantised & from 1000 feet it looks just like spin except you are not limited to two polarisation states.

    I have no idea about the practicalities of implementing that. (As the wikipedia page states, single-mode fibres are useless, you need to use short-range multi-mode fibre, and I'm guessing the that the more modes you try and use, the lower the range you'll get out of it). But in practice, similar results are obtained by multiplexing different wavelengths in a fibre, and that works fine on single-mode fibre, so until we start using the full 100THz bandwidth of fibre, I can't see much demand for anything more complicated.

    I'm struggling with the concept of doing this in free space though. The EM radiation can be completely described using the momentum / spin basis (or frequency/direction/polarisation for the classical case), it seems obvious that there is no hidden blood to be squeezed out of the stone. And once you strip away all the implementation details, if you have N connections from the antenna(e) to the electronics, then you can only extract N independent linear combinations. I'm quite prepared to believe that there might be cleverer or simpler implementations of the linear algebra than traditional MIMO, maybe using OAM does that, but if that's all that's going on the selling of it as a fundamental advance sounds misleading.

    Thank you.

    "And once you strip away all the implementation details, if you have N connections from the antenna(e) to the electronics, then you can only extract N independent linear combinations."

    That is a very nice summary of the situation. I also could buy into OAM possibly giving "cleverer or simpler implementations of the linear algebra", but even if that is the case, there really just doesn't seem to be anything else there.

    The claim: "an angular-momentum SISO link to accommodate an arbitrary number of independent transmission channels on one and the same frequency" ... sounds well beyond misleading to me. It's wrong. It's good to see in articles that people are speaking up about these claims and the hype, but I keep thinking that because it doesn't just get dismissed , that somehow there is another side to this I'm missing. For instance Tommaso believes for some reason there is something here. I'd like to keep an open mind, but at this point it just seems so clear that they are wrong I'm straining to even imagine what could counter all the basic explanations given by critics. I guess in time it will all shake out.

    dorigo
    Hi David,

    unfortunately what I believe does not have a lot of weight, since my knowledge of the field is sketchy. I prefer to avoid arguing in favour of the ideas of Tamburini and Thide' as I would probably do more harm than good. I have asked them to comment but of course they are quite busy implementing these ideas in practice, while here we are still discussing things at an academic level.

    Cheers,
    T.
    Okay then let's stick with the academic level, and I'll try to reach across from engineer to physicist and word this as a particle physics problem. Hopefully then you can join the conversation as it is your field of expertise.

    Imagine that there were indeed an infinite number of extra degrees of freedom at each point in the photon field. How would this affect the standard model? For one, it would mean positronium would decay instantly. Actually, the branching ratios of any interactions with photons in the final state would change drastically. No?

    This 'extra branches' effect would even over power the 'reduction' by the ~ 1/137 factor from the fine structure constant of adding in another photon line in any interaction diagram with a photon, due to the sum over available photon modes when calculating the cross section. Therefore naive perturbation would give the result that adding another outgoing photon always makes it more probable. This means perturbation theory would break down and not even work with QED anymore.

    In such a situation, this reminds me of a rough argument a professor told me once about why the electron had to be a field: Thermodynamics demands it. A field has infinitely more degrees of freedom than a particle, and therefore in thermodynamic equilibrium the energy split evenly means the electron's share would be finite / infinite = 0 average kinetic energy, with all the energy going into the photon modes. So in such a situation there would be an entropic dragging force whereby any particle's kinetic energy would be sacrificed to the stray electromagnetic fields as that is higher entropy. The local "net momentum = zero" frame of the photon field would become a preferred frame in this sense. Since we don't see this, these thermodynamics arguments are telling us the electron must be a field too.

    Increasing the photon field's degree of freedoms at a point from two to infinity would just recreate this situation. Electrons, even as a field now would be dwarfed by the degrees of freedom available to the photon field, therefore in the equilibrium limit it would get zero kinetic energy. It would feel an entropic drag force when moving relative to the cosmic microwave background.

