Optical Traps as Force Transducers: The Effects of Focusing the

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Optical Traps as Force Transducers: The Effects of Focusing the Trapping Beam through a Dielectric Interface Aristide C. Dogariu* and Raj Rajagopalan† School of Optics & Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Florida 32816, and Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611-6005 Received April 7, 1999. In Final Form: November 30, 1999 The use of optical forces generated by a focused laser beam for manipulating single polymer chains and colloidal particles has begun to receive increasing attention in polymer and colloid science and in biophysics. Devices based on such optical traps are increasingly used for probing the elasticity of single polymer chains and for studying the structure and mechanical behavior of biopolymers and their relation to biological activity. Colloidal forces and viscoelastic behavior of the suspending fluids can also be examined through the use of optical traps as force transducers and from the dynamics of optically bound probe particles. Such applications require a precise understanding of the trap potential and its dependence on the distance between the trap center and the glass/liquid interfaces that exist in experimental arrangements. This paper addresses some of the issues relevant to the above class of applications of optical traps using wave optics calculations. It is shown that the use of simple geometric optics corrections for the change in focus caused by a dielectric interface could lead to errors in the estimation of the measured forces. The occurrence of secondary traps and corresponding deviations from the commonly assumed harmonicity of trap potentials are also identified.

1 Introduction The use of radiation pressure to trap micron-sized (or smaller) particles using laser beams was pioneered by Ashkin1 about 30 years ago, and the developments since then have been reviewed recently by Ashkin.2 Optically induced forces in the case of a spherical particle caught in a single, cylindrical Gaussian beam are illustrated in Figure 1. Optical manipulation of the particles using radiation pressure can be achieved in a number of ways. One of the simplest possibilities is to balance the weight of the particle using radiation pressure for particles in a size range smaller than the wavelength of the light used. In this case, the trap is called a levitation trap. For many applications with larger particles, one can take advantage of backward radiation pressure to form what are known as tweezer traps, as illustrated in Figure 2. The tweezer traps can exert forces that are over 2 or 3 orders of magnitude higher than the weight of the particle and are useful in confining the particles when the gravitational force on the particle plays a negligible role. As noted by Ashkin,2 the tweezer traps are more tolerant of irregularities in particle shapes than levitation traps. Although the development of optical traps was motivated originally by the interest in trapping atoms, the use of traps for holding and manipulating micron-sized objects has caught the imagination of scientists, as evident from the extensive use of optical traps in biology (see ref 2). While the initial focus in particle trapping was largely on spatially fixing small objects (such as biological cells) or on simply manipulating their positions (hence the name tweezer trap), there has emerged a serious interest in the past few years in using optical traps as force transducers, * To whom correspondence should be addressed at the University of Central Florida. † University of Florida. (1) Ashkin, A. Phys. Rev. Lett. 1970, 24, 156. (2) Ashkin, A. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 4853.

Figure 1. Optical trapping of neutral particles. Illustration of the gradient and scattering forces experienced by a neutral particle in a Gaussian laser beam. TEM00 stands for the Gaussian intensity profile in the trapping beam. The notations Fgrad and Fsca stand for the gradient force (the focus of the present paper) and the scattering force (see discussion in Concluding Remarks). Whether the particle is drawn into the beam (i.e., trapped) or pushed away depends on the relative magnitudes of Fa and Fb, which depend on (among other things) the refractive index contrast between the particle and the medium. (Adapted from Ashkin.2)

that is, to use optical forces to measure other forces of interest in colloid and polymer science. Such a use of optical traps requires that one pay more careful attention to the details of the optical forces in a trap and to the perturbations in forces caused by the dielectric discontinuities that occur in typical experimental arrangements (e.g., the glass/ water interface between a microscope slide and the suspension). Examples of the types of issues of interest in this context are 1. What is the influence of the glass/solution boundary on the optical forces and the force distribution? 2. What is the influence of such a boundary on the shift in the focal point and on the trap location?

10.1021/la9904004 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/21/2000

Optical Traps as Force Transducers

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2 Optical Traps as Force Transducers: Two Examples

Figure 2. A schematic representation of a tweezer trap. Point f is the focal point of the beam. (Adapted from Ashkin.2)

3. How is the harmonicity of an unperturbed trap potential affected because of the above dielectric discontinuity? 4. How does the force constant of the trap change as a function of the distance from the dielectric discontinuity? 5. What influence do these results have on the experimental determination of the surface forces? The purpose of this manuscript is to address some of these issues. The manuscript is organized as follows: Section 2 begins with two examples of the use of optical traps for force measurement in colloid or polymer systems in order to set forth the rationale for studying questions of the type posed above. Our primary objective here is to illustrate the emerging uses of optical tweezers and to motivate the subsequent theoretical calculations we present in Section 4 for the field and force distributions in traps. Therefore, we do not concern ourselves with the other technical details of the illustrated applications. However, because the use of optical tweezers as force transducers is relatively new, we have written this section in a manner suitable for readers unfamiliar with the use of tweezer traps. This is then followed, in Section 3, with the theoretical formalism we use for calculating the effects of dielectric discontinuities. This section also serves as a brief background on how the trapping force arises from electrostatic considerations. Section 4 then addresses the influence of the dielectric discontinuity on the intensity distribution in a focused beam and presents our results. We discuss the effect of the discontinuity on the shape of the intensity profiles in both the radial and axial directions, the shift in the focal point, and the location of the trap center as well as the implications of these to force measurements. Illustrative results are presented for (i) the shifts in real focal point relative to the predictions based on geometric optics, (ii) the presence of secondary trapping locations and the deviation of the secondary trapping potentials from harmonicity, (iii) the magnitude of the stiffness of the primary trap and the extent of its variation as the focal point is shifted, and (iv) the implications of these to the use of optical traps as force transducers. Some comments on related issues and a few concluding remarks follow in Section 5, along with examples of additional issues that require attention.

