During summers, while working at
Ninewells Hospital, which is the largest
research hospital in the United Kingdom,
we made a sonic screwdriver. ...It beams
out enough ultrasound to pick up a
4-inch diameter, half-inch thick rubber
disk and spin it around (and we can
switch the direction of rotation and the
rate of spinning). Other members of the
team have begun work to raise the
operating frequency by two orders of
magnitude, which shrinks everything in
size, miniaturizing the device. —
And by using resonators the power
requirements also shrink.
Why? Well, on the one hand, the
combination of Magnetic Resonance
Imaging with focused ultrasound clearly
had further potential for medical
treatment, but needed further
development, which we contributed to. On
a more academic level, our results
validated, for the first time directly,
something quite general: the
theoretically predicted ratio of the
orbital angular momentum to linear
momentum in a propagating beam.
There were also some other interesting
predictions about these sorts of beams
(e.g.,
negative radiation pressure in
higher-order Bessel Beams) that we were
able to confirm and clarify. Also, our
beams serve as rapid, sensitive tests of
system aberrations (wibbly wobbly
beamy-weamy stuff that we like to have
control over).
Moreover, this work provides a model
system that fits nicely into teaching,
where I have to work hard to convince
students that there really isn't
anything "orbiting" in the
stationary states that we find in
solving the Schrödinger equation for the
hydrogen atom, even though those states
can have orbital angular momentum.
What does orbital angular momentum mean,
when nothing is orbiting?
At a soccer stadium, a beach ball might
be sent around the stadium by fans
“doing the wave” but if you’re a fan,
you generally would bat the ball
laterally, in
order to transfer momentum in that
direction: that’s
cheating! If,
instead, you start with a fan with a
stationary ball overhead and the fan
simply moves up and down, then you have
to ask where does the transverse
momentum come from.
By making
analogous
states with acoustic waves, these ideas
become clearer. For example, beneath the
blue hockey puck that we acoustically
levitate in the video, is an array of
one thousand piezoelectric actuators,
each simply moving up and down.
Surprisingly, this transfers angular
momentum if we tailor the phase shift
between adjacent actuators.
In the left-hand figure above, color
represents the relative phase lags we
impose on our array of actuators, as we
move around a circle. Note that since
phase is a periodic variable (ranging
from zero to 360°), the horizontal line
is not a
discontinuity. It's the helicity of the
resulting wavefront (shown at right)
that allows these waves to transfer
angular momentum. Our work looks
specifically at the ratio of the orbital
angular momentum to the linear momentum
carried by a propagating beam, as we
move increase the “pitch” of these
helical waves (which, again, is a result
of the phase profile we impose upon our
array of actuators). Importantly, the
same “phase factor” describes the
stationary states of hydrogen that carry
“orbital” angular momentum. It also
describes laser modes that carry
“orbital” angular momentum.
By the way, one of the (several)
advantages of doing this with acoustic
waves rather than light waves is that
acoustic waves aren't polarized, and
this eliminates one source of confusion
that may arise in the optical case.
— In thinking about optics, most
people associate the angular momentum of
light with circular polarization, which
is really something else entirely:
that sort of
angular momentum comes from a rotating
polarization (and is called
"spin" angular momentum or
"intrinsic" angular momentum).
What we are interested in, instead, is
orbital angular momentum, which is
(quite generally) associated with
helicity of the wavefront.
See:
"The sonic screwdriver," G. C. Spalding, C. Démoré, A.
Volovick, Z. Yang, Y. Hertzberg, M.
MacDonald, A. Cochran,
Proceedings SPIE 8097,
8097-58.
"Mechanical evidence of the orbital
angular momentum to energy ratio of
vortex beams," C. Démoré, Z. Yang, A.
Volovick, S. Cochran, M. MacDonald,
G. C. Spalding,
Physical Review Letters 108,
194301.
Also:
Gabe Spalding & Kishan Dholakia
have many times taught a related short course
for professionals, to enable you
to:
-
•assess a variety of approaches to beam shaping and wavefront correction
-
•explain simple protocols for optimizing some beam types of broad interest
-
•describe various aspects of data analysis for some wavefront correction algorithms
-
•identify key options for enhanced degrees of beam control, resolution, and sensitivity
