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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 24 — Nov. 28, 2005
  • pp: 9695–9701
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Switching and measuring a force of 25 femtoNewtons with an optical trap

Alexander Rohrbach  »View Author Affiliations


Optics Express, Vol. 13, Issue 24, pp. 9695-9701 (2005)
http://dx.doi.org/10.1364/OPEX.13.009695


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Abstract

An experiment is described, where an ultra-weak force of 25 femtoNewtons (fN = 10-15 N) displaces a 0.53μm latex bead captured in a soft optical trap. The displacement is measured by exploiting the phase anomaly of the trapping beam, resulting in a small phase shift of the scattered light. The 25fN force stems from the radiation pressure of a nearly collimated second laser beam, which is switched by an accousto-optical modulator. To the best of my knowledge, this is the smallest switchable force which has ever been directly measured.

© 2005 Optical Society of America

1. Introduction

Although photons are massless, they can exert a pressure on matter, by transferring their momentum h/λ, which is the inverse incident wavelength quantized by Planck’s constant. Although one photon carries a tiny momentum of 6.626 × 10-28 Ns at a wavelength of λ = 1 μm, this momentum sums up to a gigantic radiation pressure in the center of hot stars. This pressure works against the gravitational forces, which would induce a collapse of the star. In principle the photon momentum can be exploited for an interplanetary transport of spaceships with solar sails. Large, photon reflecting surfaces (sails) with low mass, are slowly, but almost frictionless accelerated by the photon flux emitted by the sun. This situation changes completely in the earth’s atmosphere, where collisions with air molecules dissipate the kinetic energy of the objects accelerated by radiation pressure. The question arises by how far these forces can be scaled down to laboratory situations.

Already 1970 Ashkin [1

1 . A. Ashkin , “ Acclearation and trapping of particles by radiation pressure ,” Phys. Rev. Lett. 24 , 156 – 159 ( 1970 ). [CrossRef]

] dragged micron-sized latex spheres with an Argon laser and measured radiation forces in the picoNewton (pN) range by finding the equilibrium of optical and friction forces. However, the smaller and lighter the optically accelerated particles, the more pronounced becomes the viscous drag force γ·v and thermal random forces Fth (leading to Brownian motion). For example, a 1μm latex bead can take up a kinetic or thermal energy of 8 kBT (kBT ≈ 4·10-21Nm at T = 300 K, kB = Boltzmann constant) after a few seconds [2

2 . H. A. Kramers , “ Brownian motion in a field of force and the diffusion model of chemical reactions ,” Physica VII , 284 – 304 ( 1940 ). [CrossRef]

]. By moving over a distance of e.g. Δx = 30nm against an external potential W(x), the effective thermal force can easily become Fth = x2W(x) · Δx = 1 pN. The gravity G = m·g of this bead is only 10 fN. A molecular motor transporting this 1μm bead along fibers in a living cell exerts forces of 3-5 pN in this Brownian storm of thermal forces. Forces in the pN regime have been successfully measured with optical tweezers (see review of Neuman and Block [3

3 . K. C. Neuman and S. M. Block , “ Optical trapping ,” Rev. Sci. Instr. 75 , 2787 – 2809 ( 2004 ). [CrossRef]

]), whereas direct force measurements in the low fN regime are very difficult to perform. With a direct measurement of a force Fmes I refer to the equilibration with a single calibrated force Fcal = Fmes.

Where do we find fN forces? For instance, the entropic force that arises when the entropy is reduced by stretching e.g. a DNA polymer with a persistence length of Lp = 50 nm is Fent = kBT/Lp = 80 fN. A 1μm small bacterium has to overcome a friction force of Ffr = γ·v = 95 fN at a propagation speed of v = 10 μm/s in water. The gravitational attraction between two 3 mm glass spheres close to each other is about 10 fN. The repulsive electrostatic force of a 0.5 μm latex sphere in de-ionized water is about 20 fN in a distance of 1.5 μm to the surface due to the auto-dissociation of water. In his famous experiments to prove the quantization of electric charge [4

4 . R. A. Millikan , “ A new modification of the cloud method of. determining the elementary electric charge and the most. probable value of that charge ,” Philosophical Magazine 19 , 209 – 228 ( 1910 ).

