I was sent a link today to an interesting article about some research done at the University of Central Florida. Researchers have concocted a class of optical beams which appear to follow a curved trajectory in free space propagation. A theoretical picture of the behavior of such a light field on propagation is shown below:
The horizontal axis represents the transverse profile of the beam, while the vertical axis represents the propagation direction. Lighter colors, of course, indicate a brighter field. As one can see, the brightest beam in the wavefield, as well as all the secondary ones to its left, are curving to the right as they propagate!
This result, which appeared in a recent issue of Physical Review Letters, is a very clever, seeming violation of common-sense rules involving light propagation. In everyday experience, light travels in straight lines unless it encounters some sort of material obstacle; this beam seems to curve without any obstacle involved. For the more theoretically inclined, this beam seemingly violates Ehrenfest’s theorem, which suggests that the ‘center of mass’ of the beam cannot accelerate in free space. (By ‘center of mass’ I’m referring to the ‘center of energy’ of the optical field.)
Ehrenfest’s theorem, however, provides a hint of how to explain this effect. Is the overall energy of the light field accelerating to the right? In the plane of origin, the field has the functional form of the Airy function:
As suggested by the figure above, the Airy function is highly asymmetric: It has a maximum field amplitude at the origin, vanishes rapidly to the right of the origin, but oscillates and decays as s to the 1/4th power to the left of the origin. This slow decay of the function to the left essentially means that the function is of infinite extent in that direction.
How does this explain the curving of the light field? Just like any light field, this field spreads out as it propagates. The largest peak in the Airy function does indeed curve to the right, but this is balanced by the infinite ‘left leg’ of the Airy function spreading to the left. This ‘left leg’ is comparatively dimmer than the largest peak of the function, so visually we tend to ignore all the ‘smeared-out’ energy diffracting to the left and only notice the relatively bright peak curving to the right.
This clever construction creates an illusion of ‘acceleration’ by subtly shifting the infinite amount of energy on the left side of the Airy pattern. In practice, we cannot create a perfect Airy pattern, which would require an infinite amount of energy in the ‘left leg’ of the pattern. But a finite approximation of an Airy function (with a Gaussian envelope, for instance) can be made to ‘curve’ for a significant distance.
There are a number of unusual behaviors one can generate by using hypothetical ‘infinite energy’ beams. Another well-known example are so-called ‘non-diffracting’ beams, which maintain their original shape and size without spreading on propagation. A perfect non-diffracting beam requires infinite energy and again cannot be made, but approximations to Bessel beams will propagate over appreciable distances without significant diffraction.
Update: Another, more intuitive way to understand the curving light beam: If we make a scale with a sufficiently long left side, we could in principle balance a little bunny rabbit against a tank:
The bunny is seemingly insignificant compared to the mass of the tank, but because it is so far along on the balance bar compared to the tank, it can have a big influence with very little mass. Similarly, the energy in the ‘left leg’ of the optical beam seems at first blush to be insignificant compared to the bright peak of the beam. However, because that energy is stretched out far, far to the left of the origin, its contribution to the ‘center of mass’ is large enough to ‘balance’ against the bright peak.
What if the bunny hops?
“What if the bunny hops?”
Duh – bunnies on scales NEVER hop — I think Einstein proved that. 🙂
Will the bunny die like Schrodinger’s cat if the tank is removed? I don’t like harming animals just for the sake of science!
Rest assured, no bunnies were harmed in the writing of this post. 😛
Phew! I can call off PETA now.
I like the analogy, I too have been known to choose tanks in my analogies. There must be some unwritten pedagogical rule that tanks are the heaviest object we can imagine having direct experience with 🙂
adawes: I’m not sure why I chose a tank; you’re probably right that it falls in that category of ‘heavy-but-not-so-heavy-as- to-sound-ridiculous-or-inconceivable.’
As in: comparing bunnies to tanks is fine, but comparing bunnies to supertankers is just silly! 🙂