    Any of these lines of thought in particle physics should make it clear: Any claims that OAM provide extra (infinite) degrees of freedom for the electromagnetic field are therefore in stark contrast to the experimental evidence backing the Standard Model.

    The OAM modes are not pointlike degrees of freedom hinted at by the angular momentum density at a point in classical EM. The OAM modes are spatial distributions of the actual internal degrees of freedom: the spin angular momentum. This is made painfully obvious by the fact that the OAM modes can be constructed from distributions in the usual polarization basis.

    If you don't wish to discuss classical EM, can we at least agree on, or discuss, these particle physics points?

    Ok, finally I think I understand what is going on in free-space. Or maybe I've just drunk so much coffee I'm a rambling lunatic.

    The OAM being an angular momentum has discrete states. There is no magic make-quantum-effects-macroscopic here, it is just the fact that if you trace a (non-zero) vector field around a closed loop, the vector has to rotate a multiple of 2pi.

    E.g., Place N circularly polarised antennae at equal intervals around a circle. Then the linear algebra is for mode number k, rotate (either in space, or by time delay) the signal from antenna number n by 2*pi*k*n/N and sum. (That's at the receiving end, invert at the transmitter).

    Probably there is clever design that combines all the elements into one integrated whole.

    This makes sense of the statement 'SISO, single frequency, single polarisation' claim:

    It's "single" in that the antennae has to be integrated as a single precisely aligned unit (but with multiple connections). "Single frequency, single polarisation" is not so much a feature, it's a limitation (assuming that the linear combinations are done by physical construction).

    The linear version (N antennae in a row, receiver matching the interference pattern from the transmitter) or grid version (N by M in a grid) are easier to understand, possibly circular symmetry will make the engineering easier.

    A note on the practicalities:

    On a point-to-point radio link you want to make both ends as directional as possible, to maximise the coupling of the transmitter and receiver, and minimise the coupling to everything else ("antenna gain" is the engineering term) - hence the dish antennae.

    What is possible is limited by mechanical considerations : it gets windy up a tall pole, temperatures change, kids swing on things, and stuff tends to flex. 3 degrees is a typical real-life figure for the directionality.

    Any scheme (circular, linear, grid) that relies purely on passive geometric alignment of interference patterns over a long range is a non-starter for outdoor usage. At a minimum you would need a closed-loop motorised alignment system - these days it's probably simpler just to do digital signal processing to fix the misalignment.

    PS. This patent application appears to support my rather wild inferences: http://www.google.com/patents/WO2012084039A1?cl=en

    "It's "single" in that the antennae has to be integrated as a single precisely aligned unit (but with multiple connections)."

    That's the problem; the only way to currently make sense of this is basically start stretching till we redefine terms. Multiple antennae elements, multiple connections -- this isn't SISO. I'm looking forward to Bo Thide's followup.

    "The OAM being an angular momentum has discrete states. There is no magic make-quantum-effects-macroscopic here"

    Well, kind of. Yes, just like in the sense that there are discrete spherical harmonics even in classical field theory. However regarding the appeal to angular momentum density in Maxwell's equations, this is not a discrete phenomena.

    Thanks for your extra comments. I think we're starting to stretch pretty far to make sense of this, and we're going to just have to wait for the followup to get any additional information out of this.

    The killer is that just as with circular v. linear polarisation, it appears that you can choose a basis combining modes of +/-k OAM instead into orthogonal modes each of zero net OAM.

    This makes it really clear that this is just traditional MIMO with everything in a ring instead of in a line or in a grid. Possibly with the linear algrebra done in the antennae complex rather than downstream in the electronics - there is no obstacle to doing that for carefully laid out linear or rectangular MIMO either - it is just interference patterns (or if you put everything in focal planes of dish reflectors, just focussing an image...)

    dorigo
    Just mentioning here that I attached at the end of the thread a message from Bo Thide'
    T.
    To be pedantic,

    if you have N connections from the antenna(e) to the electronics, then you can only extract N independent linear combinations

    would be better expressed as "the information that you can carry on N connections is just N times the Shannon bound of one connection". Although referring back to the original text "one and the same frequency (and polarization)", if you take that literally (zero bandwidth) it does just reduce to finite dimensional linear algebra on the amplitudes.