], Millikan switched a 100–200 fN small electric force FE = E·n·q to counterbalance the gravitational force m·g on 3–4 μm sized oil drops, loaded with n elementary charges q. And coming back to the radiation pressure, the daily sun light, defined by the solar intensity Isol = 1370 W/m2 exerts a force

Frad=I0·n/Qpr·(a2π)
(1)

of only 20 fN on a small absorbing dust particle of 2a = 75μm in diameter in the atmosphere. I0 = Isol is the intensity incident on the particle’s cross-section (a2π), c/n is the speed of light in a medium with refractive index n and Qpr is the pressure efficiency, which is 1 for an absorbing particle. I have setup an arbitrary collection of ultra-small forces to give the reader a feeling for magnitudes in the low femtoNewton range.

Fig. 1. Experimental setup. A collimated IR beam (1064nm) is focused by NA = 1.2 water immersion lens. A 535nm latex sphere is trapped weakly and scatters the incident IR light coherently. The detection lens dipping into the fluid droplet collects both the scattered and unscattered light, which have a phase difference of Δϕ to each other. A quadrant photo diode (QPD) records the interference pattern, which changes with Δϕ. The fluid droplet consists of a fluorescein solution. A 488nm laser is switched on, resulting in an axial displacement of the trapped bead by Δz and an phase shift Δϕ(Δz) of the IR light due to radiation pressure. The fluorescence light emitted along the 488nm beam profile is recorded by the imaging system.

Force measurements in the low femtoNewton range are mentioned regularly in the literature [5–9

5 . D. Rudhardt , C. Bechinger , and P. Leiderer , “ Repulsive depletion interactions in colloid-polymer mixtures ,” J. Phys.: Condens. Matter 11 , 10073 – 10078 ( 1999 ). [CrossRef]

], but attention has to be paid, whether these studies present a direct measurement or an estimation of the apparatus’ resolution for force measurements.

So far no report has been found in the literature where such small forces could be controlled and measured directly. In this paper I demonstrate how a force of 25 fN on a small probe is switched and measured directly with help of an ultra-weak optical trap and an interferometric position detection system. The force can be switched on and off, such that no extrapolation of a system inherent force distribution is necessary. Only two parameters need to be determined for the measurement: the stiffness κz of the optical trap [10

10 . A. Rohrbach , “ Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory ,” Phys. Rev. Lett. 95 , 168102 ( 2005 ). [CrossRef] [PubMed]

], which is obtained by signal autocorrelations 〈Sz(t)·Sz(t+τ)〉 of the probe’s position z(t), and secondly, the shift Δz of the probe’s mean position, measured before and after the femtoNewton force was switched on. The measured force is then Fz = κz·Δz. The displacement Δz is obtained via the tiny phase change of the trapping beam scattered at the probe.

2. Instrumentation and experiment

Which femtoNewton force can be quickly switched on and off and leaves the system in thermal equilibrium in both the on- and off-state? A conservative force, which can be easily controlled, results from the radiation pressure of a collimated laser beam on a non-absorbing sphere. A 488nm Argon laser beam was switched with a accousto-optical modulator (AOM) leading to an optical power of P = 11 mW at the dielectric probe, which was a 0.535 μm latex bead with refractive index ns = 1.591+0.0003i [11

11 . C. Jung and B. K. Rhee , “ Simultaneous determination of thickness and optical constants of polymer thin film by analyzing transmittance ,” Appl. Opt. 41 , 3861 – 3865 ( 2002 ). [CrossRef] [PubMed]

].

