SEMINAR 1b - 1. LETNIK, II. STOPNJA

Optical tweezers

Author: Matevž Majcen Hrovat

Mentor: prof. dr. Igor Poberaj

Ljubljana, May 2013

Abstract

The seminar introduces the physics of optical trapping and a research device that uses this property.

Optical tweezers are able to optically trap micrometre and sub-micrometre dielectric particles. They are

used for manipulation of microscopic particles and for measurement of forces between them. Forces in the

range between 100 𝑓𝑁 and 100 𝑝𝑁 are typically being measured. A standard experimental setup and

optical potential calibration procedures are presented. Finally two experiments optical tweezers are

presented.

Contents 1 Introduction ................................................................................................................................................ 3

2 Optical forces ............................................................................................................................................. 3

2.1 Rayleigh approach ............................................................................................................................... 4

2.1.1 Scattering force ................................................................................................................................. 4

2.1.2 Gradient force ............................................................................................................................... 4

2.2 Ray optics approximation .................................................................................................................... 6

3 Optical tweezers ......................................................................................................................................... 7

3.1 Measuring the position of particle in the trap ...................................................................................... 9

4 Force measurement calibration................................................................................................................. 10

4.1 Viscous drag force calibration ........................................................................................................... 11

4.2 Brownian motion calibration ............................................................................................................. 11

5 Experiments .............................................................................................................................................. 12

5.1 Force induced DNA melting ............................................................................................................. 13

5.2 Molecular motors .............................................................................................................................. 13

6 Conclusion ................................................................................................................................................ 14

7 Literature .................................................................................................................................................. 14

1 Introduction

Light can exert force on matter by scattering, absorption, emission or reradiation of light. With the

invention of laser it was experimentally possible to demonstrate this force. Ashkin was the first to observe

and report the scattering and gradient force on micron sized objects in 1970 [1]. More than a decade and a

half later Ashkin and his colleges made a three-dimensional optic trap with a laser beam [2]. Today this is

known as optical tweezers.

Optical tweezers are an important scientific tool in physics, biology and biochemistry. They use

highly focused laser beam to trap and manipulate dielectric particles from tens of nanometres to 100 𝜇𝑚 in

diameter. The force on particles is in the range between 100 𝑓𝑁 − 100 𝑝𝑁, depending on the difference

between refractive indices of a particle and medium. The ability to measure such small forces has opened

many new experiments in biophysics.

2 Optical forces

The main optical force comes from strong electric field gradient (often called gradient force). Strongest

gradient is achieved with a highly focused laser beam. With tight focusing a three-dimensional

confinement can be achieved, if the refractive index of the particle is greater than surrounding medium

(usually water). The gradient force pushes the object in the directions of the gradient of electric field.

Object is trapped in the region of the highest field strength.

Optical forces are usually defined by the equation:

𝐹 = 𝑄 𝑛1 𝑃

𝑐0 (1)

where 𝑄 is a dimensionless factor, 𝑛2/𝑛1 is ratio of indices of particle and medium and 𝑃 is laser power.

When the force is defined like this, the main factors are 𝑛1𝑃/𝑐0, which is the incident momentum per

second of laser beam in a medium with index of refraction 𝑛1 and factor 𝑄. Force is only applied when

the incident momentum per second changes either its direction or its amplitude. Factor 𝑄 describes this

change. In a system with constant P and 𝑛1, which is often the case, only factor 𝑄 determinates the force

on an objects. Factor 𝑄 is dependent on the wavelength, polarization, mode structure, ratio of indices and

the geometry of the particle. Radiation pressure force from a ray of momentum per second 𝑛1𝑃/𝑐0 is

largest when 𝑄 = 2. This corresponds to ray reflecting perpendicular on a perfectly reflective mirror.[3]

In optical traps there are two optical forces competing between trapping and pushing the particle

out of the trap. The gradient force keeps the particle trapped while the scattering force pushes the particle

out of the trap in the direction of the beam.

If the radius of a particle is much smaller than the wavelength of light, the particle is treated as a

Rayleigh particle. So the electric dipole approximation or otherwise known as Rayleigh approach is used.

In cases where the radius of the particle is much greater than the wavelength, a simple ray optics approach

is sufficient.

2.1 Rayleigh approach

Rayleigh approach is used when radius of a particle is much smaller than the wavelength of light (a<<λ).

As such we can approximate the particle as a point dipole and Rayleigh scattering approximation is used.

