Simplified Description of Optical Forces Acting on a Nanoparticle in The

Zemanek et al. Vol. 19, No. 5/May 2002/J. Opt. Soc. Am. A 1025

Simpliﬁed description of optical forces acting on a nanoparticle in the Gaussian standing wave

Pavel Zema´nek and Alexandr Jona´sˇ Institute of Scientiﬁc Instruments, Academy of Sciences of the Czech Republic, Kra´ lovopolska´ 147, 612 64 Brno, Czech Republic

Miroslav Lisˇka Brno University of Technology, Faculty of Mechanical Engineering, Technicka´ 2, 616 69 Brno, Czech Republic

Received March 23, 2001; revised manuscript received October 16, 2001; accepted October 16, 2001 We study the axial force acting on dielectric spherical particles smaller than the trapping wavelength that are placed in the Gaussian standing wave. We derive analytical formulas for immersed particles with relative refractive indices close to unity and compare them with the numerical results obtained by generalized Lorenz– Mie theory (GLMT). We show that the axial optical force depends periodically on the particle size and that the equilibrium position of the particle alternates between the standing-wave antinodes and nodes. For cer- tain particle sizes, gradient forces from the neighboring antinodes cancel each other and disable particle con- ﬁnement. Using the GLMT we compare maximum axial trapping forces provided by the Gaussian standing- wave trap (SWT) and single-beam trap (SBT) as a function of particle size, refractive index, and beam waist size. We show that the SWT produces axial forces at least ten times stronger and permits particle conﬁne- ment in a wider range of refractive indices and beam waists compared with those of the SBT. © 2002 Optical Society of America OCIS codes: 140.7010, 170.4520, 260.2110, 260.3160, 290.4020, 290.5850.

1. INTRODUCTION for very small particles—so-called Rayleigh particles— whose radius fulfils a р␭/20, where ␭ is the trapping Within the past two decades optical trapping has proved wavelength in the medium.21,22 Such a small particle be- to be an invaluable method for noncontact manipulation haves as an induced elementary dipole, and the optical of nano-objects1,2 and micro-objects,1,3 measurement of extremely weak forces,4,5 and study of single molecule forces acting on it can be divided into two components, properties6,7 and surface properties.8–11 The most fre- gradient and scattering forces. The gradient force comes quently used trapping set-up is based on a single laser from electrostatic interaction of a particle (dielectric) with beam tightly focused by an immersion microscope objec- an inhomogeneous electric field, and the scattering force 1 results from the scattering of the incident beam by the tive of high numerical aperture. This classical single- 21,23 beam trap (SBT) set-up has been gradually modified by object. ␭ using interference of co-propagating laser beams,12 self- For particles larger than /20, a more complex concept aligned dual beams,13 or optical fibers.14,15 Several laser involving the Maxwell stress tensor of the electromag- beams,16,17 diffractive optics,18 and time sharing of a netic field surrounding the particle must generally be single beam16,19 were employed to create several optical used. This requires knowledge of the total field outside traps. The common characteristic of all the above meth- the confined particle. The original theory based on 24 ods is that the axial force exerted on the object is smaller plane-wave scattering has been gradually modified so than the radial one. Recently it has been shown that the that it can be applied to the case of a spherical or sphe- optical trapping can also be achieved in the standing roidal particle placed into an arbitrary field 25–27 wave created by interference of two counter-propagating distribution. It is commonly referred to as the gen- coherent beams.20 In this case the axial force is stronger eralized Lorenz–Mie theory (GLMT). 28 than the radial one because of strongly inhomogeneous A recently presented intuitive approach assumes that optical intensity distribution in the periodic structure of the electric field creates such a large gradient force that the standing-wave nodes (minimums) and antinodes the contribution of the scattering force to the total optical (maximums). Therefore particle confinement is achieved force is negligible. Consequently, only the gradient force even in weakly focused or aberrated beams. An interest- controls the behavior of an irradiated dielectric object. ing problem that arises here is to study the properties of The stress tensor contains only the terms depending on this type of optical trap [standing-wave trap (SWT)] and the electric field, and the problem is then reduced to the compare them with the classical SBT for different param- evaluation of the change in electrostatic energy after in- eters of the trapped object (size, refractive index) and sertion of the dielectric into the inhomogeneous electro- trapping beam (waist size). static field. Because the dielectric is assumed to be Theoretical description of optical forces is simple only weak, its placement into the field does not change the ini-