The trapping force Ftrap is obtained by calibrating both the optical trap and the detection system. The axial force Ftrap = κz · Δz is calibrated by the axial stiffness κz = γz and the detection system is calibrated by the axial signal sensitivity gz = σsz / σz = σsz·(κz/kBT)1/2 of the particle’s position signal Sz(z) = gz·z, which is linear to the position z over a reasonable distance [12

12 . A. Rohrbach , C. Tischer , D. Neumayer , E. L. Florin , and E. H. K. Stelzer , “ Trapping and tracking a local probe with a Photonic Force Microscope ,” Rev. Sci. Instrum. 75 , 2197 – 2210 ( 2004 b). [CrossRef]

]. The mean square displacement of the trapped particle σz2 :=〈σz(t)2〉 follows the equipartition theorem κz·σz2 = kBT. The resulting half width σsz of the signal histogram (Fig. 2(b)) is obtained by a Gaussian fit exp[-(Szsz)2]. The autocorrelation time τz is the time the particle needs to relax inside the optical trap in axial direction and is provided by the signal autocorrelations 〈Sz(t)·Sz(t+τ)〉 = 〈|Sz(0)|2〉 exp(-τ/τ zγ = 6πaη is the viscous drag far from an interface (η viscosity of water). Therefore the relation between the particle’s position z(t) and it’s position signal Sz(z(t)) is

Sz(Z)gz·Z=σsz·[γ/(τz·kBT)]1/2·Z.
(2)

Therefore the radiation force from the 488nm laser is provided by the force equilibrium

Frad=Ftrap(Δz)=κz·Δz=κz·ΔSz/gz=ΔSz/σsz·(κz·kBT)1/2
(3)

The axial displacement of the particle Δz is proportional to the difference ΔSz in the mean position signals

ΔSz=SonSoff=1ΔTonΔTonSz,on(t)dt1ΔToffΔToffSz,off(t)dt
(4)

The position signal Sz(t) = 〈S〉 ± σsz(t) fluctuates by σsz(t) = gz·σz(t) around the mean value 〈S〉 due to thermal motion (see Fig. 2(a)). The integration times were ΔTon ≈ ΔToff ≈ 4s at a sampling rate of 50 kHz. The temperature was T = 21°C.

The signals Sz are acquired by integrating the interference intensity over the full area of the QPD [12

12 . A. Rohrbach , C. Tischer , D. Neumayer , E. L. Florin , and E. H. K. Stelzer , “ Trapping and tracking a local probe with a Photonic Force Microscope ,” Rev. Sci. Instrum. 75 , 2197 – 2210 ( 2004 b). [CrossRef]

], the signal sensitivity gz(NAdet) can be increased by decreasing the numerical aperture of the detection lens NAdet [13

13 . A. Rohrbach , H. Kress , and E. H. K. Stelzer , “ Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture ,” Opt. Lett. 28 , 411 – 413 ( 2003 ). [CrossRef] [PubMed]

, 14

14 . J. K. Dreyer , K. Berg-Sorensen , and L. Oddershede , “ Improved axial position detection in optical tweezers measurements ,” Appl. Opt. 43 , 1991 – 1995 ( 2004 ). [CrossRef] [PubMed]

], which we reduced to NAdet = 0.3. Sz is the integrated interference intensity between the incident field E i and the scattered field E s(z) which varies with the position of the probe z.

Sz(z)=Ei+Es(z)2dA=(Ei2+Es(z)2+EiEs(z)cos(ϕiϕs(z)))dA
(5)

Close to the focus, the phase difference Δϕ(Δz) = ϕis(Δz) ~ Δz is proportional to the particle’s displacement Δz due to the Gouy phase shift [15

15 . A. Pralle , M. Prummer , E.-L. Florin , E. H. K. Stelzer , and J. K. H. HÖrber , “ Three-dimensional position tracking for optical tweezers by forward scattered light ,” Microscopy Research and Techniques 44 , 378 – 386 ( 1999 ). [CrossRef]

, 16

16 . A. Rohrbach and E. H. K. Stelzer , “ Three-dimensional position detection of optically trapped dielectric particles ,” J. Appl. Phys. 91 , 5474 – 5488 ( 2002 b). [CrossRef]

]. The Gouy phase shift leads to the reduced momentum carried by a focused beam in axial direction. Via a phase shift of π it leads to a stretching of the mean wavelength in the focal area.

