Chapter 6 – Forces  211

from an object. Also, if refraction occurs at the point of a photon emerging from an optically

transparent particle, there is a change in beam direction and intensity, and thus a change in

momentum, which results in an equal and opposite force on the particle. Standard optical

tweezers utilize this effect as a gradient force in the focal plan of a light microscope.

KEY POINT 6.3

Basics of optical tweezers:

1 If a refractile particle changes the direction of a photon, then a force acts on it

according to Newton’s third law, since photons have momentum.

2 The intensity to generate optical forces large enough to overcome thermal

forces at room temperature is high and so requires a laser.

3 If a laser beam is brought to a steep focus, the combination of scatter and

refractive force results in a net force roughly toward the laser focus.

To understand the principles of optical trapping, we can consider the material in the par­

ticle through which photons propagate to be composed of multiple electric dipoles on the

same length scale as individual atoms. A propagating electromagnetic light wave through the

particle imparts a small force on each electric dipole, which time averages to point in the dir­

ection of the intensity gradient of the photon beam. A full derivation of the forces involved

requires a solution of Maxwell’s electromagnetic equations (see Rohrbach, 2005); however,

we can gain qualitative insight by considering a ray-​optic depiction of the passage of light

through a particle (Figure 6.1a).

The photon energy flux dE in a small time dt of a laser beam parallel to the optic (z) axis

of total power P propagating through the particle is given by

(6.2)

d

d

E

P t

cdp

=

=

where

c is the speed of light

p is the total momentum of the beam of photons, such that dp is the small associated

change in momentum in time dt

If we assume that the lateral optical trapping force F arises mainly from photons traveling

close to the optic axis, which is exerted as photons exit the particle at a slightly deviated

direction from the incident beam by a small angle θ, then F is given by the rate of change of

photon momentum projected onto the x-​axis:

(6.3)

F

p

t

p

t

P

c

=

≈

=

sin ^d

d

d

d

θ

θ

θ

where we assume the small-​angle approximation if θ is measured in radians. Typically, for

optical tweezers, θ will be a few tens of milliradians, equivalent to a few degrees (see Worked

Case Example 6.1).

Typically, a single-​mode laser beam (the so-​called TEM00 that is the lowest-​order fun­

damental transverse mode of a laser resonator head and has a Gaussian intensity profile

across the beam) is focused using a high NA objective lens onto a refractile, dielectric par­

ticle (typically a bead of diameter ~10⁻⁶ m composed of latex or glass) whose refractive index

(~1.4–​1.6) is higher than that of the surrounding water solution (~1.33) to form a confocal

intensity volume (see Chapter 4). Optical trapping does not require symmetrical particles

though most often the particles used are spheres. A stably trapped particle is located slightly

displaced axially by the forward scatter momentum from the laser focus, which is the point