    David Matorg claims that I have declined to address the critiques. I have most definitely not! A reply to Mr Brown will come, as soon as we have time. However, there have been other things, including influenza, that has required our full attention lately...

    Thank you!
    I apologize for my impatience, and look forward to hearing details.

    Excellent! I have some questions about implementation practicalities, if you have time to answer them I would be very appreciative.

    The picture in my mind of OAM involves rotational symmetry around a fixed-in-advance center-line of proprogation. In a situation where that is not the case (e.g., receiver and transmitter have unpredictable changes in relative position and orientation), can you avoid "massive, time- and energy consuming digital postprocessing"?

    Again my mental picture of a fixed center-line suggests precise alignment constraints. How does misalignment effect OAM communication, in particular what is the mixing of different OAM wave-modes due to misalignment? Ideally it would be useful to have the answer in terms of comparison with say, fixed-geometry MIMO implemented using a line of N equally spaced antennae using k*n*2*pi/N phasing (k = mode number, n = antennae sequence number, this in combination with the transmitter-receiver separation & frequency fixes the antennae spacing in the obvious manner).

    The Earths atmosphere is often a birefringent transmission medium for radio waves (rain or snow can cause this), the eigenmodes are typically vertical and horizontal (although wind shear will disturb that). How much is
    OAM transmission effected by birefringence (again the main issue is the mixing of different modes)?

    dorigo
    I received the following text from Bo Thide':
    ----------------

    Edward Brown (EB) writes that he has "got the impression" that we should
    have claimed that the orbital angular momentum (OAM) degrees of freedom
    represent some additional intrinsic property of the photon/EM field.
    The fact of the matter couldn't be further from the truth.  Instead the
    truth is that we have claimed precisely the opposite and stated
    repeatedly that OAM is an *extrinsic* property and we continue to state
    this in our papers, books and talks.

    EB seems to be either uniformed or misinformed of our claims and is
    therefore advised to read first hand what we have written on this subject
    and not rely on second-hand information.  Read, for instance, the article

     Tamburini, F., and D. Vicino. "Photon Wave Function: A Covariant
     Formulation and Equivalence with QED." Physical Review A 78, no. 5
     (November 18, 2008): 052116. doi:10.1103/PhysRevA.78.052116.

    in particular the note referenced as [27] where intrinsic vs. extrinsic
    is discussed.

    In Section 9.A.2 "Photon Picture" in the book chapter

     Thide', B., N. Elias II, F. Tamburini, S. M. Mohammadi, and
     J. T. Mendonça. "Applications of Electromagnetic OAM in Astrophysics
     and Space Physics Studies." In Twisted Photons: Applications of Light
     With Orbital Angular Momentum, edited by Juan P. Torres and Lluis Torner,
     pp. 155-178. Weinheim, DE: Wiley-Vch Verlag, John Wiley and Sons, 2011.

    we discuss this again (the book chapter in question can be downloaded from
    www.researchgate.net/publication/229864540_Applications_of_Electromagnetic_OAM_in_Astrophysics_and_Space_Physics_Studies
    ).  This book chapter also discusses other properties of EM/photon
    angular momentum. A perusal of the text and viewing of the figures and
    tables of this book chapter, as well as other chapters in this book,
    should prevent misinterpretation of what we claim and do not claim.

    We also discuss the intrinsic vs. extrinsic issue in

     Tamburini, F, B Thide', E Mari, A Sponselli, A Bianchini, and
     F Romanato. "Reply to Comment on 'Encoding Many Channels on
     the Same Frequency through Radio Vorticity: First Experimental
     Test.'" New Journal of Physics 14, no. 11 (November 7, 2012):
     118002. doi:10.1088/1367-2630/14/11/118002.

    where we yet again try to straighten out some other misunderstandings.