With this we can use Rayleigh scattering. The particle behaves as a point dipole with polarization

𝑝 = 𝛼 𝐸 , 𝛼 = 4𝜋휀0𝑎3

휀2−휀1

휀2+2 휀1 = 4𝜋𝑛1

2휀0𝑎3

𝑛2−1

𝑛2+2 (2)

where 𝑛 =𝑛2

𝑛1 relative refractive index, a radius of particle and 𝛼 electronic polarizability.

2.1.1 Scattering force

One of the forces on the particle is the scattering force. An atom or a molecule absorbs and soon after that

emits a photon. Simple explanation of this is that photons that are emitted from the particle are scattered in

all directions, while all the incident photons are travelling in forward direction. By conservation of

momentum there must be a forward force, caused by change in photon momentum.

The scattering force is given by:

𝐹 𝑠𝑐𝑎𝑡 (𝑟 ) =𝑛1 𝜎𝑠

𝑐0𝐼(𝑟 ) 𝑧 , 𝜎𝑠 =

8𝜋

3 𝑘4𝑎6

𝑛2−1

𝑛2+2

2

(3)

𝐹𝑠𝑐𝑎𝑡 𝑟 =8𝜋 𝑛1

3 𝑐0𝑘4𝑎6

𝑛2−1

𝑛2+2

2

𝐼 𝑟 =𝑘4𝛼2

6𝜋𝑐𝑛13휀0

𝐼 𝑟 (4)

where 𝐼 is the intensity, 𝜎𝑠 is the scattering cross-section and 𝑧 direction of the beam [4].

Figure 1: Direction of scattering force is in the direction of the beam[9]

2.1.2 Gradient force

The second force on a particle is gradient force. Lorentz force on a point dipole is

𝐹 𝑑𝑖𝑝𝑜𝑙𝑒 = 𝑝 ∙ ∇ 𝐸 +𝑑𝑝

𝑑𝑡× 𝐵 (5)

With Eq. 2 and the equation changes to

𝐹 𝑑𝑖𝑝𝑜𝑙𝑒 = 𝛼 𝐸 ∙ ∇ 𝐸 + 𝛼𝑑𝐸

𝑑𝑡× 𝐵 (6)

With Maxwell's equations and some vector identities, we arrive at the form

𝐹 𝑑𝑖𝑝𝑜𝑙𝑒 = 𝛼 1

2∇𝐸 2 +

𝑑

𝑑𝑡 𝐸 × 𝐵 (7)

The second term is the time derivative of the Poynting vector, times a constant, and the Poynting vector

oscillates sinusoidally. Because our time for measurements is much greater than the period of oscillation

(frequency of light is ~1014 Hz, and we observe the particle in the order of milliseconds or greater), the

second term vanishes. This leaves us with

𝐹 𝑑𝑖𝑝𝑜𝑙𝑒 =𝛼

2∇𝐸 2 =

𝛼

2∇𝐼 (8)

When a dielectric particle is treated as a dielectric dipole, the force is linearly proportional to the gradient

of intensity. Because of this the particle moves towards the region of higher intensity. This is why this

force is named gradient force.

The particle is trapped when the scattering force is smaller than the gradient force in the axial

direction. The particle is displaced a little downstream of the laser focus.

Figure 2: If the gradient of electric field is too small the particle is flung out of the beam (left). When the gradient is large enough

the gradient force overpowers scattering force and the particle is trapped in the beam. [5]

2.2 Ray optics approximation

Ray optics approximation is used when the wavelength of light is much smaller than the diameter of the

trapped particle. Simple quantitative explanation of ray optics is by imagining two rays travelling through

a particle as shown on Figure 3. Each ray travels in a straight line in media with homogeneous refractive

index. The change in direction can only be achieved by reflection or refraction. This happens in a medium

with inhomogeneous refractive index or on a border between two different refractive indices. Reflection

and refraction occurs according to Fresnel equations.

Figure 3: This figure show optical forces on a dielectric sphere. The trapping rays are 𝒂 and 𝒃. Because of refraction the direction

and in consequence the momentum of light changes. This change generates forces 𝑭𝒂 and 𝑭𝒃. The sum of this two forces is

restoring in axial (A, B) and transverse (C) direction so that focus (𝒇) of two ray and the centre of the sphere coincide. [6]

Each photon of light carries a momentum which is determined by the its wavelength

𝑝 =𝐸

𝑐0=

𝑕

𝜆 (9)

where E is energy of each photon. When a beam changes direction some of its momentum is transferred

to the object. The change of momentum of photons per seconds is equal to the force 𝐹 acting on a particle.