Fig. 2. Axial bead displacements and histograms. (a) The 535nm latex bead fluctuates inside the weak optical trap. By switching on an additional radiation pressure after about 5 sec, the mean position of the bead changes by 70nm. (b) The histograms illustrate the ratio between the mean displacements due to thermal noise and due to the ultra-weak external force. The widths of the histograms σsz are nearly the same in both cases. (c) The radiation force can also be obtained from the difference in the corresponding potentials.

To estimate the number of photons transferring their momentum on to the probe, the intensity I0 of the 488nm laser beam at the probe was determined. As illustrated in Fig. 1, a diluted fluorescein solution (mass ratio 2·10-5) was used to measure the beam waist s of the beam. After fluorescence excitation, the beam profile was imaged onto a CCD camera (see Fig. 3). The axial extent of the focus, i.e. the extent of the detection plane (focal plane) is about 1μm, whereas the Rayleigh length of the beam is much longer, ensuring that no artifact is introduced by the integration of the detection lens. By fitting a Gaussian beam profile to the fluorescence intensity I(r) = I0·exp(-r2/s2), a half width of s = 7μm at I0/e was obtained (Fig. 3 left). The trapped probe was well centered relative to the 488nm beam, as indicated in Fig. 3 right. Here the fluorescence increase due to coherent scattering behind the sphere is 70%, integrated over the depth of field of the trapping lens.

3. Results and discussion

The total power P488 = 11 mW in the focal plane is 62% of the power measured behind the coverslip without the trapping lens. P is distributed over the whole focal plane according to

P488=0I(r)·2πrdr=I02π0exp(r2/s2)·rdr=I0·s2·π
(6)

From Eq. (6) we find I0 = P488/(πs2) = 71.5 MW/m2, which corresponds to a power of Pgeo = I0·a2π = 16.1 μW distributed over the cross section of the sphere, which is Cgeo = a2π = 0.226 μm2. In other words Pgeo/(hν) ≈ 5.3·1013 photons per second hit the sphere’s cross section.

The resulting radiation force can be measured by using eq.(3). Totally 11 measurements were performed at two strengths of the optical trap (λ trap = 1064nm). At a very low trapping laser power Ptrap = 1.5mW in the focal plane, a radiation force of Frad = 25.6 ± 2.2 fN was obtained (N=5 measurements). This force is the product of a mean trap stiffness of κz = 0.37 pN/μm and a mean displacement Δz = 69nm. At a second, still low laser power Ptrap = 4.4mW nearly the same radiation force of Frad = 22.3 ± 5.7 fN was measured (N=6), which is the product of a mean trap stiffness of κz = 1.12 pN/μm and a mean displacement Δz = 20nm. The consistency and the relative small errors underline the reliability of the measurements. The uncertainty in the force measurement due to thermal fluctuations is F + ΔF = κz·(Δz ± σz) with σz2 = kBT/κz). The uncertainty ΔF increases with (kBT κz)1/2 at a constant external force Frad and can be reduced by using a weak optical trap, i.e. a small trapping stiffness κz, and, of course, by a low temperature T. Since resolution is a question of definition, I chose that ΔF ≤ F or σz ≤ Δz, which was approximately fulfilled in this experiment (Δz ≈ σz ≈ 70nm). By defining a different resolution limit or by achieving an even smaller trap stiffness, forces smaller than 25fN can be measured.

Fig. 3. The Gaussian profile of the weakly focused laser beam is visible due to fluorescence emission in the fluorescein solution (left image). The intensity profile reveals a beam waist of 2s = 14μm at a 1/e decay. Right: A bead trapped in the center of visible laser beam produces an increase in electric field density behind the bead due to scattering. In this region the fluorescence intensity is increased by 70% as indicated by the profile and the photo. Integration time was 20ms.

As a further verification, a similar radiation force Frad (< 15% deviation) was determined by the gradient of the potential difference as illustrated in Fig. 2 right. Furthermore, the correctness of the detector calibration gz via the relaxation time τ z (eq.(2)) was tested by moving a fixed bead over a distance Δz through the focus by a piezo-stage. The ratio gz = ΔSz / Δz provided nearly the same value.