0740-3232/2002/051025-10$15.00 © 2002 Optical Society of America 1026 J. Opt. Soc. Am. A/Vol. 19, No. 5/May 2002 Zemanek et al. tial field distribution appreciably. This allows use of an Assuming that the incident beam is polarized mainly in analytic expression for the optical force acting on spheri- the transverse direction (paraxial beam), we obtain, using cal particles in an isotropic Gaussian field or on a cube in the relation between the optical intensity and the field axially symmetric fields without further restriction on the vectors, particle size. This method will be called electrostatic approximation (ESA) in this paper. ϵ I0͑r͒ ͗S͑r, t͒͘T The application of this approach is limited to cases where the scattering force can be neglected. It has been ϵ ϫ ͗E0͑r, t͒ H0͑r, t͒͘T , shown22 that if the Gaussian standing wave (GSW) is used for the optical trapping, the scattering force acting Ӎ z ͗n ⑀ c͉E ͑r, t͉͒2͘ on the Rayleigh particle is proportional to the difference 0 2 0 0 T ϭ ⑀ ͉ ͉2 ϭ in the energy fluxes of counterpropagating waves. If the z0n2 0c E0͑r͒ /2 z0I0͑r͒. (4) two waves have the same intensity, the scattering force is zero. A similar trend can also be expected for larger par- Here z0 is the unit vector in the direction of the light ticles. Thus the ESA framework is well suited to the propagation and c is the speed of light. The equation for evaluation of the optical forces in the GSW. the energy change consequently takes the final form

␣ n2 ⌬W͑r͒ ϵ ͗⌬W͑r, t͒͘ ϭϪ ͵ I ͑r͒dV. (5) 2. ELECTROSTATIC APPROXIMATION T 2 c 0 APPLIED TO A SPHERICAL DIELECTRIC V1 PARTICLE PLACED IN THE GAUSSIAN STANDING WAVE The time-averaged force can be expressed by the diver- gence theorem A. General Description of the Optical Forces by Using Electrostatic Approximation F͑r͒ ϵ Ϫٌ͗⌬W͑r, t͒͘ Let us assume that a dielectric object is placed into an in- T ␣ ␣ homogeneous electromagnetic field in a dielectric liquid. n2 n2 (If the sources of the field are kept fixed, the change in to- ϭ ͵ ٌI ͑r͒dV ϭ ͵ nI ͑r͒dS, (6 2 c 0 2 c 0 tal energy of the field due to the insertion of the object can V1 S1 be written as the difference between the energy of the field with the inserted medium and the initial energy where n is the outward unit normal to the surface ele- with the liquid dielectric29: ment dS.

1 B. Analytical Formulas for a Spherical Dielectric ⌬ ϭ ͵ ͑ Ϫ ϩ Ϫ ͒ W ED E0D0 HB H0B0 dV, (1) Particle Placed in the Gaussian Standing Wave 2 V Classical optical trapping requires a focused laser beam with a spot diameter comparable to the trapping wave- where E (D) and E0 (D0) are the electric field vector (elec- length. To achieve this, high-quality immersion objec- tric displacement) and H (B) and H0 (B0) are the mag- tives are usually used. In this case the beam goes netic field vector (magnetic induction) after and before in- through many dielectric interfaces (optical elements insertion of the object, respectively. The integration is over side the objective, immersion oil layer, coverslip, and wa- the whole space occupied by the field. During observa- ter layer), and the actual field distribution can differ con- tion of the object we cannot follow the rapid optical fre- siderably from the theoretical model, usually represented quencies; therefore, only the time-averaged values are ac- by a Gaussian beam.30 However, since we want to cessible. It can be shown (see Appendix A) that if the present here only the basic properties of the SWT, we object and the liquid are nonmagnetic, the time-averaged change in the field energy can be rewritten as