    Furthermore, we have of course never claimed that one can measure
    EM observables in a single point.  Quite the contrary! In fact, EM
    observables (the 10 Poincare'-group observables energy, linear momentum,
    angular momentum, boost momentum, as well as other, non-Poincare' group
    observables) are carried by the EM field in the form of volumetric
    densities that must me measured (integrated) over a certain spatial
    volume to become observable and therefore usable in radio science,
    astronomy, radio communications, radar applications, etc.  This is
    true for today's radio where the linear momentum density (the Poynting
    vector) is typically integrated over a (very thin) cylinder, such as
    a half-wavelength dipole or dipole-like antenna.  In our experience,
    there are many radio engineers (and even some physicists) who are not
    aware of the fact that this is a non-local measurement.  Therefore we
    have put extra emphasis on this in our publications.

    This non-local measurement (integration) of linear momentum density
    is essentially 1D and associated with *force* action, and a single
    antenna of this type can of course not be used for measuring angular
    momentum which is a 2D or 3D observable associated with *torque* action.
    Therefore "antennas" for measuring angular momentum will be very different
    from the antennas used for measuring linear momentum.  Work is going on
    to construct such torque-based antennas but to the best of our knowledge
    none has been demonstrated publically yet.  How EB can call it "bizarre"
    is therefore beyond us.

    If EB had taken the trouble to read our conference paper

     Thide', B., F. Tamburini, H. Then, C. G. Someda, E. Mari, G. Parisi,
     F. Spinello, and F. Romanato. "Angular Momentum Radio." In Complex
     Light and Optical Forces VIII, 8999:89990B–89990B–11. San Francisco,
     CA, USA, 2014. doi:10.1117/12.2041797.

    he would have realised that his "impressions" of our work is heavily
    biased and distorted also on the local vs. non-local issue, and that his
    allegations are contradictory to actual facts.

    As additional reading, we like to recommend the very insightful
    and elucidating essay

     Dyson, F. J. "Why Is Maxwell's Theory so Hard to Understand?" In
     James Clerk Maxwell Commemorative Booklet. Edinburgh, Scotland, UK:
     James Clerk Maxwell Foundation, 1999.

    by one of the masterminds behind QED. The essay is downloadable from
    http://www.ma.hw.ac.uk/iciam99. Particularly the sections in the second
    half of this essay that discuss what is observable and what is not.

    Most good textbooks on electrodynamics cover the physics of EM angular
    momentum.  An excellent one by another of the QED masterminds is

     Schwinger, Julian, Lester L. Deraad, Kimball A. Milton, Wu-Yang Tsai,
     and Joyce Norton. Classical Electrodynamics. Illustrated edition. Perseus
     Books, 1998.

    who treats electromagnetic angular momentum in a way should be easy to
    understand by anybody with a university background in physics, including
    electromagnetism.  In case a more detailed step-by-step derivation of
    the properties of EM angular momentum is needed, Chapter 4 of

     Thide', Bo. Electromagnetic Field Theory. Second edition.
     (An unlocked draft version is freely downloadable from
     http://www.plasma.uu.se/CED/index.html but will be slightly edited
     before it goes to print.)

    might be useful.

    Finally we like to draw the attention to the fact that this also works
    at the quantum single-photon level.  As has been shown experimentally,
    each individual photon does not carry only the (intrinsic) property spin
    angular momentum (SAM), but can be prepared to carry also the (extrinsic)
    property orbital angular momentum (OAM). It can even be entangled in
    OAM. See

     Mair, Alois, Alipasha Vaziri, Gregor Weihs, and Anton
     Zeilinger. “Entanglement of the Orbital Angular Momentum
     States of Photons.” Nature 412, no. 6844 (July 19, 2001):
     313–16. doi:10.1038/35085529.

