paper 26

7
 Dynamics of a large dielectric particle in a dual optical trap Michael Lawrence Castañares, Lester Geonzon, Michael Gavriel Reyes, Adonis Flores Renante Violanda, and Rommel Bacabac*  Medical Biophysi cs Group, Physics Department, Un iversity of San C arlos Talamban, Cebu City, Philippines *Corresponding author: [email protected] om Abstract This study presents the behavior of a large particle, 2.5 μm bead, in a dual optical trap. The system is modeled using a Langevin equation with a localized gradient force due to the beam trap. The position distribution of the bead is investigated as the separation between the two beams is increased. The experimental results confirm our simulation, which shows that there exists a critical range of beam separation, where the particle tends to be pulled strongly  by either traps as demonstrated by th e shift and skew on the particle’s position distribution. For large separation, the system reduces to a single beam optical trap. Keywords: optical tweezers, nonlinear optics, computation, Brownian motion,  biophysics, R unge-Kutta method PACS: 42.65.Jx, 45.10.-b, 87.50.wf 1. Introduction Small colloidal particles (or micron-sized dielectric spheres)  can be trapped using laser light passing through a microscope objective [1-3]. Otherwise known as “optical tweezers”, the technique is now widely used in the field of Biophysics, for the manipulation of small molecules to living cells. A stable trap is achieved by th e use of an intense laser of a few Watts coupled with a high numerical aperture objective [4]. For large micron-sized particles, the trapping force can be derived using geometric optics [5]. And for small displacements from the focus, the behavior of the trap can be approximated with a harmonic potential.  A double potential well is constructed by introducing another beam into the system. The question arises as to how a single particle would behave between two optical traps. Recently, a bistable three dimensional optical trap was constructed using dual beam optical tweezers with similar trap stiffness [10]. It was demonstrated that a small 0.6 μm bead jumps between the two stable states. However, it is not known whether large particles would behave similarly in a dual trap. In this paper, the behavior of a large particle in a dual trap is investigated. The system is modeled using a Langevin equation with localized gradient force. This equation is numerically solve d using the Runge-Kutta method to determine  particle’s position distribution for different beam separation distances. The results of the simulation is compared to the motion of a 2.5 μm polystyrene bead moving between the 830nm and 1064nm optical traps with similar stiffness (0.3 pN/ μm). The position distribution of the bead is then examined as the beam separation is increased from 40nm to 4.57 μm. 2. Theory The Langevin equation is a common model used to describe the random behavior (Brownian motion) of microscopic particles in a viscous fluid. The motion of a particle in a dual optical trap particle is  (1) where  is the drag force,  is the gradient force by beam1 positioned at a distance d  from the origin ( while  is another gradient force by beam 2 that is fixed at the origin and    is the random/Langevin force which causes the Brownian motion of the particle such that  [11]. The motion is overdamped due to the high viscousity of the medium. Also, for polystyrene beads the Reynold s number is very small such that the inertial term  is negligible. The configuration of the system is presented in figure 1. For small displacements from the focus, the optical potential created by the beam is approximated to be harmon ic. Thus the gradient force, which is r esponsible for bringing the particle to the beam focus, is    (2)

description

PHYSICS

Transcript of paper 26

  • Dynamics of a large dielectric particle in a dual optical trap

    Michael Lawrence Castaares, Lester Geonzon, Michael Gavriel Reyes, Adonis Flores

    Renante Violanda, and Rommel Bacabac* Medical Biophysics Group, Physics Department, University of San Carlos

    Talamban, Cebu City, Philippines

    *Corresponding author: [email protected]

    Abstract This study presents the behavior of a large particle, 2.5 m bead, in a dual optical trap. The system is modeled using a Langevin equation with a localized

    gradient force due to the beam trap. The position distribution of the bead is

    investigated as the separation between the two beams is increased. The

    experimental results confirm our simulation, which shows that there exists a

    critical range of beam separation, where the particle tends to be pulled strongly

    by either traps as demonstrated by the shift and skew on the particles position distribution. For large separation, the system reduces to a single beam optical

    trap.