The force on a object hit with N photons is

𝐹 = 𝑁𝐸 𝑛1

𝑐0 ∆𝑡=

𝑃 𝑛1

𝑐0 (10)

The scattering force and the gradient force can be calculated using Fresnel equations

𝐹𝑠 =𝑛1𝑃

𝑐0 1 + 𝑅 cos 2𝜃 − 𝑇2 cos (2𝜃−2𝛷)+𝑅 cos 2𝜃

1+𝑅2+2𝑅 cos 2𝛷 =

𝑛1𝑃

𝑐0 𝑄𝑠 (11)

𝐹𝑔 =𝑛1𝑃

𝑐0 𝑅 sin 2𝜃 − 𝑇2 sin (2𝜃−2𝛷)+𝑅 sin 2𝜃

1+𝑅2+2𝑅 cos 2𝛷 =

𝑛1𝑃

𝑐0 𝑄𝑔 (12)

The sum force of these two forces is

𝐹𝑡 =𝑛1𝑃

𝑐0 𝑄𝑠

2 + 𝑄𝑔2 =

𝑛1𝑃

𝑐0 𝑄𝑡 (13)

Figure 4: Here is shown the magnitudes of 𝑸𝒈, 𝑸𝒔 and 𝑸𝒕 and direction of gradient in YZ plane. The length represents the force,

if focus of the beam was where the arrow begins. (A) shows gradient, (B) scattering and (C) total forces on the sphere. In (C) the

line between 𝑬 and 𝑬′ indicates where 𝑸𝒕 is purely in y direction [6].

If a particle's diameter is comparable to the wavelength of light used for trapping, neither

Rayleigh nor ray optics approach should be used. In this intermediate regime an electromagnetic

interaction with mater must be applied. The time average force can be represented by:

𝐹 = 𝑇𝑖𝑗 𝑛𝑗 𝑑𝑆𝑆

(14)

𝑇𝑖𝑗 = 휀0𝐸𝑖𝐸𝑗 +1

𝜇0𝐵𝑖𝐵𝑗 −

1

2 휀0𝐸𝑖𝐸𝑗 +

1

𝜇0𝐵𝑖𝐵𝑗 𝛿𝑖𝑗 (15)

where 𝑛𝑗 is outward normal unit vector and the integral is is over a surface enclosing the particle [7].

With this approach all six components of electromagnetic field must be obtained at the surface of the

object. This makes calculations much more difficult.

3 Optical tweezers

A basic setup is shown in Figure 5 (Left). The main components are: a laser, a beam expander, some

optics used to steer the beam location (Beam steering) in the sample plane, a microscope objective,

condenser, a position detector and a microscope illumination source coupled to a CCD camera [8]. The

trapped particle is in the sample plane.

To achieve the greatest gradient, and with that the greatest force, the laser beam must be focused

on a smallest possible beam width. By expanding the laser beam to fill the whole objective pupil of the

microscope the smallest spot size is achieved. To get the smallest spot an objective with the greatest

numerical aperture (NA) is used. To further increase NA the microscope objective is immersed in water or

oil as shown in Figure 5 (Right). Typically objectives for optical tweezers have a NA between 1,2 to 1,4.

[9]

Figure 5: (Left)A basic setup of optical tweezers with basic components.(Right) Scheme of objective immersed in oil. [8]

To manipulate objects in three dimensions, steering the beam focus in axial and transverse

direction must be possible. The transverse movement in achieved with two lenses labelled "Beam

Steering" in Figure 5(Left). This is achieved by laterally moving Beam steering lenses. To get axial

movement, axial displacement of the initial lens is adjusted. With these two actions the movement of

trapped particle in three dimensions is possible. Usually for faster movement acousto-optic deflector

(AOD) is used. This is a transparent crystal in which sound waves are generated by piezoelectric

transducer. Laser beam, which is diffracted on the sound waves, diffracts in accordance to Bragg's law.

The direction of refracted laser depends on frequency of piezoelectric transducer, i.e., frequency of sound

waves. Manipulating the frequency gives lateral control to the particle in the plane perpendicular to the

beam. The position of the beam can be changed up to 100.000 times a second and the step size is smaller

than 1 𝑛𝑚.

Optical tweezers are usually coupled with optical microscope. The sample is illuminated by a

separate light source using dichroic mirrors. This light is than directed at a CCD camera and the signal can

be transmitted to the computer or a monitor. With this the trapped particle can be tracked [8].