According to Mie-theory, a sphere with index ns = nR+i·nI = 1.591+0.0003i and radius a = 268nm has the scattering function S(θ=0) = 7.715-10.548i leading to a scattering efficiency Qsca = 4/(k·a)2Re{S(0)} = 1.4653, k = 2π n/λ, λ = 488nm. This would mean that 147% of the incident photons are scattered at the sphere (extinction paradox) [17

17 . C. Bohren and D. R. Huffman , Absorption and scattering of light by small particles ( Wiley Science Paperback, New York , 1998 ). [CrossRef]

]. However, due to diffraction, i.e. the redistribution of optical energy and momentum, more photons are scattered in forward direction (θ=0) due to interference with the unscattered field. The resulting pressure efficiency without absorption is Qpr - Qabs = 0.180, where

Qpr=Qabs+QscaQscan2k02a20π12(T1(θ)2+T2(θ)2)cosθsinθ
(7)

Here n is the refractive index of the medium (water) and Qabs is the absorption efficiency. T1(θ) and T2(θ) are the angle-dependent Mie-scattering functions parallel and perpendicular to the plane of incidence[18

18 . H. C. van de Hulst , Light Scattering by Small Particles ( Dover Publications, Inc., Leiden , 1957 ).

]. Using Eq. (1), we obtain a theoretical radiation pressure of Frad = 12.8fN, if Qabs = 0. This theoretical value is only half as large as the measured value for the radiation force. It is therefore likely that absorption takes place at the sphere. The momentum uptake ~ Qabs due to absorption and incoherent scattering must be about the same as due to coherent scattering ~(Qpr - Qabs).

Therefore my colleague Holger Kress (EMBL Heidelberg) proposed and proved the spontaneous adhesion of fluorescein fluorophores at the surface of the bead (Bangs Lab Inc.). Assuming in the simplest case a dense monolayer of fluorophores, we expect a number N of absorbing fluorophores on the front and back side of the bead with geometrical cross-section Cgeo = a2π. With a geometrical cross-section r2π of the fluorophores, the number of molecules is N = 2a2π / r2π = 2.26·106. The diameter of a fluorescein fluorophore is about 2r = 0.5nm and the molar extinction coefficient of fluorescein is about Cext,fl = 72000 (mol/l)-1cm-1 (see Sigma-Genosys, www.fisheroligos.com/tec_flc.htm), which is Cext,fl = 72000 m2/NA = 0.012 (nm)2 (NA = Lohschmidt number, Csca,fl = 0 due to incoherent, isotropic scattering). This results in a total absorption cross-section of Cabs = N·Cext,fl = 0.034 μm2 or an absorption efficiency Qabs = Cabs / Cgeo = 0.15, which is about the same as Qpr - Qabs = 0.18 due to coherent scattering at the sphere. Again, according to this estimation, a latex sphere with diameter 535nm takes up the same amount of light momentum as a 0.5nm thin monolayer of fluorescein molecules around the sphere. The emitted fluorescence of the dye monolayer is small in comparison to the increased excitation and fluorescence behind the sphere due to coherent scattering of the excitation light.

Some years ago Matsuo et al. [6

6 . Y. Matsuo , H. Takasaki , J. Hotta , and K. Sasaki , “ Absorption analysis of a single microparticle by optical force measurement ,” Journal of Applied Physics 89 , 5438 – 5441 ( 2001 ). [CrossRef]

]
tried to exploit the difference between the measured radiation pressure efficiency on a particle and the computed scattering efficiency to estimate the absorption of the particle. However, in this study unique measurements were complicated due to a more complex scattering at the 10-fold larger spheres and the close proximity of two interfaces.

Finally I want to remark that also the absorbance of the 488nm light by fluorophores inside the fluorescein droplet should have an effect on the final momentum transfer, because it mainly reduces I0 at the sphere. An estimate of this reduction is complicated due to the diffusion of fluorophores through the beam.

Although different ratios of momentum uptake due to resonant and non-resonant scattering at the (coated) sphere may result in a radiation force of 25fN, the force switching and measurement remains unaffected by this ratio.

