9.6 Theory of the dipole force 197shown in Fig. 9.11. As the laser beam is moved the particles remain
trapped in the region of high intensity. The microscope is used to view
the object through a filter that blocks the laser light. The objects are
immersed in water and mounted on a microscope slide in a standard way.
The liquid provides viscous damping of the motion.^35 Optical tweezers^35 Laser radiation can levitate small ob-
jects in air, but this is less straightfor-
ward than the manipulation of objects
floating in water.
works, not just for simple spheres, but also for biological cells such as
bacteria, and these living objects can withstand the focused intensity
required to trap them without harm (the surrounding water prevents
the cells from heating up). For example, experiments have been carried
out where a bacterium is tethered to the surface of a glass microscope
slide by its flagellum (or ‘tail’) and the body is moved around by optical
tweezers. This gives a quantitative measure of the force produced by the
microscopic biological motor that moves the flagellum to propel these
organisms (see Ashkin (1997) and Lang and Bloch (2003) for reviews).
This section has introduced the concept of a radiation force called the
gradient force, or dipole force, and the next section shows that a similar
force occurs for atoms.^36
(^36) This analogy is not just of pedagog-
ical interest—the first experiments on
optical tweezers and the dipole-force
trapping of atoms were carried out in
the same place (Bell Laboratories in the
USA).
9.6 Theory of the dipole force
Actually, this section does not just derive the dipole force on an atom
from first principles but also the scattering force, and so demonstrates
the relationship between these two types of radiation force. An electric
fieldEinduces a dipole moment of−er= 0 χaEon an atom with a
(scalar) polarizability 0 χa. The interaction energy of this dipole with
the electric field is given by
U=−
1
2
0 χaE^2 =1
2
er·E, (9.34)whereEis the amplitude of the electric field andUis used here to denote
energy to avoid confusion with the electric field. This expression comes
from the integration of dU=− 0 χaEdEfromE=0toE=E(the factor
of 1/2 does not occur for a permanent electric dipole). Differentiation
gives thez-component of the force as
Fz=−∂U
∂z= 0 χaE∂E
∂z, (9.35)
and similarly forFxandFy. Radiation of angular frequencyω,prop-
agating along thez-direction, can be modelled as an electric fieldE=
E 0 cos (ωt−kz)̂ex.^37 The gradient of the energy gives thez-component
(^37) This particular field is linearly polar-
ized parallel tôex, as in Section 7.3.2,
but the results derived here are quite
general.
of the force as^38
(^38) This classical treatment gives the
same result as the quantum mechan-
ical derivation when the electric field
varies slowly over the typical dimen-
sions of an atomic wavepacket (λdB
λlight). Under these circumstances,
classical equations of motion corre-
spond to equations for the expectation
values of the quantum operators, e.g.
therateofchangeofmomentumequals
the force corresponding to
d〈p〉
dt
=−〈∇U〉.
This is an example of Ehrenfest’s theo-
rem in quantum mechanics. The quan-
tum mechanical derivation of the dipole
force is given in Cohen-Tannoudjiet al.
(1992).
Fz=−ex
{
∂E 0
∂zcos (ωt−kz)+kE 0 sin (ωt−kz)}
. (9.36)
The two parts of this force can be understood using either the classical
or the quantum mechanical expressions for the dipole that were derived
in Chapter 7. The classical model of the atom as an electron undergoing