In this section, we discuss two examples of the use of optical radiation forces for measuring forces in polymer and colloid physics, so that the relevance of the questions posed in Section 1 is placed in context. The examples are chosen so that they are recent and, at the same time, illustrate the use of optical forces in the lateral direction (i.e., along the radial direction of the beam) as well as in the normal direction (along the axis). Since the primary focus here is to illustrate how the optical trap is used as a force transducer, the physical and chemical aspects of the applications discussed are kept to a minimum. In both cases illustrated below, one measures the displacement of the trapped particle from the trap center using a suitable detection technique; then the force acting on the particle is obtained simply from the balancing optical force, given by the product of the trap stiffness and the displacement, as demonstrated by Simmons et al.3 2.1 Radial Force Measurement. As illustrated in a number of recent studies, the mechanical behavior of single polymer chains can be examined using optical traps.4-6 In such uses of optical traps, one end of a polymer chain is typically either grafted onto or physisorbed on a small spherical particle (say, a latex “bead”), which is then trapped using an optical trap. The other end, or another segment, of the polymer chain is held in its place by attaching it to a surface (typically a microscope slide) or by other suitable means (e.g., by attaching it to a second bead held in place by another trap). The mechanical behavior and the spring constant of the chain can then be determined by measuring the displacement of the latex bead from the trap center, as the displacement of the bead in combination with the force constant of the trap potential (assumed to be harmonic) allows one to determine how much the chain is stretched as well as what the force experienced by the chain is. Force-versus-distance measurements thus obtained are very useful for understanding issues such as the elasticity of the chains or, in the case of biopolymers, the relation between biochemical actions and mechanical behavior of the polymer. As an example, we consider the recent work of Wang et al.,7 who used an optical trap to study the mechanical motor action of RNA polymerase (RNAP) as it moves along a DNA chain while carrying out transcription.8 Here, RNAP, fixed to a cover glass, transcribes a DNA chain, whose “downstream” end is attached to a polystyrene bead (see Figure 3). The bead is held in a stationary optical trap, as shown in the figure, but is drawn away from the trap center by RNAP. The force acting on the bead from the transcription action is countered by the restoring optical force arising from the displacement of the bead from the trap center. The displacement of the bead at which the restoring force balances the “transcription force” provides the force at which transcription stops. Such (3) Simmons, R. M.; Finer, J. T.; Chu, S.; Spulich, J. A. Biophys. J. 1996, 70, 1813. (4) Finer, J. T.; Simmons, R. M.; Spulich, J. A. Nature 1990, 386, 113. (5) Veigel, C.; Bartoo, M. L.; White, D. C. S.; Sparrow, J. C.; Molloy, J. E. Biophys. J. 1998, 75, 1424. (6) Kellermayer, M. S. Z.; Smith, S. B.; Granzier, H. L.; Bustamante, C. Science 1997, 276, 1112. (7) Wang, M. D.; Schnitzer, M. J.; Yin, H.; Landick, R.; Gelles, J.; Block, S. M. Science 1998, 282, 902. (8) RNAP, an enzyme, carries out the synthesis of an RNA copy of the template DNAsan essential step in gene expressionsby moving along the template as it synthesizes the RNA molecule in its entirety. The characterization of the enzyme movement through the steadystate velocity as a function of an applied force provides fundamental information on the enzyme mechanism.

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Figure 3. Force measurement on a DNA chain attached to glass surface using an optical trap. The cartoon illustrates the experimental geometry. The ellipsoid on the cover glass is RNAP. The polystyrene bead is held under tension by the optical trap, and its subsequent displacement in the x-direction under low tension is detected interferometrically. As transcription proceeds, DNA (attached to the bead) is drawn through the polymerase at successive times (t1, t2, t3, ...). See text for more details. Variations of this method are described in Wang et al.7

measurements of force-versus-velocity behavior provide critical information on the RNAP transcription mechanism, an unresolved feature of which is the nature of biochemical reaction steps that lead to movement.7 Static measurements using a stationary optical trap exerting a variable force have been reported by Yin et al.,9 and dynamic measurements using a feedback mechanism are described in Wang et al.7 Additional experimental details may be obtained from these references. An important point of interest in the present context is that the use of optical forces allows one to apply mechanical loads opposing the transcription-induced motion at a level sufficient to probe the enzyme mechanism but not too large to disrupt the enzyme structure. The above example also illustrates the need to understand optical forces exerted on trapped particles by a focused laser beam in the vicinity of a dielectric discontinuity (caused by a glass/ water interface in the example discussed; see Figure 3). 2.2 Axial Force Measurement. Polymer additives have been used routinely in practice for centuries for preparing commercial products such as paints, pharmaceuticals, cosmetics, and food products. Recent developments in advanced materials and synthetic organic chemistry have further enhanced the use of polymers in a number of emerging areas such as fabrication of engineered surfaces, polymer composites, tribology, biomedical implants, and artificial drug delivery systems, to name a few.10 Underlying all such applications are issues concerned with (i) the conformational dynamics of polymer chains at interfaces, (ii) the dependence of the conformational behavior on the molecular architecture of the chains, chemical properties of the surface, and solvent quality, and (iii) the resulting interfacial forces and mechanics. Jaganathan11 has recently shown that an optical trap can be used to measure very weak polymerinduced forces between a flat surface and a small particle. As in the case of the example discussed in Section 2.1, the tweezer trap here functions as a harmonic potential well characterized by a trap potential φtr ) (ktr/2)(h htr)2, where h is the distance along the axis of the beam, htr is the location of the trap center, and ktr is the “stiffness” of the trap potential. In the present case, we restrict (9) Yin, H.; Wang, M. D.; Svoboda, K.; Landick, R.; Block, S. M.; Gelles, J. Science 1995, 270, 1653. (10) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997. (11) Jaganathan, A. Direct Measurement of Polymer-Induced Forces. M.S. Thesis, University of Florida, Gainesville, FL, 1999.