    OAM can hardly be more quantum than that.
    HenryB
    Thank you very much for taking the time to reply.
    I am glad to hear that you are not claiming OAM is an additional intrinsic freedom.  Your post here appeared to state that to me, in order to reach your conclusions.  I'm glad we cleared that up.
    EB seems to be either uniformed or misinformed of our claims and is therefore advised to read first hand what we have written on this subject and not rely on second-hand information.
    I find this odd.  I responded to what you wrote and posted here, at this site.  To be honest, even after your recent post, I still do not know how to make sense of claims like:

    "the rotational degrees of freedom, is multi-mode, allowing an angular-momentum SISO link to accommodate an arbitrary number of independent transmission channels on one and the same frequency (and polarization)."

    What is your definition of SISO?
    If there is only a single input (even if from a connection to a very complicated antenna shape), you only have access to a single variable versus time.  This cannot give you an arbitrary number of independent transmission channels covering the same frequency band.

    If there are multiple antenna elements, and multiple connections, I would not call that SISO.
    Maybe I am misusing the term, and taking it too literal.  What does the "Single" in "Single Input" or "Single Output" mean to you?
    Furthermore, we have of course never claimed that one can measure EM observables in a single point.  ... EM observables ... are carried by the EM field in the form of volumetric densities that must me measured (integrated) over a certain spatial volume to become observable and therefore usable in radio science ...
    We seem to be talking past each other here.  I talked about a point because you began with discussing densities.  So fine, let's integrate the density over a finite volume.  The issue now merely shifts from a "point", to the far field limit such that the density changes negligibly over the integration volume.

    Since we agree that OAM is extrinsic, then in this limit if you measure the angular momentum, effectively all you are measuring IS the linear momentum (P) in the volume, and then doing L = r x P where r is the mean vector to your integration volume from the arbitrary origin that was chosen.

    As we agree, the OAM modes only exist as a spatial distribution of the actual fundamental photon properties.  Therefore, having these carry separate information will not work in the far field limit.

    I see in a response to criticism in the literature that you went out of the way to explain that the angular momentum density falls off as 1/r^2 just like the linear momentum density.
    http://iopscience.iop.org/1367-2630/14/11/118002/pdf/1367-2630_14_11_118...

    Well yes, of course.  But that is answering the wrong question.  To demonstrate that you can create "an arbitrary number of independent transmission channels", you need to show that you can distinguish these modes.  That is what is lost in the far field.  That is what the critique is.

    It is only when you can still distinguish the spatial distribution from source that this OAM channel idea is possible.  And if you can still distinguish the spatial distribution, you are not in the far field limit.
    Work is going on to construct such torque-based antennas but to the best of our knowledge none has been demonstrated publically yet.  How EB can call it "bizarre" is therefore beyond us.
    How are we talking past each other here?
    I reread what I wrote, and it still sounds clear to me that what I am calling bizarre is your definition of a single antenna.  I said that when responding to your claim that "OAM can increase the spectral capacity using only a single antenna at the transmitting and receiving ends."

    To have multiple OAM modes carrying different data, you will need more than one antenna element.  Do we really disagree on this?

    Again, what is your meaning of "single" in "a single antenna"?
    If you have a giant array of multiple antenna elements that you control the signals to, but these are all on a single mast, do you then call that a "single antenna"?

    I am asking questions because what you are saying is not making sense to me (and apparently others as well).  Instead of explaining, you seem to mostly be talking about something else (missing my point) or saying in many varied phrases that my understanding of your position is wrong ("uniformed or misinformed of our claims", "is heavily biased and distorted", "his allegations are contradictory to actual facts", etc.).

    The whole reason I am asking questions is BECAUSE what you are saying appears convoluted and distorted to me.  So I am already aware that I am likely arguing against something other than what you intend to present.  But be aware that, despite your intentions, this is what your presentations appear like and is possibly why there are so many critics.  I am reaching out not to dismiss your work, but to ask for clarification on what appears to be statements that cannot fit together.
    Ok, just for fun here's a concrete example...