    Keywords: optical tweezers, nonlinear optics, computation, Brownian motion,

    biophysics, Runge-Kutta method

    PACS: 42.65.Jx, 45.10.-b, 87.50.wf

    1. Introduction Small colloidal particles (or micron-sized dielectric spheres) can be trapped using laser light passing through a

    microscope objective [1-3]. Otherwise known as optical tweezers, the technique is now widely used in the field of Biophysics, for the manipulation of small molecules to living cells. A stable trap is achieved by the use of an intense

    laser of a few Watts coupled with a high numerical aperture objective [4]. For large micron-sized particles, the

    trapping force can be derived using geometric optics [5]. And for small displacements from the focus, the behavior of the trap can be approximated with a harmonic potential. A double potential well is constructed by introducing

    another beam into the system. The question arises as to how a single particle would behave between two optical

    traps. Recently, a bistable three dimensional optical trap was constructed using dual beam optical tweezers with

    similar trap stiffness [10]. It was demonstrated that a small 0.6 m bead jumps between the two stable states. However, it is not known whether large particles would behave similarly in a dual trap.

    In this paper, the behavior of a large particle in a dual trap is investigated. The system is modeled using a

    Langevin equation with localized gradient force. This equation is numerically solved using the Runge-Kutta method

    to determine particles position distribution for different beam separation distances. The results of the simulation is compared to the motion of a 2.5 m polystyrene bead moving between the 830nm and 1064nm optical traps with similar stiffness (0.3 pN/m). The position distribution of the bead is then examined as the beam separation is increased from 40nm to 4.57 m.

    2. Theory The Langevin equation is a common model used to describe the random behavior (Brownian motion) of

    microscopic particles in a viscous fluid. The motion of a particle in a dual optical trap particle is

    (1)

    where is the drag force, is the gradient force by beam1 positioned at a distance d from the origin

    ( while is another gradient force by beam 2 that is fixed at the origin and is the

    random/Langevin force which causes the Brownian motion of the particle such that [11]. The motion is overdamped due to the high viscousity of the medium. Also, for polystyrene beads the Reynolds number is very small such that the inertial term is negligible. The configuration of the system is presented in figure 1.

    For small displacements from the focus, the optical potential created by the beam is approximated to be harmonic.

    Thus the gradient force, which is responsible for bringing the particle to the beam focus, is

    (2)

  • where is the trap stiffness which depends on the intensity and wavelength of the beam. Experimetally, the trap stiffness is determined using the equipartition theorem [12].The average displacement of the particle from the

    equilibrium position is related to the trap stiffness by

    ,

    (3)

    where is the Boltzmann constant. Due to the probabilistic nature of the Langevin force, the equation of motion is solved numerically. A numerical approximation of the particles position is obtained using the fourth order Runge-Kutta method. Given a general differential equation,

    (4)

    an approximate solution by Runge-Kutta method is expressed as:

    (5)

    where is particles position at a later time , position from the previous time and s are real coefficients.

    3. Methodology 3.1 Experimental Setup

    Figure 1 shows the schematic of our optical tweezers setup. The trapping area is built around an inverted

    microscope (Carl Zeiss Axiovert 200) with a differential interference contrast (DIC) system. To slightly overfill the

    back aperture of the oil-immersion objective (NA=1.4), both lasers with wavelengths 1064nm (IRCL-700 1064,

    CrystaLaser, USA) and 830nm (model IQ1C140/6017 diode laser, CrystaLaser) are expanded using beam

    expanders. A fast steering mirror (FSM 300, Newport, USA) controls the 1064nm laser beam in the sample plane. The condenser delivers light to the sample and collects scattered beam from the optical trap. The samples are imaged

    using a CCD camera (WAT-902H, Watec, Japan) located at the camera port of the microscope. Samples of

    polystyrene beads (Bangs Laboratories, Inc.) with diameter 2r=2.5m in a distilled water are prepared in a

    micropscope slide with a cover slip. The slide is then placed on a two-axis nanopositioning stage (Mad City Labs,

    Inc., USA). A 0.083 microns/pixel calibration of the image is obtained by using a 0.01mm/line calibration slide. The

    relation of the calibration factor along the x and y axis is linear.