Dogariu and Rajagopalan

Figure 4. Polymer-induced “steric” force between a glass surface and silica sphere, both with physisorbed poly(ethylene oxide) chains, measured using an optical trap. The electrostatic and van der Waals contributions to the total force have been subtracted from the measurements (see Jaganathan11).

ourselves to positional fluctuations of the trapped particle in the axial (z) direction of the trapping beam (i.e., normal to the interacting surface). The total force experienced by a particle close to a surface is the sum of the trap force, ftr, and the surface force, fs, arising from the presence of the interacting surface. Since the surface force is balanced exactly by the trap force at the most probable position hm of the particle, one has fs(hm) ) -ftr(hm) ) ktr(hm - htr). The determination of hm experimentally (e.g., using scattering of an independently generated evanescent wave field by the trapped particle) thus allows one to determine the surface force (also referred to as “colloidal” force) at hm directly.11,12 The combination of evanescent waves and a tweezer trap for measuring the total force has already been demonstrated by Sasaki et al.,13 although they did not pursue the measurement of particle/wall surface forces.14 As the trap stiffness can be adjusted by changing the power of the trap laser to magnitudes of the order of 10-3 pN/nm (about 4 orders of magnitude smaller than the typical stiffnesses of the cantilevers of atomic force microscopes), the use of tweezer traps allows one to not only measure very small forces directly but also to probe “soft” polymer interfaces. A typical example, for the interaction force between a glass slide and a silica particle, both with a thin layer of physisorbed poly(ethylene oxide) chains, is shown in Figure 4 as an illustration.11 The use of tweezer traps also allows one to probe otherwiseinaccessible high-energy regions and, moreover, to measure nonconservative forces,15 since the forces are measured directly. Moreover, the measurement of the dynamics (12) Clapp, A. C.; Ruta, A. G.; Dickinson, R. Rev. Sci. Instrum. 1999, 70, 2627. (13) Sasaki, K.; Tsukima, M.; Masuhara, H. Appl. Phys. Lett. 1997, 71, 37. (14) Evanescent wave light scattering in combination with twodimensional (lateral) trapping of the particles has also been used to measure colloidal forces. This approach has been called total internal reflection microscopy (TIRM) [Walz, J. Y. Curr. Opin. Colloid Interface Sci. 1997, 2, 600; Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93] or evanescent wave light scattering microscopy (EWLSM) [Tanimoto, S.; Matsuoka, H.; Yamauchi, H.; Yamaoka, H. Colloid Polym. Sci. 1999, 277, 130]. However, the trap laser in this case is not used as a tweezer trap (see Section 1), and the method of determining the colloidal force does not require a knowledge of trap-induced forces; therefore, the issues of interest in the present paper are not relevant in the case of TIRM/ EWLSM. The trapping beam in TIRM/EWLSM remains (essentially) cylindrical and traps the particle laterally while allowing it to execute Brownian motion along the axis. The readers interested in TIRM/ EWLSM may consult the above references for details. (15) Sasaki, K.; Horio, K.; Masuhara, H. Jpn J. Appl. Phys. 1997, 36, 721.


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caption). If the trap potential at such distances also deviates significantly from harmonicity because of the dielectric discontinuity caused by the presence of the glass surface, the colloidal force or other force-dependent properties calculated on the basis of harmonicity could be seriously in error. Issues such as these form the motivation for the examination of the influence of dielectric discontinuities reported in this paper. 3 Theoretical Background

Figure 5. Total potential (i.e., trap potential + colloid potential) measured at different trap locations relative to the glass surface. The left-most curve corresponds to a trap location very close to the glass surface. The right-most curve is the total potential at a distance somewhat farther from the surface, where the van der Waals attraction dominates the surface force. Still farther from the surface, the total potential would become harmonic again (not shown). The distances from the surface are relative to an arbitrary reference and are not absolute distances (Rajagopalan, unpublished).

of Brownian fluctuations of the particle held in the trap (or the response of the particle to a blinking or oscillating trap) can be used to study interfacial microrheology.16,17 Although not necessary in all cases, the analyses of data in many of the above instances assume that the trap potential remains harmonic as the trap is brought close to interfaces where a sudden jump in dielectric constant occurs. It is often assumed further that the trap stiffness remains unchanged as the trap is brought close to the interface. Such assumptions, if seriously at odds with reality, can substantially affect the calculated forces or other quantities that depend on the details of the force. As an example, we show in Figure 5 a series of measured total potentials, that is, the sum of the trap potential and the surface potential, as functions of h, the distance between the particle and the planar glass surface, for a charged silica particle interacting with a charged glass surface. Each total potential shown corresponds to a fixed trap position relative to the surface. Far from the glass surface, the total potential, which is the same as the trap potential since the colloidal potential is essentially zero at that distance, is essentially harmonic for all practical purposes (not shown in the figure). Similarly, when the particle is very close to the glass surface, the total potential is again harmonic since the particle/surface potential is steep and repulsive in the close vicinity of the glass. On the other hand, at some intermediate distances, the van der Waals force between the particle and glass becomes significant for the data shown, and if the variation of the van der Waals force with distance is larger than the variation of the trap force, the total potential experienced by the particle can deviate significantly from harmonicity and can even have double minima (see Figure 5 and the (16) Jimenez, J.; Ma, M.; Rajagopalan, R. Brownian Fluctuation Spectroscopy of Complex Fluids and Polymer Layers. Paper presented at the APS Centennial Meeting, Atlanta, GA, March 1999. (17) Nhan, D. Brownian Fluctuation Spectroscopy Using Atomic Force Microscopy. M.S. Thesis, University of Houston, Houston, TX, 1998. See also, Ma, H.; Jimenez, J.; Rajagopalan, R. Langmuir 2000, in press.