1 ͗⌬ ͑ ͒͘ ϭϪ ⑀ ␣ ͵ ͗ ͑ ͒ ͑ ͒͘ W r, t T 2 E r, t E0 r, t TdV, (2) 2 V1

␣ ϭ ⑀ ⑀ Ϫ ⑀ ϭ 2⑀ where 1 / 2 1, i ni 0 is the permittivity of the particle (i ϭ 1), the liquid (i ϭ 2), and the vacuum (i ϭ 0). Symbols n1 and n2 represent the refractive indices of the particle and the liquid, respectively. The integration is now performed over the volume V1 of the particle. If the particle is not strongly polarized, i.e., n1 Ӎ n2 , the perturbed electric ﬁeld E can be approximated by the unperturbed one E0 and we obtain Fig. 1. Orientation of axes in the standing-wave apparatus.

1 The z axis follows the direction of the reﬂected wave. Positive z0 ͗⌬ ͑ ͒͘ ϭϪ ⑀ ␣ ͵ ͉͗ ͑ ͉͒2͘ W r, t T 2 E0 r, t TdV. (3) means that the beam waist of radius w0 is located in the reﬂected 2 V1 wave. Zemanek et al. Vol. 19, No. 5/May 2002/J. Opt. Soc. Am. A 1027

2 have chosen a moderately focused Gaussian beam to krB 1 1 ␾ ͑ ͒ ϭ Ϫ ͩ Ϫ ͪ reach a compromise among the exact description, speed of GSW rB , zB 2kzB calculation, and availability of analytical results. 2 Ri Rr Let us assume that the GSW is created by interference ϩ zB z0 of an incident wave and a wave reflected at a dielectric Ϫ arctanͩ ͪ surface (see Fig. 1), and let us omit any multiple scatter- zR ing between the object and the surface. Let the orienta- Ϫ zB z0 tion of the z axis be parallel to the direction of the re- Ϫ arctanͩ ͪ ϩ ␺. (10) flected beam. Therefore the ‘‘trapping force,’’ which acts zR against the incident wave, has positive sign in this coor- Here, I is the on-axis intensity at the position of the dinate system. The incident and the reflected Gaussian 00 22 beam waist and is related to the total power of the inci- beams can be written as ϭ ␲ 2 dent Gaussian beam by I00 2P/ w0 . 2 Because we assume that the dielectric object has w0 rB E ͑r , z ͒ ϭ E expͩ Ϫ ͪ expͫ ik͑z ϩ z ͒ spherical shape, it is useful to rewrite radial and axial cy- i B B 00 w w 2 B 0 i i lindrical coordinates (rB , zB) referenced to the beam by ␹ ␾ 2 ϩ using spherical coordinates (r, , ) referenced to the i krB zB z0 Ϫ Ϫ i arctan ͩ ͪͬ, sphere center, which is shifted from the center of the cy- 2 Ri zR lindrical coordinate system by rs laterally and zs axially: r 2 ϭ r2 sin2 ␹ ϩ r 2 ϩ 2rr sin ␹ cos ␾ ), z ϭz ϩ r cos ␹. 2 B s s B s w0 rB ͑ ͒ ϭ ␳ ͩ Ϫ ͪ ͫ Ϫ ͑ Ϫ ͒ In this case the final forms of the energy change [Eq. (5)] Er rB , zB E00 exp 2 exp ik zB z0 wr wr and the axial force [Eq. (6)] are 2 Ϫ i krB zB z0 ␣ a ␲ Ϫ ϩ i arctanͩ ͪ Ϫi␺ͬ, n2 ⌬ ͑ ͒ ϭϪ ␲ ͵ ͵ ͑ ͒ 2 ␹ ␹ 2 Rr zR W rs , zs 2 IGSW rB , zB r sin d dr, 2 c 0 0 (7) (11)

