Chapter 6 – Forces  213

at which the gradient of the intensity of the focused laser light in the lateral xy focal plane of

the microscope is zero.

If the particle is displaced laterally from the focus, then the refraction of the higher-​

intensity light fraction through the particle close to the focus causes an equal and opposite

force on the particle, which is greater than that experienced in the opposite direction due to

refraction of the lower-​intensity portion of the laser beam. The particle therefore experiences

a net restoring force back to the laser focus and hence is “trapped,” provided any external

force perturbations on the particle do not displace it beyond the physical extent of the optical

tweezers.

In practice, stable optical tweezers require a diffraction-​limited focus; photons entering

the focal waist of the confocal volume at a steep angle relative to the optical axis result in

high-​intensity gradients across the trap profile and so contribute the most to the optical

restoring force. To achieve this steepness of angle requires a high NA objectives lens in the

range ~1.2–​1.5 often combined with marginally overfilling the back aperture of the objective

lens with collimated incident laser light. The actual size of the optical tweezer trapping

volume is determined by the spatial extent of the diffraction-​limited interference pattern in

the vicinity of the laser focus, which laterally (xy) has a width of ~λ, whereas axially (z) this is

more like two to three times times λ (see Chapter 4). This implies that the intensity gradient

is reduced by the same factor. Combining this reduction in axial gradient stiffness with a

weakness of the axial trapping force due to forward scatter radiation pressure results in axial

trap stiffness values (i.e., a measure of the restoring force for a given small displacement of

the particle) that are smaller than the lateral stiffness by a factor of ~3–​8, depending on the

particle size and specific wavelength used.

6.3.2  OPTICAL TWEEZER DESIGNS IN PRACTICE

Typical bead diameters are ~0.2–​2 μm, though optical trapping has been demonstrated on

gold-​coated particles with a diameter as small as 18 nm (Hansen et al., 2005). The wave­

length used is normally near infrared (NIR) of ~1 μm, the choice being made on the basis

of optimization of trap stiffness and size while minimizing sample photodamage. Some

damage is due to a localized heating effect from laser absorption either by the water solvent

or chromophores in the biological sample, at a level of ~1–​2 K for every 100 mW of NIR

laser power. However, the most likely cause of biological damage is due to the generation of

free radicals in water through single-​ and multiphoton absorption effects found at high local

intensities at the focus of a trap, which can bind indiscriminately to biological structures.

The choice of wavelength used is a compromise between two competing absorption

factors. One is that absorption of electromagnetic radiation by water itself increases sharply

from visible into the infrared, peaking at a wavelength of ~3 μm. However, natural biological

chromophores can absorb strongly at visible light wavelengths, as well as increasing the like­

lihood for generating free radicals; therefore, a wavelength of ~1 μm is a good compromise.

At wavelengths between 1 and 1.2 μm, there is also a small local dip in the water absorption

spectrum, which makes Nd:YAG (λ =​ 1.064 μm) and Nd:YLF (λ =​ 1.047 μm) crystal lasers

attractive choices (Figure 6.1b).

In most applications, optical tweezers are coupled to a light microscope. An NIR laser

beam is expanded usually to marginally overfill the back aperture of a high NA objective

lens, which is steered by upstream optics to rotate the beam through the back aperture,

resulting in lateral displacement of the optical trap at the focal plane in a microscope flow cell

(Figure 6.1c). Steering of the optical trap can be done using mirrors positioned in a conjugate

plane to the objective lens back aperture. However, it is common in many applications to use

higher bandwidth steering with acousto-​optic deflectors (AODs), discussed in the following

text. The laser beam for generating a conventional gradient force optical trap can be split

before reaching the sample, either using a space-​dividing optical component such as a glass

splitter cube or by time-​sharing the beam along different optical paths in the microscope

setup to generate more than one optical tweezers (Figure 6.1d). Time-​sharing is most popu­

larly obtained by passing the initial beam through AODs.