Before presenting our results on the effects of dielectric discontinuities on the intensity distribution in an optical trap, we outline the basic theoretical formalism necessary for calculating the field distribution and forces of interaction. This outline also serves to illustrate the origin of the trap force exerted by a focused laser beam. 3.1 Electrostatics of a Dielectric Body Embedded in Another Dielectric Continuum. For studying the field distribution and its effects on a dielectric body imbedded in another dielectric medium, one can consider the equivalent problem of (isotropic) dielectrics in electrostatic fields. (The isotropy condition can be relaxed when more complex particles need to be trapped.) When a dielectric material, such as a particle being trapped, is placed in an external electric field, even if the basic constituents of the material remain neutral, dipole electric moments are induced in each of them. This leads to an overall polarization of the medium, which perturbs the external field and gives rise to forces in the dielectric; however, energy conservation laws can be formulated on the basis of Maxwell’s equations. The electrostatic energy associated with a dielectric medium with permittivity 1 placed in an external electric field (of initial distribution) Ei is given by

Wi )

∫V Ei‚Di dV

1 2

(1)

1

where Di is the electric induction; the integration covers the whole space V1. When a body of arbitrary shape with volume V2 (V2 , V1) and dielectric permittivity 2 is introduced in this medium through an isothermal process, the presence of this body changes the field distribution over the whole volume V1. The resulting change in the energy between the initial and final states can be written as

∆W ) Wf - Wi )

∫V Ef‚Df dV - 21∫V Ei‚Di dV

1 2

1

1

(2)

The energy difference ∆W is equal to the work done in the isothermal process and, therefore, ∆W is in fact the change in the Helmholtz potential as the body V2 is brought from infinity to the location it occupies. If it is assumed that no other distribution of charges is present or induced at the interface between 1 and 2, the use of the Gauss theorem leads to

∆W )

∫V (1 - 2)Ei‚Ef dV

1 2

2

(3)

In the case of isotropic media, for which the electric polarization vector is P ) 0χeE, where 0 is the permittivity of vacuum and χe, the dielectric susceptibility, is defined as χe ) ( - 0)/0, eq 3 can be written as

∫V Ei‚P dV

1 ∆W ) 2

2

(4)

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if 1 ) 0. Important simplifications can be made for small differences in dielectric permittivities, that is, if 2 ) 1 + δ. In this case, Ef ≈ Ei and, to within infinitesimal corrections,

∫V δ|Ei|

1 ∆W ) 2

2

2

dV

(5)

for 1 ) 0, the case of vacuum, since in this case δ ) 0χe. The above development shows the origin of the attraction or repulsion experienced by a body of permittivity 2 as it interacts with a field in a medium of permittivity 1. Note that if 1< 2, the total energy decreases, meaning that the inserted body experiences an attractive potential. The opposite case, namely, a repulsive potential, occurs when 1 > 2. In Section 4, we use eq 5 or its variant shown below to determine the energy experienced by a particle in a field perturbed by a dielectric interface. 3.2 Trap Potential. It is useful to examine first how the trapping potential is calculated in the absence of any dielectric discontinuities prior to the encounter of the laser beam with the particle. In the classical approach to the trapping of homogeneous particles, one considers a certain distribution of electric field intensity I(r) in the trapping beam and then uses eq 5 to determine the three-dimensional trapping potential at location r′ of the particle, that is,

∫V I(r′- r) dr

δ ∆W(r′) ) 2

(6)

2

In their original treatment, Ashkin et al.18 considered the initial intensity distribution to be that of a focused Gaussian beam and calculated the force associated with the potential ∆W(r). This is the particular situation in which the energy density of the beam of power P is given by

I(r) )

( ) (

) (

)

x2 + y2 z2 2P exp exp π$2 2$2 2$2ζ2

(7)

where $ is the beam waist, ζ is a parameter known as the Rayleigh range of the beam and is a measure of the focusing asymmetry (i.e., the difference between the Gaussian profile in the x-y plane and the Gaussian profile in the z-direction), and P ) π$20cE20/4 with c being the speed of light. Here, E0 is the electric field at the beam waist, the center of which is taken as the origin x ) y ) z ) 0. The beam waist is defined as the radial distance at which the intensity is 1/e2 times the intensity at the origin, and, for a Gaussian beam, it is given by $ ) λ(0.518/NA), where λ is the wavelength and NA is the numerical aperture of the focusing optics. The parameter ζ is defined as the axial distance at which the beam radius becomes x2$ and is given by ζ ) 1.627/NA. From eqs 6 and 7, one can obtain the axial dependence of the potential energy of a dielectric body of volume V2 located at position z′ as

( )

1 2P δ ∆W(z′) ) 2 π$2

(

2

+y ∫V exp -x 2$ 2 2

(

)

2

×

)

(z - z′)2

exp -

2$2ζ2

dx dy dz (8)

which, for the particular case of a dielectric sphere of radius R and ζ = 1, becomes (18) Ashkin, A.; Dziedzic, J. M.; Biorkholm, J. E.; Chu, S. Opt. Lett. 1986, 11, 288.