For in vivo experiments the choice of wavelength is crucial. A low absorption is recommended to

lessen the damage to biological material. Biological material absorbs small wavelengths (visible light -

< 700 𝑛𝑚), but longer wavelengths are absorbed by the water (wavelengths of > 1200𝑛𝑚) (Figure 7). So

a compromise must be made between absorption in water and biological material. Most often an infrared

laser is used (usually Nd:YAG laser -𝜆 = 1064 𝑛𝑚).

Optical tweezers must be able to accurately move the sample in steps of nanometre. For the large

movements, steps of several hundred nanometres or more, a mechanical motorized stage is enough. For

finer movements piezoelectric positioning gives us the desired accuracy in movement and positioning.

Figure 6: The graph shows relative transparencies of water, deoxyhemoglobin (𝑯𝒃) and oxyhemoglobin (𝑯𝒃𝑶𝟐). Here is shown

the importance of selecting laser with correct wavelength. For wavelengths greater the 2𝝁𝒎 water strongly absorbs light [12]

Simple optical tweezers can be upgraded to trap multiple particles by creating multiple traps. By quickly

switching between two or more positions with a single beam, multiple particles can be trapped

simultaneously. The switching frequency can be up to 100𝑘𝐻𝑧. It is also possible to create multiple

beams and in this way create multiple traps. This is achieved with computer generated holograms [13].

3.1 Measuring the position of particle in the trap

To determine optic forces an accurate measurement of position of particle in optical trap is

necessary. The simplest way to detect the position of laser exiting the sample plane is to guide the laser

on to the quadrant photodiode (Figure 6).

Figure 7: With the signal from photodiode the position of the particle in the trap is possible [10].

The beam is deflected when the particle is displaced. The deflected beam causes a change of currents in

the quadratic photodiode and with the analysis of the four currents the displacement can be accurately

measured to 1𝑛𝑚. Accuracy is mostly limited by the noise in electronic equipment - preamplifier.

4 Force measurement calibration

Usually dimensions of a bead is between both Rayleigh and ray approximation. So before determining the

force, the determination of the correlation between force and displacement between the centre of the

sphere and focus of the laser must be made. First approximation is that the relation is linear, which is good

for small displacements as shown in Figure 8. Force in optical tweezers can be described by the equation:

𝐹 = −𝑘𝑥 (16)

where constant 𝑘 is trap stiffness. By calibrating the system the trap stiffness is obtained and determining

force can be extrapolated by measuring the distance between focus of the laser and the centre of the sphere.

Figure 8: Computed forces on a sphere. The beam profile is Gaussian. The left graph shows transverse forces on a sphere as a

function of transverse displacement. The right graph shows axial forces on a sphere as a function of axial displacement [14]

4.1 Viscous drag force calibration

In the viscous drag force calibration, a known force is applied on a trapped particle and the displacement

is measured, than the correlation between force and displacement is known. The force on a sphere in a

viscous medium is well known as Stokes' law.

𝐹𝑣𝑖𝑠 = 6𝜋𝜂𝑎𝑣 = 𝛾𝑣 (17)

where 𝜂 is viscosity, 𝑎 radius of the sphere, 𝑣 speed of medium and 𝛾 is viscous drag coefficient. A

known force can be applied, if the velocity of the liquid can be measured. By measuring the displacement

of the bead at different forces, i.e., different speeds of medium, the traps stiffness is obtained.

4.2 Brownian motion calibration

Second possible calibration method that is often used is the measurement of the Brownian motion off a

trapped bead. A single bead is trapped. The movement of a trapped bead is the effect of forces due to

thermal fluctuations in the medium. The bead is governed by the Langevin equation

𝑚𝑥 + 𝛾𝑥 + 𝑘𝑥 = 𝐹 𝑒𝑥𝑡 (𝑡) (18)

The size of the particle and the fact that it is in an aqueous environment makes the system highly

overdamped. With this fact the inertial term can be dropped [22].