where E00 is the electric field amplitude at the beam ␣ ␲ waist position, k is the wave number in the medium, ␳ is n2 ͑ ͒ ϭ ␲ ͑ ͒ 2 ␹ ␹ ␹ the reflectivity of the surface, ␺ is the phase shift after re- Fz rs , zs 2 ͵ IGSW rB , zB a sin cos d . 2 c 0 flection [the Fresnel reflection coefficient has the form (12) ϭ ␳ Ϫ ␺ rm exp( i )], and z0 is the distance of the beam waist from the surface (mirror). Positive (negative) z corre- 0 These integrals can be expressed analytically only if one sponds to beam waist created in the reflected (incident) of the following approximations is employed. wave (see Fig. 1). Symbol w0 is the beam waist size and wi and wr are the widths of the incident and the reflected beams at a given z , respectively. R and R are the ra- B i r Ӷ Ӷ ␭ dii of the wave-front curvature for the incident and the re- 1. a w0 , a (Rayleigh Particle) flected waves, respectively, The particle is so small that the intensity can be consid- ered constant over the integration volume, and therefore Ϯ 2 1/2 it can be taken out of the integral in Eqs. (11) and (12). ͑zB z0͒ ͑ ͒ ϭ ͫ ϩ ͬ wi/r zB w0 1 2 , Consequently, we obtain zR

2 zR 2 n2 R ͑z ͒ ϭϯ͑z Ϯ z ͒ͫ 1 ϩ ͬ, (8) ⌬W͑r , z ͒ ϭϪ ␣␲a3I ͑r , z ͒, (13) i/r B B 0 Ϯ 2 s s GSW s s ͑zB z0͒ 3 c ϭ 2 and zR kw0 /2 is the Rayleigh length. Under these assumptions we can write for the total intensity distribution 2 n2 (F͑r , z ͒ ϭ ␣␲a3ٌI ͑r , z ͒. (14 s s 3 c GSW s s ⑀ n2 0c ϭ ͉ ϩ ͉2 IGSW͑rB , zB͒ Ei͑rB , zB͒ Er͑rB , zB͒ 2 Equation (14) is identical to the Harada et al. result for the gradient force in a paraxial beam21 if we use the sub- 2 w0 stitution ϭ ͓Ϫ͑ 2 2͔͒ I00 2 exp 2rB /wi wi 2 2 2 2 ␣ ϭ m Ϫ 1 Ӎ 3͓͑m Ϫ 1͒/͑m ϩ 2͔͒, (15) w0 ϩ 2␳I exp͓Ϫ͑r 2/w 2͔͒ 00 B i which, however, is correct only for m Ӎ 1. The difference wiwr between the left-hand and right-hand sides of Eq. (15) is ϫ ͓Ϫ͑ 2 2͔͒ ␾ exp rB /wr cos GSW 7% and 14.7% for m ϭ 1.1 and m ϭ 1.2, respectively. 2 Therefore the validity of ESA within the Rayleigh ap- w0 ϩ ␳2 ͓Ϫ͑ 2 2͔͒ proximation can be extended into a region of higher re- I00 2 exp 2rB /wr , (9) wr fractive index by substituting Eq. (15) into Eq. (14). 1028 J. Opt. Soc. Am. A/Vol. 19, No. 5/May 2002 Zemanek et al.