( )

∆W(z′) ) -

( )∫ ( )

R z′2 2P δ exp - 2 0 F2 × 2 π$ 2$ F2 z′2 exp - 2 dF ∼ exp - 2 (9) 2$ 2$

( )

where F is the radial distance normal to the beam (with the origin at the center of the beam waist). The Gaussian distribution of the potential in eq 9 is a direct consequence of the Gaussian intensity profile in eq 7 (and the assumption that ζ = 1) and is commonly used in representing the trap force, which is obtained from the negative gradient of ∆W. Note that the force is linear in z′ in the vicinity of the trap center, thus leading to the harmonicity of the trap potential. Tight focusing of the trap beam (i.e., a sharp distribution in the intensity I(r)) is not necessary for trapping particles when the difference in the permittivities (i.e., δ) is large, since, as one can see from eq 6, large enough energies can be achieved for sufficiently large δ. For such cases, an approach similar to the above can be developed for Gaussian beams (in the so-called “paraxial description”19), and the resulting intensity can be written as

I(r) )

( ) (

)

xj2 + yj2 2P 1 exp π$ 1 + (2zj)2 1 + (2zj)2

(10)

where the intensity is presented in terms of the normalized coordinates xj ) x/$, yj ) y/$, zj ) z/ζ$2. Equation 10 is valid for λ , $ and may not be appropriate for tightly focused beams, which are usually needed when trapping particles in liquid suspensions. However, for trapping of dielectric particles in air,20 the above approximation should be sufficient. A particularly simple situation occurs when particles much smaller than the wavelength are trapped. In this case, ∆W(r) ∼ I(r), and the actual force acting to restore the particle position is simply proportional to the local gradient of the electric field intensity. For example, the axial restoring force for a small particle is then simply

F(z′) ∼

∂∆W(z′) ∂I(F,z′) ∼ ∂z′ ∂z′

(11)

where I(F, z′) can be evaluated as described previously. However, in more general cases, the above simplification is inaccurate and insufficient. In what follows, we consider particles of sizes comparable to the wavelength of the radiation used in a focused laser beam that is brought in through a dielectric discontinuity. 3.3 The Effect of a Dielectric Interface. The above discussion, which focuses on the trapping of particles in a homogeneous medium when the intensity profile in the beam remains undisturbed by the presence of any dielectric interfaces before the beam encounters the particle, sets the stage for our task in this paper. In the following section, we consider the presence of dielectric discontinuities and present results for the beam intensities and trapping potentials. This task requires two steps: (i) First, the field distribution in the beam (after it passes through a dielectric interface) is calculated using the Fresnel refraction principle. The boundary condition for this comes from the standard field distribution in a homogeneous medium (typically glass in the cases of (19) Born, M.; Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light; Cambridge University Press: Cambridge, UK, 1998. (20) Omori, R.; Kobayashi, T.; Suzuki, A. Opt. Lett. 1997, 22, 3816.


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interest here), along the lines described above. (ii) Then, we use the resulting field distribution and eq 6 derived above to obtain the trapping potentials. 4 Results and Discussion As emphasized earlier, in applications involving particle trapping, focusing the trapping laser beam through an interface is unavoidable. Good index-matching can be achieved between the microscope objective and the cover slide, but there is always a refractive index mismatch on the other side, that is, between the inner surface of the cover slide and the fluid with the trapped particle. When electromagnetic radiation is focused through an interface between media with different dielectric properties, spherical aberrations (phase deterioration) are produced and lead to a series of important distortions in the fields especially near the focusing region, as we shall demonstrate in this section. Numerous applications of high-resolution microscopy, in which the distortions introduced by interfaces strongly affect image quality, have given rise to intensive studies of the effects of dielectric discontinuities on light focused through interfaces. The literature dealing with the problems related to focusing the electromagnetic radiation is extensive and well-established.21 The influence of a dielectric interface has been studied subsequently by an angular spectrum representation of plane waves, and formal solutions as well as asymptotic dependences have been obtained.22 Moreover, the focusing effects have been illustrated numerically using a boundary condition in the form of current distribution.23 Recently, the spherical aberration caused by a dielectric interface has also been studied extensively, again in the context of high-resolution microscopy.24-26 Although numerous applications of laser trapping similar to the ones described in Section 2 involve the focusing of the trapping beam through an interface, the resulting distortions and asymmetries of the electromagnetic forces and their effects on trapping forces have been largely overlooked so far.27 4.1 Shift in Focus from Geometric Optics. To contrast the results we report in the following subsection, it is useful to first consider the shift in focal point caused by a dielectric discontinuity as predicted by the simpler, geometric optics calculations. When a converging spherical wave is incident on a dielectric interface n1/n2 as shown in Figure 6, the phase of the transmitted field is altered. The condition for the fields to add constructively at the new focus O1 is that the phase of all rays, such as a or b in Figure 6, be the same. This implies that the sum of the initial phase at the lens aperture and the phase accumulated in propagation be the same for all rays. Applying the Fermat principle and the Snell law of refraction,19 one can easily show that the shift ∆fGO in the focus can then be written in terms of the distance d of the interface from the “unperturbed” focus as

{ x

∆fGO(d) ) d 1 -

n2 n1

1-

( ) [(

1L 4 f

2

) ]}

n1 n2

2

-1

(12)

(21) Wolf, E. Proc. R. Soc. London 1959, A253, 349. (22) Gasper, J.; Sherman, G. C.; Stamnes, J. J. J. Opt. Soc. Am. 1976, 66, 955. (23) Ling, H.; Lee, S.-W. J. Opt. Soc. Am. 1984, A1, 965. (24) To¨ro¨k, P.; Varga, P.; Laczik, Z.; Booker, R. J. Opt. Soc. Am. 1995, A12, 325. (25) Wiersma, S. H.; To¨ro¨k, P. J. Opt. Soc. Am. 1996, A13, 320.