𝐹𝑒𝑥𝑡 𝑡 = 𝛾 𝑑𝑥

𝑑𝑡+ 𝑘𝑥 (19)

𝐹𝑒𝑥𝑡 (𝑡) is the thermal force. The time average of this force is zero and the power spectrum 𝑆𝐹 is constant

𝑆𝐹 = ℱ2(𝑓) = 4𝛾𝑘𝐵𝑇 (20)

where ℱ(𝑓) is the Fourier transform of 𝐹(𝑡). The result, if Fourier transformation is applied to the both

sides of Eq. (19), is:

2𝜋𝛾 𝑓𝑐 − 𝑖𝑓 𝑋 𝑓 = ℱ(𝑓) (21)

where 𝑓𝑐 = 𝑘/2𝜋𝛾. The result of combining Eq. (20) and Eq. (21) is

𝑆𝑋(𝑓) = 𝑋2 𝑓 =𝑘𝐵𝑇

𝛾𝜋2(𝑓𝑐2+𝑓2)

(22)

This is the power spectrum of frequencies of a trapped bead in optical tweezers [15].

For low frequencies, smaller then 𝑓𝑐 , the power is approximately constant. In the high frequencies region,

above 𝑓𝑐 , the amplitude falls with the square of the frequency. With the measurement of the movement of

the bead, due to thermal fluctuating, the power spectrum and from this the corner 𝑓𝑐 can be obtained. From

this graph the constant value at lower frequencies is easily obtained:

𝑆 0 = 𝑆0 =𝑘𝐵𝑇

𝛾𝜋2𝑓𝑐2 (23)

With values 𝑆0 and 𝑓𝑐 = 𝑘/2𝜋𝛾 acquired from the graph, it is possible to determine 𝑘 and 𝛾

𝑘 =2 𝑘𝐵𝑇

𝜋 𝑆0 𝑓𝑐 , 𝛾 =

𝑘𝐵𝑇

𝜋2 𝑆0𝑓𝑐2 (24)

The value of 𝛾 = 6𝜋𝑎𝜂 is only true for idealized conditions, which are not always satisfied. A more

accurate value of 𝛾, for a given system, is extrapolated from the calibration.

5 Experiments

Figure 9: Typical power spectrum due to thermal

fluctuations of a trapped bead. The corner frequency

was 𝒇𝒄 = 𝟐𝟎𝟔𝟓 ± 𝟓 𝑯𝒛 . The peak at 𝒇𝒅𝒓𝒊𝒗𝒆 =𝟑𝟐 𝑯𝒛 is the drive frequency which moved

sinosuidally with amplitude 𝑨 = 𝟏𝟓𝟎 𝒏𝒎 . The

object was sampled with frequency 𝒇𝒔𝒂𝒎𝒑𝒍𝒆 =

𝟔𝟓𝟓𝟑𝟔 𝑯𝒛. The temperature was 𝑻 = 𝟐𝟒, 𝟒 °𝑪 [16]

Optical tweezers are quite versatility tool in many scientific areas such as biology and biochemistry. They

are used for trapping and manipulating dielectric spheres, living cells, viruses and small metal particles.

Other applications are the measurement of small forces, confinement and organization of multiple objects,

tracking of movement, altering of large structures, etc. By attaching long polymer molecule on a dielectric

sphere even manipulation and measuring physical properties of polymer molecules is possible [17].

5.1 Force induced DNA melting

By applying force on a strand of double-stranded DNA (dsDNA) a force induced DNA melting occurs. In

this process the dsDNA changes to ssDNA (single strand DNA) and elongates by a factor of ~1,7. A

dsDNA is attached to two beads. One bead is held fixed by a glass micropipette and the second one is

captured with optical tweezers. In this way it is possible to study the force-versus-extension behaviour of a

single DNA molecule. From this elasticity and structure parameters can be deduced accurately [18, 19].

Figure 10: Theoretical data for dsDNA and ssDNA and measured data of dsDNA. It appears the stretching is a transition between

dsDNA and ssDNA [15].

At smaller forces one can measure physical properties of DNA, such as persistent length and elasticity. At

force 𝐹 ≈ 65 𝑝𝑁 an abrupt change occurs. The dsDNA strand lengthens and in the end overstretches to

~1,7 its original length. If the force is relaxed, DNA quickly and reversibly shrinks to its normal length.

5.2 Molecular motors

A lot of work has been done to measure forces, step size and speeds created by molecular motors (such as

kinesin, myosin, RNA polymerase etc.) [20]. Molecular motors are molecules that use energy to create

motion. One such motor is RNA polymerase. It is an enzyme that constructs RNA strands from DNA in a

process referred as transcription. Measuring the stalling force and motor speed is achieved by attaching

one part of DNA on a bead trapped with a laser and molecular motor on a fixed bead as shown in Figure

11(Left). Then the glass micropipette is moved to a specific location or until the desired force on the bead

in the optical trap is measured. Such experiments show that RNA polymerase has a mean production

speed of 23 base pairs per second and can apply forces up to 25 𝑝𝑁 until stalled [20].