    Take 4 linear polarisation antennae arranged in a plus sign. At a fixed frequency, there are 8 real modes (2 phases per antennae, or if you prefer, 4 complex modes).

    This can be expressed in an OAM basis:

    * Two circularly polarised modes, OAM zero.
    * For each of two circular polarisations, modes of OAM +1 and -1, built out of that circular polarisation.
    * Two modes of net OAM zero, each built by combining a +2 and -2 OAM mode.

    Nothing to stop a 4 way MIMO implementation choosing that OAM basis, and if it doesn't, moving the antennae a bit will probably create it. Very likely a WIFI unit near you is using OAM right now - unless they are carefully positioned, it's unlikely that 3 or more antennae will completely cancel the OAM modes!

    I presume in general 2n+1 dipole antennae in a circular arrangement gives you a basis consisting of modes of integer OAM between -n and n, at each OAM two modes built from opposite circular polarisations, while 2n+2 also gives you two extra net 0 OAM modes built from OAM +/- (n+1).

    And as always, reiterate that if you want, linear algebra can be done with analog electronics, or waveguides, or funky antennae design (or just by measuring along a different axis, for that matter).

    PS. I just realised the answer to something that was bothering me: OAM and directional distribution of intensity are not independent.

    Measuring OAM relative to a given axis, different modes propagate in different directions. E.g., for point-to-point transmission, measuring OAM along the axis between transmitter and receiver, in an OAM basis, only the modes with OAM zero (measuring OAM around the axis) propagate in the direction from transmitter to receiver.

    The modes of OAM k have intensity scaling as the +/- k power as you tilt away from the axis. (In free space the negative powers are non-physical, if you delete the axis - e.g., an air-cored optical fibre, presumably you use them). [This is all just the fact that the representations of the 2d rotation group, expressed using complex numbers, are just integer powers...]

    If you're wanting to use OAM to carry multiple modes from one transmitter to one receiver, then: OAM +/- 1 along the transmission direction will require antennae diameter similar to normal MIMO dimensions. Higher modes are going to require much larger antennae complexes.... and not wasting power in the 'wrong' directions will be a challenging engineering task. [Ironically my previous fears about extreme directionality were misplaced - that applies to the non-physical negative powers....]

    Thinking about the the negative power modes told me what to google for.
    They're the near field around a helical conductor.
    Turns out OAM has been in use for decades, just without the hype.
    Usually known as "normal mode helical antennae".

    And a nice and easy mathematical description.... At some distance from the source, take a (finite segment of) a plane transverse to the direction to the source. Use complex coordinates z. Use complex numbers for the phase and intensity of the radiation from the source, as a function of position.

    Some (see below for the rest) of that function can then be expressed as a power series in z plus a power series in its complex conjugate zbar. OAM mode k is just z^k, mode -k is zbar^k.

    This makes a few of things obvious:

    (1) the OAM modes do not propagate along their axis, and are heavily suppressed near it. The only way to get any significant power near the axis will be to use a large highly directional antennae with high internal field loading. If the antennae leaks anything in the wrong direction, then that k-th power scaling will shove all the signal out that direction. [A proper analysis of the leakage is going to require 3d spherical harmonics, not this 2d approximation.]

    (2) Far from the apparent claim that the modes keep nicely separate, any misalignment mixes them awfully. Shifting the axis from zero to point 'a' is just the transform z -> (z+a). The k-th power of z+a then describes the mixing of mode k from the misalignment.

    (3) And don't think going off-axis will save you from the fact that OAM propagates badly. If a > z then the mixing (z+a)**k is dominated by the constant - i.e., off-axis you can't distinguish the OAM modes from the zero OAM modes (you just get phase shifting insignificantly with position).

    Far from giving you an infinite number of modes transmitted from A to B in free space, OAM is completely useless. Broadcast antennae might well twist phase or polarisation with direction, formally that is OAM, but no-one will care.