Figure 6. The focal shift ∆f that occurs when light is focused at a depth d through a dielectric interface between homogeneous media with n1 > n2.

where L and f are, respectively, the diameter and the focal length of the objective. The position of the real focus O1 is located between the interface and O at a distance of d - ∆fGO, and it is this distance that is actually controlled by the external micropositioning system of the trapping device. Equation 12 shows that, on the basis of geometrical optics arguments, a linear dependence is obtained between the mechanical displacement of the virtual focal distance d and the shift ∆fGO in the position of the real focus. It is worth noting that, usually, the slope of ∆fGO (d) is much smaller than unity. It is also important to note that the above simple geometric arguments can only address the shift of the focus (albeit approximately); more complex distortions introduced by focusing through an interface are revealed only when a full-wave treatment is used to describe the electromagnetic field propagation, as shown below. 4.2 Field Propagation and Distribution. We first focus on the calculation of the effect of the dielectric discontinuity on the field (intensity) distribution in a laser beam after it passes through the interface. To this end, we consider a laser beam of known power arriving at an interface through a homogeneous medium of refractive index n1, as shown in Figure 6. The interface separates the medium of refractive index n1 (say, a glass) from the second, continuous medium of refractive index n2 (say, a liquid such as water). In any homogeneous medium, the electric and magnetic fields can be described in an integral representation as a superposition of plane waves.21 The Fresnel refraction law describes the passage of the fields through any dielectric interface. We use the fields calculated in the medium of index n1 as boundary conditions to obtain a rigorous vector solution for the diffraction problem using a second integral representation24 of the electromagnetic field in the second medium (i.e., of refractive index n2). The resulting electric field distribution in the second medium is then given by

I(F, z) )

∫0Ωxcos φ1 sin φ1 exp[-idk(n1 cos φ1 -

n2 cos φ2)](τs + τp cos φ2) J0(kn1F sin φ1) exp(ikn2z cos φ2) dφ1 (13) where Ω is the angle of the convergence cone illustrated in Figure 6, J0 is the Bessel function of the first kind and zeroth order, k ) 2π/λ is the wavenumber, and d is the depth of the real focus from the interface (see Figure 6). The integration in eq 13 extends over the half-angle convergence cone of the objective. Equation 13 permits the evaluation of intensity along the axial direction z as (26) Wiersma, S. H.; Visser, T. D.; To¨ro¨k, P. Pure Appl. Opt. 1998, 7, 1237. (27) Recently, a ray-optics calculation has been used to estimate the trapping efficiency of commercial, high-numerical-aperture microscope objectives by Gu, M.; Ke, P. C.; Gan, X. S. Rev. Sci. Instrum. 1998, 68, 3666.

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well as in the transversal plane at any radial position r (anisotropy is assumed for the field distribution in this plane). The angles of incidence and refraction in the two media are denoted by φ1 and φ2, respectively, in eq 13 and are related to each other through

φ2(φ1) ) sin-1

(

)

n1 sin φ1 n2

(14)

The coefficients τs and τp are the Fresnel refraction coefficients and are functions of n1, n2, φ1 and φ2:

τs ) 1 +

n1 cos φ1 - n2 cos φ2 n1 cos φ1 + n2 cos φ2

(15)

τp ) 1 -

n2 cos φ1 - n1 cos φ2 n2 cos φ1 + n1 cos φ2

(16)

and

Contour plots of intensity distribution in the x-z plane thus calculated are presented in Figures 7a and b for the case of focusing a beam of wavelength 633 nm through a dielectric interface n1/n2 ) 1.51/1.33, corresponding to the common situation of a glass/water interface encountered in most typical applications. An objective with the focal distance of 130 µm and a numerical aperture of 1.25 is considered in this example. The case d ) 0 corresponds to focusing in the medium n1 in the absence of the interface and, as expected for a spherically symmetric distribution of the initial electric field, the symmetry is fully preserved. However, when the interface is situated at a distance of d ) 30 µm (from the unperturbed focus), a strong disturbance of the electric field is produced, as evident from Figure 7b. In Figure 8, we show the axial intensity distribution for the same situation used in Figure 7. Two major observations are worth noting. First, as the distance d increases from 0 to 60 µm, the position of the maximum intensity, that is, the real focus, shifts toward the interface, as also predicted by geometrical optics arguments and illustrated schematically in Figure 6. However, we shall see in the following subsection that the difference between the focal shift predicted by the wave optics calculations and the one from the simpler, geometrical optics result is nonnegligible and could be important in practice. The second implication of the result shown in Figure 8 is that a complex interference pattern develops and causes an increasing asymmetry in the intensity distribution. The implication of such asymmetry to the resulting trapping potentials and forces is taken up first in the following subsection. 4.3 Trap Potential and Force. A key aspect of the field distributions shown above is how these rather complex intensity distributions affect the trapping energy given by eq 5. To illustrate this, we show a few examples of the axial trapping potentials in Figure 9 for three different cases of virtual focus d for focusing through a glass/water interface. An important implication of the results shown is that, for typical powers used in trapping experiments, secondary minima in the trap potentials could result due to the considerable axial fluctuations of the intensity of the light beam. The positions and magnitudes of these secondary trapping wells also depend on other factors, such as the particle size and the refractive index contrast between the trapped particle and the suspending medium. Nevertheless, Figure 9 shows that such secondary trapping wells can have depths that are