Ӷ Ӎ ␭ 2. a w0 , a (Intermediate Particle in Moderately Focused Beams) This assumption means that over the integration volume the intensity does not change materially in the transverse direction, and therefore purely transverse-coordinate- dependent terms can be taken out of the integral in Eqs. (11) and (12). As the particle size is comparable to the wavelength, there is an intensity change in the axial direction that cannot be neglected over the integration volume. If all the terms connected with Eq. (9) are transformed into the spherical coordinates (see Appendix B) Ӷ and the assumption a w0 is applied, we obtain the following expressions:

n2 1 ⌬ ͑ ͒ ϭϪ␣ ͫͩ ͓Ϫ͑ 2 2͔͒ W rs, zs P 2 exp 2rs /wi c wi Fig. 3. Behavior of two polystyrene spheres of slightly different radii (a ϭ 0.3␭, 0.35␭) placed symmetrically with respect to the 1 4 standing-wave node. The following parameters were used for ϩ ␳2 ͓Ϫ͑ 2 2͔͒ ͪ 3 exp 2rs /wr a ϭ ␭ ␳ ϭ ␺ ϭ ␲ ϭ ␮ w 2 3 the force calculation: w0 , 1, 3 /2, z0 0 m, m r ϭ ϭ ␭ ϭ 1.95, P 1W, vac 1064 nm. ␳ ϩ exp͓Ϫr 2͑1/w 2 ϩ 1/w 2͔͒ k3w w s i r i r The combined behavior of both terms has the effect that the center of a particle of radius less than 0.3576␭ ϫ ͑ Ϫ ͒ ␾ ͬ sin 2ka 2ka cos 2ka cos s , (16) moves toward the place of maximum optical intensity, ϩ ␺ ϭ ␲ ϭ that is, GSW antinode (2kzs 2 N, N 1, 2,...), ␣ while the center of a particle of radius between 0.3576 2n2 P ͑ ͒ ϭϪ ␳ ͓Ϫ 2͑ 2 ϩ 2͔͒ Ͻ ␭ Ͻ Fz rs , zs 2 exp rs 1/wi 1/wr a/ 0.6148 is pushed to the intensity minimum, c k wiwr ϩ ␺ ϭ ␲ ϩ ␲ that is, GSW node (2kzs 2 N, N ϫ Ϫ ␾ ϭ 1, 2,...), and so forth. An intuitive rule for this ten- ͑sin 2ka 2ka cos 2ka͒sin s , (17) dency can be stated: The sphere in the GSW moves so ␾ ϭ ␾ where s GSW(rs , zs). This approximation enables that it covers the maximum number of GSW antinodes. us to perform the integration analytically even for off-axis This effect is illustrated in Fig. 3, where two pairs of Þ positions of the sphere center (i.e., rs 0). polystyrene spheres of slightly different radii (a According to Eq. (17) the sign of the force is determined ϭ 0.3␭, 0.35␭) are placed symmetrically with respect to ␾ by the product of two terms. The first one, sin s , comes the node (the sphere centers are depicted by ϩ and ϫ and from the spatial intensity profile described by the inter- their surfaces by continuous and dashed circles, respec- ference term in Eq. (9), and it determines the positions of tively). The upper plots show the position of the spheres zero, minimum, and maximum optical forces along the op- with respect to the GSW nodes (lighter regions) and anti- tical axis. The second term, sin 2ka Ϫ 2ka cos 2ka,may nodes (darker regions). The lower plots show the axial be set equal to Ga and depends on the ratio of the particle force acting on these objects for various distances of the size and the trapping wavelength in the medium. Its objects from the mirror. The arrows in the top plots show plot is shown in Fig. 2. Extremes of function Ga can be the direction of the total optical force acting on the ␭ ϭ ϭ found for aextr / M/4, M 1, 2,.... The zero value of sphere. The solid small circles depict the stable equilib- Ga is obtained for the following values of the sphere ra- rium position of the sphere, which is called the trap posi- ␭ ϭ dius: azero / 0, 0.3576, 0.6148, 0.8677, 1.1194,.... If a tion. The optical trap position is shifted by ϳ␭/4 from an- sphere of this particular size is placed into the GSW, the tinode to node if the smaller particle is exchanged with change in the field energy does not depend on the sphere the larger one. position in the GSW. Consequently, the axial optical force acting on the sphere is zero for all particle positions in the GSW, and obviously this particle cannot be trapped. Ӎ Ӷ 3. a w0 , a zR Let us consider the slightly more general case of a sphere that is so large that the intensity changes in the lateral direction cannot be neglected over the integration volume. To obtain analytical results we assume that the beam widths wi(z) and wr(z) do not change with respect to the particle size (see Appendix B) and that the particle is situ- ated on the beam axis. Improved results with respect to the previous case can be expected only in a narrow range of particle sizes. Following Eqs. (11) and (12) the energy Fig. 2. Profile of the size-dependent term Ga in the analytical form of the axial force acting on a dielectric sphere in the GSW. change and the axial force can be expressed as Zemanek et al. Vol. 19, No. 5/May 2002/J. Opt. Soc. Am. A 1029