6 Conclusion

The seminar shows that optical tweezers have many useful applications in many different fields of study,

especially in biology and biochemistry. Many systems have small characteristic forces and weak

interaction, and it is only possible to measure those forces with optical tweezers. Optical trapping can

work in concert with other microscopy techniques such as single-molecule fluorescence [21]. With the use

of Gaussian beams in higher modes, which must be radially asymmetric, the controlled rotation of

particles is possible. With improvement of optical tweezers and development of new trapping methods a

whole plethora of new experiments will be possible that are not feasible right now.

7 Literature

[1] Ashkin A., "Acceleration and trapping of particles by radiation pressure" Phys. Rev. Lett. 24, 156-159 (1970)

Figure 11: (Left) Schematic diagram of an

optical tweezers experiment to measure the

transcription forces generated by E. coli RNA

polymerase. (upper right) Tether length in units

of base pair (bp) plotted as function of time.

Some pauses are indicated with red arrows.

(Lower right) Histogram (Blue bars) of average

velocities from individual transcription records.

A Gaussian fit to these data (red solid line)

supplies the mean velocity of 𝟐𝟑 ± 𝟏𝟏 𝒃𝒑/𝒔

[20].

[2] Ashkin A., Dziedzic J.M. ,Bjorkholm J.E., Chu S., "Observation of a single-beam gradient force optical trap for

dielectric particles" Opt. Lett. 11, 288-290 (1986)

[3]http://www.optics.rochester.edu/workgroups/novotny/courses/OPT463/STUDENT_PAPERS/tweezers.pdf

(10.3.2013)

[4]http://ucilnica.fmf.uni-lj.si/mod/resource/view.php?id=5774 (2.3.2013)

[5] http://www.uni-leipzig.de/~pwm/web/?section=introduction&page=opticaltraps (8.3.2013)

[6] Ashkin A. "Forces of a single beam gradient laser trap on a dielectric sphere in the ray optics regime" Biophys. J.

Volume 61: 569-582. (1992)

[7] Gordon JP. "Radiation Forces and momenta in dielectric media" Phys. Rev. 8:14-21 (1873)

[8]http://en.wikipedia.org/wiki/Optical_tweezers (10.3.2013)

[9]http://www.isibrno.cz/omitec/index.php?action=sbt.html (10.3.2013)

[10]http://ej.iop.org/images/0034-4885/76/2/026401/Full/rpp316697f10_online.jpg

[11] Natan Osterman "Meritev stohastične resonance s pomočjo laserske pincete", Diploma thesis (2004)

[12] Karel Svoboda, Steven M. Block "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct.

23:247-85 (1994)

[13] Hu Zhang, Kuo-Kang Liu "Optical tweezers for single cells" doi: 0.1098/rsif.2008.0052 (published: 6 July

2008)

[14] Timo A. Nieminen et al., "Optical tweezers computational toolbox" Journal of Optics A 9, S196-S203 (2007)

[15] http://www.biophysics.org/Portals/1/PDFs/Education/williams.pdf (10.3.2013)

[16] http://www.mpipks-dresden.mpg.de/mpi-doc/julichergruppe/julicher/COOTWPDITBFP06.pdf (10.3.2013)

[17] http://www.stanford.edu/group/blocklab/Optical%20Tweezers%20Introduction.htm (10.3.2013)

[18]Smith SB, Cui Y, Bustamante C., "Overstretching B-DNA: the elastic response of individual double-stranded

and single-stranded DNA molecules" Science (1996) 271 795

[19] Mogurampelly Santosh et al. "Force Induced DNA Melting" PACS number: 87.14G-, 05.70.Jk, 5.10.-a

[20]Davenport JR, Wuite GJ, Landick R, Bustamante C. "Single-molecule study of transcriptional pausing and arrest

by E. coli RNA polymerase" Science 287:2497-2500 (2000)

[21] Matthew J. Lang, Polly M. Fordyce and Steven M. Block, "Combined optical trapping and single-molecule

fluorescence" Journal of Biol. 2: 6 (2003)

[22] Yi Deng, John Bechhoefer and Nancy R Forde " Brownian motion in a modulated optical trap" J. Opt. A: Pure

Appl. Opt. 9 (2007) S256–S263