    3.2 Particle detection and trap stiffness

    The stiffness of each laser, 830nm and 1064nm, is determined by observing the motion of the 2.5 micron bead in a

    single trap. First, a sequence of images (1000 frames) of a trapped bead is captured using a particle tracker program

    implemented in LabVIEW. The image sequence is analyzed using a pre-selected region of interest (ROI) to limit the data processing to pixels regions where the bead is moving. The position of the particle is extracted using a sequence

    of image processing routines: smoothing, edge detection, fill holes, filtering by circularity and area, and centroid

    measurement. This image processing protocol effectively extracts the beads shape by taking its outline (edge detection) and disregarding neighboring noise objects using the criteria for circularity and area. Then, the location of

    Figure 1. A visual representation of a large dielectric sphere between two optical traps (A). A box function

    with smooth cutoff (C) is multiplied to the gradient force to limit its action within the beam waist

    This results to a localized traps by beam1 and beam2 (B).

    Force (pN)

    Position (m)

    (A) (B) (C)

  • the bead is determined by taking the centroid of the binary object image. The average area of the beads is about 800

    pixels and the error of the position is 3nm.

    This uncertainty of the position is in good agreement with the study of Wuite et al. [6]. Once the location of the

    particle is determined, a histogram plot of the positions of the particle is generated. From the Gaussian distribution,

    trapping stiffness is determined using equation (3).

    3.3 Simulation on the numerical solution of the Dual trap model

    In equation (2), the gradient force is approximated as force on a spring. From figure 1, the interaction of the bead and the beam is limited to regions where refraction occurs or when a section of the bead is exposed to the beam.

    Equivalently, the interaction only occurs within the beam waist . In this paper, we attempt to model the localized interaction by a trial box function with a smooth cut off like a product of two hyperbolic functions. Thus, the

    gradient force of each beam is modified as:

    (6)

    where models the localized interaction for a trap beam of width . This expression of the gradient force is used in eq (1), to arrive at the Langevin model for a particle in a dual trap. The shape of the box function is shown

    in figure 1C for a beam waist of microns. Note that the effective beam diameter is reduced to 12 microns. This resulting gradient force is presented in figure 1B for a beam separation d=10 microns.

    The Langevin model described by eq(1) (substituted with the modified gradient force in eq(6)), is numerically

    solved using the Runge-Kutta method with the initial position of the bead set at random between the two beams. The

    values of the parameters: are 0.58, 1.37, 7, 20(1.2), 20, and 0.5, respectively. Applying the box function reduces the effective width of beams1 and beam2 to 14 and 12 microns, respectively.

    Note that beam1 exhibits a stiffer and wider trap. In this simulation, beam1 is the 1064nm laser and beam2 is the

    830nm laser. These parameters are in the order of 10E-6 which results to a micrometer displacement of the particle.

    Thirty trial runs of the simulation are made for different beam separation distances: 0 to 9 microns. Note that due to

    the random motion of the particle, only average values of the position and their deviations are meaningful. Thus,

    analysis on the resulting position histrograms is used. The skewness of the position histogram and the shift of the

    distribution are studied. A positive skewed distribution exhibits a tail from the +x axis. While a shift is defined as

    the distance of the 50th percentile of the distribution from the 1064nm steered beam. The skew and shift describes

    the effects of the two beams on the average motion of the particle.

    Figure 2. Optical tweezer setup wherein a laser system is steered and focus into the

    trapping cell. The setup consists of: a)Polarizer, b) Beam expander, c) Mirror, d) Fast

    steering mirror, e) Lens, f) Dichroic mirror g) CCD camera, h)condenser, i)objective

    and j) illumination.

    c

    Lasers

    Beam steering

    Trapping cell

    a b c

    d

    e

    j

    f

    g

  • 4. Results and Discussion The resulting position skew and shift of the position distribution from the simulation for different beam separation

    are presented in figures 2 and 3. In figure 2, all the thirty trial points for a given separation is plotted to show that

    they cluster around a certain skew values. Due to the random nature of the system, there also exist some extremums.