Figure 7. Contour plots of |E|2 in the x-z plane corresponding to radiation with wavelength λ ) 830 nm in a vacuum when the beam is focused (a) in a homogeneous medium of refractive index n1 and (b) through a dielectric interface n1/n2 ) 1.51/1.33 in the geometry presented in Figure 6, with focusing depth d ) 20 µm, focal length f ) 130 µm, and numerical aperture of the lens NA 1.25.

Figure 8. Axial intensity distributions for radiation with the wavelength λ ) 830 nm in a vacuum focused through a dielectric interface n1/n2 ) 1.51/1.33 in the geometry presented in Figure 6, with focal length f)130 µm, numerical aperture of the lens NA of 1.25, and focusing depth d as indicated.

considerably larger than the thermal energy kBT and can, therefore, serve as stable traps. The existence of stable secondary traps requires that the potentials exhibit minima in the lateral directions as well. The fact that such minima exist in directions normal to the beam can be seen from Figure 7b, for example. The contour lines in Figure 7b at z around -5 µm (where the existence of a secondary minimum is indicated in Figure 10b, for d ) 20 µm) show the existence of a minimum along the x-direction. (The axisymmetry of the field guarantees a similar situation along the y-direction.) Figure 10a presents the axial force acting on a particle caught in the trapping potential for one of the conditions used in Figure 9. In addition to revealing the presence of secondary traps, both Figures 9 and 10a illustrate that asymmetries in the trap potentials are possible, that is,


Figure 9. Interaction potentials corresponding to a silica particle with refractive index 1.42 and radius 0.75 µm that is trapped by a 100-mW laser beam with wavelength λ ) 830 nm in a vacuum and focused through a dielectric interface n1/n2 ) 1.51/1.33 for the geometry illustrated in Figure 6. The focal length is f ) 130 µm, the numerical aperture of the lens NA is 1.25, and the potentials correspond to focusing at different focusing depths d as indicated.

Figure 10. (a) Axial forces corresponding to the physical conditions used in Figure 9 for d ) 15 µm, indicating the secondary trap locations. (b) Axial forces for three different values of d to illustrate variations in positions and shapes of primary and secondary traps.

one can have significant departures from the harmonicity of the trap potential. This situation appears to be important mainly in the case of secondary minima, and our calculations show that significant errors can be introduced by a naive description of the axial restoring force such as, for instance, the one given in eq 11. It is also evident from Figure 10b that the locations and shapes of the secondary minima depend on the virtual focal distance d. Figure 10b also indicates that the stiffnesses of the primary traps remain essentially constant (within a few percent of each other) as the distance d is changed. The calculated stiffness (∼1.4 × 10-2 pN/nm) is consistent with experimental observations (∼0.7 × 10-2 pN/nm) for 1.5 µm diameter silica particles in water.11 (A more quantitative comparison cannot be made unless more accurate estimates of the laser power and the refractive index difference are known at the experimental conditions.) We emphasize, however, that the prediction and analysis of the trapping potential should be made on a caseby-case basis and must include the laser power, the

Langmuir, Vol. 16, No. 6, 2000 2777

Figure 11. Shift in focal point caused by the dielectric interface for the geometry shown in Figure 6 (λ ) 830 nm in a vacuum and n1/n2 ) 1.51/1.33). Estimation based on geometrical optics (∆fGO, continuous line); wave optics calculation of the axial dependence of the maximum in |E|2 (∆fWO, open symbols); and primary trap locations (∆Trap, filled circles), that is, estimates based on eq 5 of the actual minima in the potential energy of interaction corresponding to a silica particle (with refractive index 1.42 and radius 0.75 µm) that is trapped by a 100-mW laser beam.

refractive indices of particles and interfaces, and the size of the trapped particle, as well as the trapping location relative to the interface. As noted at the end of the previous subsection, another interesting implication of the wave optics results is the difference between the actual focal shift, ∆fWO (as obtained from the wave optics calculations) and the approximate shift ∆fGO given by geometric optics. This difference could play an important role in the use of optical traps as force transducers, depending on how the trap is used to determine the external forces of interest. For example, in the type of applications described in Section 2.1, the position of the trap is usually fixed relative to the distance from the dielectric interface, and the trap potential is explicitly determined through calibration. In such cases, the above-mentioned difference between wave optics and geometric optics calculations is not an issue. However, in the type of applications described in Section 2.2, one systematically varies the trap location htr to determine the external force of interest at various locations normal to an interacting surface. The trap location htr in such cases is determined from the mechanical displacement of the focusing objective with respect to the interface through which the trapping beam is focused (see Figure 6), and the effect of the presence of the dielectric discontinuity at a distance d is usually accounted for by using geometrical optics (i.e., a simple geometrical correction is introduced to account for the shift in focal point caused by the refractive index mismatch). Such a correction, for the physical conditions we have considered so far, is shown as a continuous line in Figure 11, that is, this line is based on eq 12. A more realistic approach is to actually calculate the position of the axial intensity maxima as given by our wave optics calculations; this is depicted by open symbols in Figure 11. However, the shift ∆Trap in the actual trapping positions, indicated by filled circles in Figure 11 (as given by F(z) ) 0), is usually different especially for particles that are comparable or larger than the width of the oscillations in the axial intensity. Not properly accounting for these effects could lead to sufficiently significant errors in the determination of forces, as illustrated in the following example. Consider a trapping laser beam focused at a virtual focal distance of d ) 20 µm from a glass/water interface. According to Figure 11, the actual focus predicted by geometric optics is at about 17.4 µm, whereas the wave optics calculations show that the real focus is at about 16.7 µm. What is important in our context, however, is