4. Conclusions

The smallest switchable force ever was measured by using an ultra-weak optical trap. Different measurements manifest a force of about 25 fN resulting from the radiation pressure of a 1 1mW laser beam (λ = 488nm) on a 0.53 μm latex sphere. About half of this force results from momentum transfer due to coherent scattering, the other half is likely due to absorption of a dense monolayer of fluorescein molecules around the sphere. With this study I try to set a new milestone concerning force measurements in the sub-picoNewton regime and I encourage other researchers to underbid this value with careful measurements.

Acknowledgments

I thank Holger Kress for helpful comments, especially for providing the concept of the absorbing monolayer around the sphere and I thank Dr. Ernst H.K. Stelzer for general support.

References

1 .

A. Ashkin , “ Acclearation and trapping of particles by radiation pressure ,” Phys. Rev. Lett. 24 , 156 – 159 ( 1970 ). [CrossRef]

2 .

H. A. Kramers , “ Brownian motion in a field of force and the diffusion model of chemical reactions ,” Physica VII , 284 – 304 ( 1940 ). [CrossRef]

3 .

K. C. Neuman and S. M. Block , “ Optical trapping ,” Rev. Sci. Instr. 75 , 2787 – 2809 ( 2004 ). [CrossRef]

4 .

R. A. Millikan , “ A new modification of the cloud method of. determining the elementary electric charge and the most. probable value of that charge ,” Philosophical Magazine 19 , 209 – 228 ( 1910 ).

5 .

D. Rudhardt , C. Bechinger , and P. Leiderer , “ Repulsive depletion interactions in colloid-polymer mixtures ,” J. Phys.: Condens. Matter 11 , 10073 – 10078 ( 1999 ). [CrossRef]

6 .

Y. Matsuo , H. Takasaki , J. Hotta , and K. Sasaki , “ Absorption analysis of a single microparticle by optical force measurement ,” Journal of Applied Physics 89 , 5438 – 5441 ( 2001 ). [CrossRef]

7 .

J. C. Meiners and S. R. Quake , “ Femtonewton force spectroscopy of single extended DNA molecules ,” Phys. Rev. Lett. 84 , 5014 – 5017 ( 2000 ). [CrossRef] [PubMed]

8 .

G. M. Wang , E. M. Sevick , E. Mittag , D. J. Searles , and D. J. Evans , “ Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales ,” Phys. Rev. Lett. 89 , -( 2002 ). [CrossRef] [PubMed]

9 .

J. Y. Shao , “ Finite element analysis of imposing femtonewton forces with micropipette aspiration ,” Ann. Biomed. Eng. 30 , 546 – 554 ( 2002 ). [CrossRef] [PubMed]

10 .

A. Rohrbach , “ Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory ,” Phys. Rev. Lett. 95 , 168102 ( 2005 ). [CrossRef] [PubMed]

11 .

C. Jung and B. K. Rhee , “ Simultaneous determination of thickness and optical constants of polymer thin film by analyzing transmittance ,” Appl. Opt. 41 , 3861 – 3865 ( 2002 ). [CrossRef] [PubMed]

12 .

A. Rohrbach , C. Tischer , D. Neumayer , E. L. Florin , and E. H. K. Stelzer , “ Trapping and tracking a local probe with a Photonic Force Microscope ,” Rev. Sci. Instrum. 75 , 2197 – 2210 ( 2004 b). [CrossRef]

13 .

A. Rohrbach , H. Kress , and E. H. K. Stelzer , “ Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture ,” Opt. Lett. 28 , 411 – 413 ( 2003 ). [CrossRef] [PubMed]

14 .

J. K. Dreyer , K. Berg-Sorensen , and L. Oddershede , “ Improved axial position detection in optical tweezers measurements ,” Appl. Opt. 43 , 1991 – 1995 ( 2004 ). [CrossRef] [PubMed]

15 .

A. Pralle , M. Prummer , E.-L. Florin , E. H. K. Stelzer , and J. K. H. HÖrber , “ Three-dimensional position tracking for optical tweezers by forward scattered light ,” Microscopy Research and Techniques 44 , 378 – 386 ( 1999 ). [CrossRef]

16 .

