214 6.3  Optical Force Tools

An AOD is composed of an optical crystal, typically of tellurium dioxide (TeO2) in a syn­

thetic tetragonal structure (also known as the crystal paratellurite). In this form, TeO2 is a

nonlinear optical crystal that is transparent through the visible and into the mid-​infrared

range of the electromagnetic spectrum, with a high refractive index of ~2.2, exhibiting a

relatively slow shear-​wave propagation along the [110] crystal plane. These crystals exhibit

photoelasticity, in that mechanical strain in the crystal results in a local change in optical

permittivity, manifest as there being a spatial dependence on refractive index. These factors

facilitate standing wave formation in the crystal parallel to the [110] plane from acoustic

vibration if a radio-​frequency forcing function is applied from a piezoelectric transducer

from one end of the crystal, with the other end of the crystal at the far end of the [110] plane

acting as a fixed point in being coupled to an acoustic absorber (Figure 6.2a). The variation in

refractive index can be modeled as

(6.4)

n z t

n

n

t

kz

,

(

) =

+

−

(

)

0

∆cos ω

where

n0 is the unstrained refractive index

ω is the angular frequency of the forcing function

k is the wave vector of the sound wave parallel to the z-​axis (taken as parallel to the

[110] plane)

The factor Δn is given by the photoelastic tensor parameters. The result is a sinusoidally

varying function of n with a typical spatial periodicity of around a few hundred nanometers,

which thus has similar attributes to a diffraction grating for visible/​infrared light. The

diffracted light is a mixture of two types, which due to Raman–​Nath diffraction can occur at

an arbitrary angle of incidence at lower acoustic frequencies (most prevalent at ~10 MHz or

less), and that due to Bragg diffraction (see Chapter 4) at higher acoustic frequencies more

typically >100 MHz, which occurs at a specific angle of incident θB such that

(6.5)

sin

i

d

i

θ

λ

λ

B

f

n v

v

f

n

n

= ⁻

+

−

(

)



^



^

2

1

2

2

2

2

2

where

λ is the free-​space wavelength of the incident light

f is the acoustic wave frequency

ni and nd are the incident and diffracted wave refractive indices of the medium, respectively

v is the acoustic wave speed

AODs are normally configured to use the first-​order Bragg diffraction peak angle θd for beam

steering, which satisfies sin(θd) =​ λ/​Λ where Λ is the acoustic wavelength. The maximum

efficiency of an AOD is ~80% in terms of light intensity propagated into the first-​order Bragg

diffraction peak (the remainder composed of Raman–​Nath diffraction and higher-​order

Bragg peaks), and for steering in the sample focal plane in both x and y requires two orthog­

onal AODs; thus, ~40% of incident light is not utilized, which can be disadvantageous if a

very high stiffness trap is desired.

An AOD has a frequency response of >10⁷ Hz, and so the angle of deflection can be rapidly

alternated between ~5° and 10° on the submicrosecond time scale, resulting in two time-​

shared beams separated by a small angle, which can then each be manipulated to generate a

separate optical trap. Often, two orthogonally crossed AODs are employed to allow not only

time-​sharing but also independent full 2D control of each trap in the lateral focal plane of the

microscope, over a time scale that is three orders of magnitude faster than the relaxation time

due to viscous drag on a micron-​sized bead. This enables feedback type experiments to be

applied. For example, if there are fluctuations to the molecular force of a tethered single mol­

ecule, then the position of the optical trap(s) can be rapidly adjusted to maintain a constant