⌬ W͑0, zs͒ lapped beam waists to get the initial field components of the standing wave. Moreover, we assumed that the ͱ spherical object was located on the beam axis so we could n2 wi 2a ϭϪ␣ ͫ ϩ ␳2͒ Ϫ employ the radial symmetry of the problem and simplify P a͑1 ͫ dawͩ ͪ 33 c ͱ2 wi the calculation. We wrote the modified code ourselves, but we do not present a detailed mathematical descrip- ͱ tion, because the method is well described in the wr 2a ϩ ␳2 dawͩ ͪͬ literature.26,27,31,33 ͱ2 wr In this study we neglected any electrostatic interac- tions between the surface and the particle as well as mul- 2 ␳ 2W tiple scattering of the incident beam. Furthermore, we ϩ ͩ sin͑2ka͒͑cos ␾ Ϫ Z sin ␾ ͒ ϩ 2 s W s assumed that the beam waist was placed on the surface wiwr k͑1 ZW ͒ with reflectivity equal to 100%. The axial positions zs of 2W3 the sphere center, where we calculated the axial forces, ϩ Reͭ exp͑i␾ ͓͒exp͑i2ka͒daw͑X ϩ Y͒ s satisfy the inequality a р z р (a ϩ ␭). C3/2 s Although the adopted simplifications (CGB, absence of spherical aberrations, diffraction, and multiple-scattering ϩ Ϫ ͒ Ϫ ϩ ͒ ͬ events) may seem drastic, this model provides at the least exp͑ i2ka daw͑ X Y ͔ͮ ͪ , (18) a correct qualitative description of the behavior of dielectric spheres in the GSW34 and an acceptable speed of calculation. Fz͑0, zs͒ ␳ 2 n2 W ϭ 4␣ P ΀Ϫ sin ͑2ka͒ ϩ 2 c wiwr 1 ZW B. Validity of the Electrostatic Approximation ϫ ␾ ϩ ␾ ͑sin s ZW cos s͒ Validity of the ESA is limited mainly by the assumptions that the object does not change the field distribution and 3 ikW that the incident beam is polarized only in the transverse ϩ Reͭ exp͑i␾ ͓͒exp͑i2ka͒daw͑X ϩ Y͒ C3/2 s direction [see Eq. (4)]. Therefore, it is desirable to study the differences between the ESA and a more precise method (GLMT) as a function of object size, refractive in- ϩ exp͑Ϫi2ka͒daw͑ϪX ϩ Y͔͒ͮ ͪ , (19) dex, and Gaussian beam waist size. Rough comparison of the GLMT and the ESA methods ␾ where X, Y, W, C, ZW , and s are defined in Appendix B. revealed that the ESA methods are applicable only if the object is smaller than 0.25␭. For larger spheres, only very small relative refractive indices must be used. Since the maximum trapping force is nonzero for particles of applicable sizes, we can compare the methods by using 3. GENERALIZED LORENZ–MIE THEORY maximum relative error of the ESA trapping force defined APPLIED ON A SPHERE PLACED IN by the formula THE GAUSSIAN STANDING WAVE AND THE SINGLE BEAM A. Calculation of Optical Forces by Using Generalized Lorenz–Mie Theory The GLMT uses the scattering procedure presented first by Mie24 who derived expressions for the field distribution outside a spherical object of arbitrary size placed into a plane wave. This original method was generalized so that it permits a mathematical description of the forces acting on spherical and oblate objects placed in a general electromagnetic field.25,27,31 For the description of a focused beam, a modification to the fundamental Gaussian beam was introduced [so called fifth-order corrected Gaussiam beam (CGB)].26,32 It uses a field expansion in ϭ the beam size parameter s 1/kw0 to the fifth order, achieves better agreement with the wave equation, and therefore provides more precise calculation of the optical forces. Because we consider the standing wave created by the interference of two counterpropagating focused la- Fig. 4. Maximum relative error of the ESA methods (in percent) ser beams, we have easily adapted the GLMT formalism as a function of the sphere radius and relative refractive index ϭ ␭ to this case. Instead of a single CGB we summed field for four beam waist sizes (w0 0.75, 1, 1.25, 2.5 ) and the follow- ␳ ϭ ␺ ϭ ␲ ␭ ϭ ϭ ␮ components of two counterpropagating CGBs with over- ing parameters: 1, 3 /2, vac 1064 nm, z0 0 m. 1030 J. Opt. Soc. Am. A/Vol. 19, No. 5/May 2002 Zemanek et al.