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the following. One can calculate, from the slopes of the lines shown in Figure 11, the change in htr as the focal point of the beam is moved by varying d. If one were to assume naively that the trap location is given by the focal point and, in addition, if one further assumes that geometric optics is sufficient to determine the focal point, Figure 11 shows that each 100 nm change in d would shift htr by 88 nm (since the slope of the geometric optics line is about 0.12). On the other hand, if one uses the more appropriate wave optics calculations and the actual location of the trap, one obtains a shift in htr of only 83 nm. Although the difference is only 5 nm, the cumulative effect of such a difference could be significant. For instance, in measurements of the type described in Section 2.2, one first positions the trapped particle about a few hundred nanometers in front of another surface as a starting point in the measurement of the interaction force between that surface and the particle. At such a distance, the trapped particle experiences only the trap force. Then, the particle is moved closer to the surface in discrete shifts, and the deviation of the particle from the trap center is then used to determine the colloidal force, as described in Section 2.2. If d is changed by, say, 500 nm in this process, the error in htr is 5 × 5 ) 25 nm. For a typical trap stiffness, ktr, of about 0.006 pN/nm, this corresponds to an error in force of 0.15 pN. This is of the same order of typical magnitudes of the van der Waals force measured close to the surface and therefore could represent a significant error in the measured force. Note that shifts in d of larger than 1000 nm may often be needed in the experiments, thus contributing to larger errors in the estimated forces than noted above. Moreover, additional errors are possible depending on the specific values of the particle size and refractive indices. Finally, for the value of d chosen for this example (i.e., 20 µm), it so happens that the location of the (primary) trap is very close to 16.7 µm, that is, at the same position as the real focus predicted by wave optics calculations (see Figure 11). However, as evident from Figure 11, for larger values of the virtual focal distance d, the shift in trap location, ∆Trap, exceeds ∆fWO. Therefore, unless the exact trap location is used for calculating the shift in htr, one can expect additional errors in the estimated colloidal forces. In summary, we have discussed several aspects of optical traps used for force measurements in polymer and colloid science and in biophysics. In doing so, we have neglected external fields such as gravitational and magnetic fields, but their incorporation for specific experimental geometries is straightforward. We have shown that, in practical situations, because of the inherent presence of a dielectric interface arising from the wall of the sample cell and the liquid containing the trapped particle, there is a strong disturbance of the light distribution. As a result, the commonly used simple Gaussian shape of the trapping potential may not be an acceptable description. More important is the presence of secondary trapping positions and the possible nonharmonic behavior of the corresponding trapping forces. Moreover, a nontrivial depen-


dence is found between the mechanical translation of the focusing optics and the corresponding shift in the trap position. 5 Concluding Remarks In addition to the issues discussed so far, there is another effect induced by the electric field incident on the trapped particle that could be important. As is well-known, light carries momentum as well as energy, and the total forward momentum transfer is proportional to the total extinction cross-section minus the forward scattering amplitude and results in a scattering force Fsca (see Figure 1). For plane waves incident on a particle, the magnitude of this radiation pressure is Fsca ) I0[Cabs + Csca(1 - cosθ)], where Cabs and Csca are, respectively, the absorption and scattering cross-sections. (The average of the cosine of the scattering angle cosθ is a measure of the ratio of the particle size to the wavelength.28) In most cases of practical interest Cabs , Csca and, from the general theory of scattering, one recalls that Csca ∼ (m2 - 1)2, where m is the relative refractive index of the particle (i.e., m ) n2/ n1). Therefore, Fsca ∼ (m2 - 1)2. On the other hand, from eq 5 it follows that the gradient force is of the order of (m2 - 1), and thus, for the case of silica in water, this force is 1 order of magnitude larger than Fsca. In the case of large particles, one expects the latter to be even lower, since the incident intensity I0 ) 2P/π$2 is not uniformly distributed over the particle’s surface (see eq 7 for the particular case of a Gaussian beam). It is worth noting that, in general, there are also components of the radiation pressure that act transversely, but their magnitudes are smaller in the case of large particles; therefore, one expects even less influence on the trapping potentials. In conclusion, we have shown that a careful interpretation of the trap location and its shift relative to a shift in the virtual focus of the trapping beam, trap field, and trap potential is required for correct analysis of force data obtained from trapping experiments in liquid cells. A number of specific issues have been identified as potential sources of errors in assessing the trap position, harmonicity, and stiffness. Additional issues of interest include questions such as What is the influence of the above in the lateral directions? and What is the extent of additional changes arising from a second dielectric discontinuity introduced by the second confining glass plate in the experimental setup? These questions will be addressed in a future communication. Acknowledgment. This work was partially motivated by our work on colloidal forces and rheology and optical characterization of dispersions, supported in part by the National Science Foundation Engineering Research Center for Particle Science and Technology (NSF ERC-PST) at the University of Florida, Gainesville, Florida. LA9904004 (28) van de Hulst, H. C. Light Scattering by Small Particles; Dover: New York, 1981.

Optical Traps as Force Transducers: The Effects of Focusing the

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