    Signals in a fibre are a bit different. Looking at the publications on OAM-in fibre they appear to be using air-core fibres (so the glass fibre is a tube). Then the analytic function can include negative powers of z or zbar (because z=0 is missing). Air-core fibres are nice in the lab (it reduces non-linearities that might spoil delicate experiments) but useless for communication (they are quite lossy, and presumably expensive).

    [Oh, the part of the phase/intensity function that is not analytic-plus-a-conjugate-analytic? I don't think they propagate at all, they are completely near field.]

    Thanks for sharing! I wasn't sure how to go about calculating the mode mixture.

    I just discovered this site recently, and it seems to be a great combination of prepared presentation (articles), and still having discussion. It is hard to capture that, and forums or online journals don't get that good combination of the two. Whoever runs this site, well done!

    dorigo
    One link of interest to English-speaking readers of this thread:

    http://www.digitalairwireless.com/wireless-blog/recent/the-future-of-rad...

    Cheers,
    T.
    logicman
    Thanks for the link Tommaso - a picture is worth ...

    the potential to pack an infinite number of steams into a single signal

    Steam radio?
    Antennae sizings: for normal point-to-point MIMO you pack N modes into an antennae area of O(N). - either a linear array, or else an AxB grid. For OAM you need a circumference O(N) for N modes, area O(N^2). (You have 2 linear transverse degrees of freedom but only one rotational degree of freedom).

    Getting low OAM modes out of radio gear is no-big-deal, normal MIMO WIFI units will do it without noticing, if you arrange the antennae correctly. High OAM modes are perfectly possible, you just need impractically big antennae units.

    Incidently, a pure state of zero OAM is spherically symmetric, which is impossible with radio waves. Every radio transmitter ever made produces non-zero OAM states (although the total OAM might be zero). But that leads to:

    Question: does the theorem that a pure state of zero OAM is impossible extend to other OAM pure states?

    Re 2.5Tb/s in fibre quoted in that article - it was done over very short distance (about 1m IIRC). A common figure-of-merit for optical transmission research is bandwidth*distance. Bog standard commercial optical networking gear beats that on bandwidth*distance by a large factor (10km * 10Gb/s is 100Tb-meters/s).

    I smell a rat in the images in that article:

    Firstly, the dish reflector, they claim the parabolic twist creates the OAM states - but what it will do at a fixed frequency is change the OAM of all signals identically (think about the tiny rotational back-reaction on the antennae). It does not split OAM zero into states of different OAM... Either they had a second untwisted signal path (either a second dish, or maybe just the direct path that doesn't bounce off the reflector), or the two channels were different frequencies or different polarisations.

    Secondly, the diagrams of the waves appear to just show different phases of circular polarisation. They definitely do not show different OAM states. In an angular momentum pure state, angular orientation is meaningless (uncertainty principle). But the two waves shown only vary by angular orientation...

    There appears to be a single special case of free-space OAM that works beautifully, and lives up to the hype. But it does not generalise.

    Take a single mode of positive OAM +k and a single mode of negative OAM -h. (+1 and -1 are the obvious choices).

    Then antennae misalignment on +k mixes with 0,1,...+k and antennae misalignment on -h mixes with 0,-1,...-h.

    But crucially there is no mixing between +k and -h from antennae misalignment and you have a lovely system where you can use 2 OAM modes times 2 polarisations with a straightforward two-dish two-polarisation antennae complex and easily keep them separate without any complex gymnastics in digital processing or whatever (& if you're clever enough the two dishes might be concentric rings).

    The only caveat that I can see is that reflections off the ground will mess things up.

    To reiterate, if a point-to-point link has more than one OAM mode of the same sign, then misalignment mixes them and you need to do funky processing to separate them again.

    And this is still MIMO in the sense of relying on the angular extent of your antennae to separate modes - this is limited range and won't help you e.g., between a satellite and ground.