max(͉F ͑a р z р a ϩ ␭͒ Ϫ F ͑a р z р a ϩ ␭͉͒) ⌬ ϭ apr s GLMT s Fz , (20) Fmax

ϭ р р ϩ ␭ where Fmax max(FGLMT(a zs a )) and Fapr is the force value calculated using a particular approximation ‘‘apr.’’ Relative errors are summarized in Fig. 4 for various beam waist sizes as a function of the sphere radius and the relative refractive index. The numbers on the contour curves mark the relative error in percent. We compared the GLMT with three ESA methods labeled ESA-int, ESA-daw, and ESA-anal calculated by using Eqs. (12), (B8), and (17), respectively. ESA-int is the р⌬ р most precise method, giving 5% Fz 22% for w0 ϭ ␭ р⌬ р ϭ ␭ 0.75 and 0.5% Fz 14% for w0 2.5 . The ESA-daw method gives more precise results than the ESA-anal one for intermediate sphere radii (0.1 р a/␭ р 0.23) and smaller m, but negligible improvement is found for greater m. The relative error of the ESA-daw р⌬ р ϭ ␭ method satisﬁes 10% Fz 34% for w0 0.75 and ⌬ р ϭ ␭ 0.5% Fz 16% for w0 2.5 . We see that the wider the beam waist size, the better the coincidence with the GLMT, but no signiﬁcant improvement was found for w0 Ͼ 2.5␭. We also checked whether intuitive substitution of ␣ by 3(m2 Ϫ 1)/(m2 ϩ 2) in Eqs. (12), (B8), and (17), according to Eq. (15), improves the coincidence between the GLMT and the ESA methods. We denote these methods ESAn-int, ESAn-daw, and ESAn-anal, and results are shown in Fig. 5. We can see that precision increased on average by more than three times, especially for higher refractive indices and intermediate particles (0.1␭рa р 0.22␭). The relative error is smaller than 5% for the majority of ESAn-int results. Contrary to the case with ESA-daw, the ESAn-daw method slightly increases the precision in a narrow region of higher m for small w0 and smaller m for wider w0 . Together with the ESAn-anal method the relative error with the ESAn-daw method is smaller than 10% for wider beam waist sizes. No simple Fig. 6. Contour plot of maximum SWT axial trapping force (in piconewtons) with respect to the z axis as a function of the relative refractive index, particle size, and beam waist radius calculated by using the GLMT. Dotted–dashed and dashed lines represent the loci of maximum and minimum values of the maximum trapping force with respect to the particle size and relative refractive index. The shaded areas represent the nontrapping regions. The following parameters were used: P ϭ ␳ ϭϭ ␺ ϭ ␲ ␭ ϭ ϭ ␮ 1W, 1, 3 /2, vac 1064 nm, z0 0 m.

tendency with respect to the beam waist size was found, but there is a region of minimum error that moves from the area of higher m for smaller w0 toward the area of ⌬ smaller m for higher w0 . The steep increase of Fz for larger a and m in Figs. 4 and 5 is caused by the fact that Fmax , which appears in the denominator of Eq. (20), decreases here.