A. Rohrbach and E. H. K. Stelzer , “ Three-dimensional position detection of optically trapped dielectric particles ,” J. Appl. Phys. 91 , 5474 – 5488 ( 2002 b). [CrossRef]

17 .

C. Bohren and D. R. Huffman , Absorption and scattering of light by small particles ( Wiley Science Paperback, New York , 1998 ). [CrossRef]

18 .

H. C. van de Hulst , Light Scattering by Small Particles ( Dover Publications, Inc., Leiden , 1957 ).

OCIS Codes
(140.7010) Lasers and laser optics : Laser trapping
(260.3160) Physical optics : Interference
(290.2200) Scattering : Extinction
(290.4020) Scattering : Mie theory
(290.5850) Scattering : Scattering, particles
(300.2530) Spectroscopy : Fluorescence, laser-induced

ToC Category:
Research Papers

History
Original Manuscript: October 18, 2005
Revised Manuscript: October 17, 2005
Published: November 28, 2005

Citation
Alexander Rohrbach, "Switching and measuring a force of 25 femtoNewtons with an optical trap," Opt. Express 13, 9695-9701 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-24-9695


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References

  1. A. Ashkin, "Acclearation and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
  2. H. A. Kramers, "Brownian motion in a field of force and the diffusion model of chemical reactions," Physica VII, 284-304 (1940). [CrossRef]
  3. K. C. Neuman, and S. M. Block, "Optical trapping," Rev. Sci. Instr. 75, 2787-2809 (2004). [CrossRef]
  4. R. A. Millikan, "A new modification of the cloud method of. determining the elementary electric charge and the most probable value of that charge," Philosophical Magazine 19, 209-228 (1910).
  5. D. Rudhardt, C. Bechinger, and P. Leiderer, "Repulsive depletion interactions in colloid–polymer mixtures," J. Phys.: Condens. Matter 11, 10073–10078 (1999). [CrossRef]
  6. Y. Matsuo, H. Takasaki, J. Hotta, and K. Sasaki, "Absorption analysis of a single microparticle by optical force measurement," Journal of Applied Physics 89, 5438-5441 (2001). [CrossRef]
  7. J. C. Meiners, and S. R. Quake, "Femtonewton force spectroscopy of single extended DNA molecules," Phys. Rev. Lett. 84, 5014-5017 (2000). [CrossRef] [PubMed]
  8. G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, "Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales," Phys. Rev. Lett. 89, - (2002). [CrossRef] [PubMed]
  9. J. Y. Shao, "Finite element analysis of imposing femtonewton forces with micropipette aspiration," Ann. Biomed. Eng. 30, 546-554 (2002). [CrossRef] [PubMed]
  10. A. Rohrbach, "Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory," Phys. Rev. Lett. 95, 168102 (2005). [CrossRef] [PubMed]
  11. C. Jung, and B. K. Rhee, "Simultaneous determination of thickness and optical constants of polymer thin film by analyzing transmittance," Appl. Opt. 41, 3861-3865 (2002). [CrossRef] [PubMed]
  12. A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, and E. H. K. Stelzer, "Trapping and tracking a local probe with a Photonic Force Microscope," Rev. Sci. Instrum. 75, 2197-2210 (2004b). [CrossRef]
  13. A. Rohrbach, H. Kress, and E. H. K. Stelzer, "Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture," Opt. Lett. 28, 411-413 (2003). [CrossRef] [PubMed]
  14. J. K. Dreyer, K. Berg-Sorensen, and L. Oddershede, "Improved axial position detection in optical tweezers measurements," Appl. Opt. 43, 1991-1995 (2004). [CrossRef] [PubMed]
  15. A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Hörber, "Three-dimensional position tracking for optical tweezers by forward scattered light," Microscopy Research and Techniques 44, 378-386 (1999). [CrossRef]
  16. A. Rohrbach, and E. H. K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91, 5474-5488 (2002b). [CrossRef]
  17. C. Bohren, and D. R. Huffman, Absorption and scattering of light by small particles (Wiley Science Paperback, New York, 1998). [CrossRef]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc., Leiden, 1957).

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