Chapter 9: Optical Tweezers

Question 1

What are optical tweezers?

Answer 1

Instruments that use the momentum transfer of photons to trap small particles. The trapped particles can then be manipulated using several different techniques and the forces acting on them can be measured.

Question 2

How small are the forces that optical tweezers can measure?

Answer 2

1-100 pN

Question 3

Optical tweezers are capable of manipulating single molecules with ________ precision.

Answer 3

Nanometer

Question 4

What are the applications of optical tweezers?

Answer 4

• Examining properties of DNA
• Observing activity and effects of binding proteins acting on DNA
• Viewing molecular motors
  (Determining energy required for these processes and interactions)

Question 5

Describe the basic premise of optical traps

Answer 5

Optical tweezers are used to trap dielectric microspheres (beads) made of polystyrene by using strongly focused laser light of a specific wavelength which transfers momentum to the microspheres. This produces a force, known as radiation pressure, which pushes the microsphere in the direction of the incident beam.

Question 6

Give the equation for the magnitude of momentum for a single photon

Answer 6

|p| = magnitude of momentum
h = Planck’s constant
λ = wavelength

Question 7

The force that arises from momentum transfer of a single laser ray is proportional to the _____ of the ray divided by the ________ of the medium.

Answer 7

Power
Velocity (speed of light/refractive index)

Question 8

How is the total momentum transferred to a microsphere estimated?

Answer 8

By modelling the bead as a plane mirror on which the light is incident. A calculation can then be done for the momentum transferred by an incident photon.

k = 2π/λ
ħ = h/2π
ω = angular frequency
E = photon energy

Question 9

Give the equation from the force exerted by a laser beam on a microsphere at a rate of N photons per second

Answer 9

(Following Newton II)

F = force
P_beam = beam power
v = velocity

Question 10

What is the typical power of optical tweezers?

Answer 10

A few milliwatts

Question 11

What are the two types of optical trapping and what are their differences?

Answer 11

• Rayleigh regime where the sphere has a diameter much smaller than the wavelength of the laser.
• Geometrical optics regime where the laser has a wavelength much shorter than the sphere diameter.

Question 12

How can biological molecules be manipulated by optical tweezers?

Answer 12

They are attached by linkers to the polystyrene beads.

Question 13

Briefly outline the Rayleigh regime

Answer 13

As the microsphere diameter is a lot smaller than the wavelength of the laser, it is assumed that it is exposed to a relatively constant oscillating electric field from the light. Hence, the dielectric material becomes polarised so it acts as a single point electric dipole. This interaction produces a gradient force which is parallel to the gradient of the intensity so the bead will always be pushed to the position of highest intensity, however, the gradient force is counteracted by a scattering force.

Question 14

For the Rayleigh regime, give the equation that relates the oscillating electric field to the electric point dipole

Answer 14

Question 15

Give the equation for the force acting on the electric point dipole (from the dielectric material following the Rayleigh regime)

Answer 15

F_g = gradient force
U = potential energy
p(r, t) = electric point dipole
E(r, t) = oscillating electric field

Question 16

Derive the equation for the average gradient force in the Rayleigh regime

Answer 16

Question 17

Give the equation for the average gradient force in the Rayleigh regime

Answer 17

F_g = gradient force
n = refractive index
ε_0 = permittivity of a vacuum
α = polarisability
I(r) = beam intensity

Question 18

What is the scattering force in the Rayleigh regime?

Answer 18

A force that arises due to the bead oscillating synchronously after the harmonically oscillating electric field that radiates in all directions. This changes the magnitude and direction of the energy flux, changing the momentum of the photons.

Question 19

Give the equation for the scattering force in the Rayleigh regime

Answer 19

Question 20

What is the Poynting vector?

Answer 20

The vector cross product between the electric field, E(r, t), and the magnetic field, H(r, t), in the electromagnetic wave. It is given by S(r, t) and represents the instantaneous energy flux crossing a unit area per unit time in the beam propagation direction.

Question 21

The averaged Poynting vector over time gives the ______ of the beam.

Answer 21

Intensity

Question 22

What is the equation that relates the Poynting vector and intensity?

Answer 22

Question 23

Give the equation for scattering force in terms of the intensity

Answer 23

Question 24

For efficient trapping, the gradient force must be ______ than the scattering force which can be done by having a very tight focus of the laser beam because this ________ the intensity gradient.

Answer 24

Bigger
Steepens

Question 25

Briefly outline the geometric optics regime

Answer 25

The diameter of the polystyrene microsphere is much greater than the wavelength of the laser beam so the refraction of individual beams is considered. If the microsphere is displaced from the focal point of the laser, refraction occurs so the laser wave vector and momentum also change due to conservation of momentum. This provides a restoring force that keeps the bead in the focal point.

Question 26

The restoring force for the geometrical optics regime is proportional to the ______ ______ ___ ______ ____ ___ ___ _____ _____. Hence, it can be described similarly to the ______ _____ equation.

Answer 26

Distance between the sphere centre and the focal point
Spring force

Question 27

Give the equation for the restoring force in the geometrical optics regime

Answer 27

F = restoring force
k = trap stiffness constant
x = displacement

Question 28

What is the accuracy of the geometrical optic regime for a micron-sized sphere?

Answer 28

10nm

Question 29

The lower the trap stiffness, the _____ the resolution of the force.

Answer 29

Lower

Question 30

How can optical tweezers be calibrated?

Answer 30

The viscous drag force, F = 6πηrv, must first be calibrated. The velocity is found by surrounding the bead in an oscillating fluid of fixed amplitude and frequency then measuring velocity as v = ωx_0 cos(ωt). This can be done for several different signals and a graph of force against signal can be plotted to find the calibration factor (gradient).

Question 31

Derive the calibration factor of optical tweezers

Answer 31