    For a given separation, the 50th percentile is used to represent the most probable value of the skew.

    From both figures 3 and 4, the beam separation is divided into three regions: strong pulling, critical, and weak

    coupling. For small displacements (3-6 microns), there are extreme positive and negative skews (fig 3A). Also, there is a positive shift in the position distribution (fig. 4A). Thus, the particle is found between the beams. These suggest

    that the particle is strongly pulled by either traps. As the beam separation increases, the distribution shifts linearly

    away from 1064nm (towards the 830nm beam). This is the behavior of a linear spring described by eq(3). Increasing

    the separation, the skew values cluster near zero. Also the uncertainty of the shift significantly increases. This region

    is termed as a critical region since the high clustering of the skew and uncertainty in the shift occurs in small interval

    of the displacement. From the distribution at this separation, two peaks are observed (fig. 4B). Thus, The particle

    jumps between the two bistable states which was previously reported by McCann et al. [10]. Further increasing the

    seperation, more positive skews (fig. 3A) are observed indicating that the particle is likely to be found near the

    1064nm beam. The shift of the distribution from the 1064nm beam starts to decrease, as if the interaction 830nm

    beam started to be weaker. This is due to the smooth cutoff of the box function of the localized gradient force. This

    region is termed as weak coupling since the 830nm only acts a small perturbation. Note that the effective beam

    widths from the simulation for 1064nm and 830nm are: 14 and 12 microns, respectively. At 9 microns separation where the beams dont overlap at the others focus, the particle could be locally found in either of the beam trap depending on the initial position of the particle as shown by the large error in the shift plot.

    -1

    0

    1

    2

    3

    0 2 4 6 8

    shif

    t (m

    icro

    ns)

    beam separation (microns)

    0

    20

    40

    60

    80

    100

    -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

    Fre

    quen

    cy

    particle position (microns)

    Figure 4. The shift of the distribution with separation distance (A) and the occurence of a bistable state at a critical

    separation of 6.7 microns (B).

    A B

    Critical region

    -5

    -3

    -1

    1

    3

    5

    7

    3 4 5 6 7 8 9

    Skew

    (a.u

    .)

    beam separation (microns)

    Critical region

    Figure 3 (Left to right) The plot of the skewness (A) for different beam separation and selected position

    histograms from the simulation with a skew of: -1.7 (B1), 0.6 (B2), and 1.35 (B3).

    A B

    1 2

    3

  • bead

    2.5m

    1064nm 830nm

    A B C

    Figure 5. The actual image of the bead in the dual trap for: 0(A), 2.41(B), and 4.65 microns(C) beam separation. At large separation distance (4.65 microns), the system reduces to a single trap

    which demonstrates the localized interaction of the bead and beam.

    bead

    The results from the simulation are verified experimentally by observing the motion of a 2.5 micron polystyrene

    bead in a dual trap. The bead is initially trapped by the 830nm diode laser with 35mW power with a trap stiffness of

    0.2-0.3 pN/m as (see figure 5A). The order of the trap stiffness still allows observable random displacements of the bead from the beams focus. Using far field diffraction, the beam waist of the 830nm and 1064nm lasers are estimated to be 0.2m and 0.3m, respectively. The 1064nm laser at 350mW with trap stiffness of 0.3-0.5 pN/m is introduced to the initial system. From figure 5, both beam spot size/waist are similar and small compared to the

    diameter of the bead. It is instructive to regard the stiffer trap (1064nm) as the original trap and the slightly weaker

    trap as a perturbation (830nm). The 1064nm beam is steered to the left at different separation distances from 0 to

    4.65 m. The position distribution of the particle is generated for each separation distance of the beam and compiled into one histogram as shown in figure 5.

