How does this not violate the conservation of momentum (and/or energy)? It seems like the outgoing photon must have some increased momentum (proportional to h/lambda), which also means it will have increased energy (~ h * c/lambda), which it must have gotten from the tractored particle. The induced multipoles ought to be a higher energy than the ground state, though, so where did the extra energy come from?
You could conserve momentum by having the photons continue after interacting, but at a higher energy level (i.e. heavier). But I can't figure out how to do that and also conserve energy.
One possibility is that you end with the object at a standstill, so while you need to move it, you don't actually need any extra momentum or energy to do so. If there was some way to borrow it maybe? Sideways motion perhaps?
Now that I have some time to read the paper, on the second page there's this sentence: "The Fourier decomposition of a PIB consists of plane wave components whose k-vectors form a cone that make an angle Theta_0 with the z-axis." The way it must get momentum is having the outgoing light be more collimated than the incoming light.
My new question is, How is this different from an Optical Trap? I'm imagining the point of the cone at the particle, but that may not actually be the case, and my skimming of the article is not enough to make sense of figure 3.
Not only is the point of the cone not at the particle; it's not even in the same space as the particle! The cone is in the Fourier transform of the beam.
A standard optical trap, applied to very small objects, works like this: the electric field causes polarization in the thing being trapped, turning it into a little dipole; it then exerts a force on the induced charges, basically pulling on each end of the dipole; that force is bigger in regions of greater field intensity; the net effect is a small force pulling the object towards where the field is bigger. (In an electromagnetic wave there is also a magnetic field, which also acts on the objects, but the effects of those average out to zero over each cycle of the wave.)
Now, really the effect of an electric field on a particle is more complicated than just turning it into a dipole. There will be higher-order stuff as well. The authors of this paper claim that if you look at the quadrupole term and consider interactions between the different terms (warning: I am not at all sure I understand this bit, and they've left the details to a supplementary section that appears not to be included in the paper on the arXiv), then with a suitably shaped beam and the right sort of particle you can get a backward pull against the direction of propagation of the wave. (So, yes, the light will have to emerge travelling more-directly-forward than it went in, so to speak. Which is ... counterintuitive.)
So it's different from an optical trap in that it relies on higher-order effects, and a more sophisticated beam shape, and it can only work on very small objects (the higher-order stuff gets averaged away for bigger ones; so I don't think you could use this technique for things like biological specimens). But it uses much the same underlying physics.
[EDITED to add: their trick also doesn't work for the very smallest objects, substantially below one wavelength. It works only in the "Mie regime", meaning with particles whose size is comparable to the wavelength.]
When trying to find out how an object is going to interact with the electromagnetic field, it's generally useful to decompose the charge distributed on the object into "multipole modes". It turns out that any arbitrary distribution of charges can be expressed as a sum of such multipoles, and the interaction of each multipole with the E&M field can be solved even though the object itself may be too complicated. Usually, only the first few multipoles are important, which allows efficient computation of the objects behavior by summing up the contribution of the first few multipoles.
The first multipole ("monopole") is just the net charge of the object. The second multipole ("dipole") is the degree to which the object has positive charge distributed toward one side and negative charge toward another.
A single isolated point charge is a perfect monopole, with zero higher-order multipole components. An object composed of two opposite charges of equal magnitude, separated by a distance, is a perfect dipole, with zero monopole and higher-order multipole components. And so on.
Apparently, the point is that objects which are electromagnetically simple (such as a single point charge) cannot be pulled by a tractor beam. Instead, the object must be complicated such that it has several large multipole components which interact in the right ways to produce a net pull.
Typo in article: There is, of course, still a potential gradient. As the abstract notes, the important thing is that there is no equilibrium point, i.e. no place where the gradient is zero. The object can be pulled all the way back to the source.
Standard "optical tweezers" pull things towards regions of higher intensity (it's the intensity gradient, not the potential gradient, that's relevant here), so you can't pull anything past where the intensity is maximal. But the technique described in the paper works even with no intensity gradient along the direction of beam propagation. (I think that if there were one, the gradient force would swamp their new force, but I'm not sure.)
Whoops, looks like I was confused about the terminology. Thanks.
In case anyone else reads this: according to the wikipedia article, the term "intensity" in "intensity gradient"--in this context--is just the strength of electric (and magnetic) field associated with the laser. The electric field, in turn, is the gradient of the electric potential [1]. Point charges will follow the potential gradient, but neutral dieletric objects (which I think is what optical tweezers are usually used on, and which usually can be modeled as dipoles) will follow the intensity gradient.
Simple dipoles can be pulled with optical tweezers, but limits on laser power mean that the distance between intensity extrema (and so the maximum distance things can be pulled) is relatively short. The hypothetical tractor beam only works on objects more electrically complicated than dipoles, but doesn't rely on following an intensity gradient and can pull all the way back to the source.
[1] This may not be quite applicable here because of the ambiguities in defining a potential for non-static electric fields like those of lasers.