    From figure 6B, the particles position distribution is slightly skewed to the right towards the 830nm beam for the small beam separation distance strong pulling region. Also, the particle is between the two beams as observed in figure 5B. At 1 micron separation, the skew is negatively large which indicates a transition to the critcal region as

    simulated in figure 3A at 6 micron separation. Steering the 1064nm beam further to the left, the shift in the particles distribution gradually decreases weak coupling until the particle is trapped at the focus of the 1064nm beam (see figure 5C).

    5. Conclusion In this paper, a robust model for the motion of a large particle is constructed by introducing a modification of the

    gradient force of an optical trap to account for localized interaction. The behavior of the particle as predicted from

    the model such as the shift and skewness of its position distribution were in good agreement with experimental data.

    For a particle in a dual beam trap, there exists a region of beam separation where the particle tends to be strongly

    pulled by one beam; i.e., for large beam separations, the system reduces to a single trap. Interestingly, another region

    exists where the bead is found equally probable at either beam. This phenomeon could be further explored to model

    the tuning of transitions between well-defined stable states.

    A B

    Figure 6 The compiled position distribution of the particle (A) and the skew values (B) as the 1064nm bead is moved

    away from the 830nm trap fixed at 24 microns. The location of the 1064nm beam and the resulting position distribution of

    the particle with beam separations: 0, 1.00, 2.41, 3.98, 4.65 microns are labeled from (1-5), respectively.

    0.00

    0.05

    0.10

    0.15

    0.20

    0.25

    0.30

    0.35

    19.0 19.8 20.6 21.4 22.2 23.0 23.8

    norm

    aliz

    ed fre

    qu

    ency

    position of the bead (microns)

    Direction of the steered 1064nm

    1

    1

    2 2

    3 4 5

    5 4 3

    -5

    -4

    -3

    -2

    -1

    0

    1

    2

    0 1 2 3 4 5

    Sk

    ew (

    a.u

    )

    separation (microns)

    Transition to

    Critical region

    A B

  • Acknowledgements This project was funded by CELLS and GELS research program. The proponents would like to acknowledge

    Mr. Florencio Bustamante and Mr. Erwin Orosco for assisting us in setting up the optical tweezers. Also the proponents would like to extend their gratitude to Dr. Perry Esguerra for sharing some useful insights in attacking

    the classical mechanics problem.

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    3. M. Woerdemann, Introduction to Optical Trapping, Structured Light Fields, Springer-Verlag, Berlin-Heidelberg, 2012.

    4. C. Schmidt, A practical guide to optical tweezers, Taylor and Francis, pp.259-279, 2006 5. N. Malagnino, et. al., Measurements of trapping efficiency and stiffness in optical tweezers, Opt. Commun.,

    214, pp 15-24, 2002. 6. G. Wuite, et. al., An integrated laser trap/flow control video microscope for the study of single biomolecules,

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    7. J.Molloy, M. Padgett, Lights, action: optical tweezers, Contemporary Physics, vol. 43, no. 4, pp. 241-258, 2002.

    8. J. Bechhoefer, S. Wilson, Faster, cheaper, safer optical tweezers for the undergraduate laboratory, Am. J. Phys. 70(4), pp 393-400, 2002.

    9. D. Chowdhury, 100 years of Einsteins Theory of Brownian Motion: From Pollen Grains to Protein Trains 2, Resonance, pp 42-54, 2005.

    10. L. McCann, et. al., Thermally activated transitions in a bistable three-dimensional optical trap, Nature 402, pp 785-787, 1999.

    11. J. Mas, et. al., Understanding Optical Trapping Phenomena: A Simulation for Undergraduates, IEEE trans.Educ, vol. 54, no.1, 2011.

    12. K. C. Neuman, S. M. Block, Optical Trapping, Rev. Sci. Instrum., 75, pp 2787-2809, 2004. 13. E.-L. Florin, et. al., Photonic force microscope calibration by thermal noise analysis, Appl. Phys. A, 66, pp 75-

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  • Appendix1:

    The Runge-Kutta coefficients s

    (7)