C. Comparative Study of the Maximum Trapping Force Fig. 5. Maximum relative error of the ESA methods (in percent) if the ␣ ϭ m2 Ϫ 1 term is replaced by 3(m2 Ϫ 1)/(m2 ϩ 2) as a in the Standing-Wave Trap and the Single-Beam function of the sphere radius and relative refractive index for Trap ϭ ␭ four beam waist sizes (w0 0.75, 1, 1.25, 2.5 ) and the following Figure 6 presents the contour plots of the maximum axial ␳ ϭ ␺ ϭ ␲ ␭ ϭ ϭ ␮ parameters: 1, 3 /2, vac 1064 nm, z0 0 m. trapping force acting on the spherical particles placed Zemanek et al. Vol. 19, No. 5/May 2002/J. Opt. Soc. Am. A 1031 into the GSW for various values of particle size, relative refractive index, and beam waist size. The numbers represent the levels of constant forces in piconewtons, and the shaded regions correspond to the negative maximum trapping force. For these parametric configurations the particle is accelerated toward the surface without a chance of its confinement in the GSW (nontrapping regions). This is caused by the gradient forces from the neighboring GSW antinodes, which pull the object in op- posite directions and thus cancel each other. Conse- quently, the weaker gradient force due to the focused beam envelope dominates and accelerates the particle to- wards the beam waist placed on the mirror. The larger the sphere, the wider is this nontrapping region. We also see that this region becomes narrower with increasing beam waist since the gradient of the Gaussian envelope of the intensity becomes smaller. Nontrapping predictions of the ESA models fit the GLMT results only for m Ӎ 1. Figure 6 shows that the difference between the consequent force-maximizing or -minimizing sphere radii is smaller than ESA predictions (equal to ␭/4) and still decreases with increasing value of the relative refractive index. Figure 7 shows the maximum trapping force in the SBT for comparison. We see that the wider the beam waist, the lower must be the relative refractive indices used to confine the object in the SBT. Direct comparison with the SWT forces (Fig. 6) acting on larger particles (outside the nontrapping regions) reveals that the SWT provides trapping forces that are at least one order of magnitude stronger. For smaller particles and wider beam waists this disproportion becomes even more pronounced.

4. SUMMARY AND CONCLUSIONS Fig. 7. Contour plot of maximum SBT axial trapping force (in In the work described in this paper the trapping proper- piconewtons) with respect to the z axis as a function of relative ties of the Gaussian standing wave (GSW) were studied refractive index, particle size, and beam waist radius calculated theoretically by using electrostatic approximation (ESA) by using the GLMT. The dotted–dashed curves represent the loci of maximum and minimum vales of the maximum trapping and generalized Lorenz–Mie theory (GLMT). On the ba- force with respect to particle size and relative refractive index. sis of the ESA method, analytical formulas for optical The dotted curves denote the borders of the trapping regions, and forces acting on dielectric particles with relative refrac- the shaded areas represent the nontrapping regions. The inset tive indices close to unity were derived. The validity of plots in the bottom two plots magnify the regions of interest with the ESA and its analytical formulas was tested by com- the same horizontal scale. The following parameters were used: P ϭ 1W, ␭ ϭ 1064 nm. parison with the numerical results obtained from the vac GLMT. Despite the drastic simplification, it turned out that the ESA methods are convenient for fast estimation sizes and refractive indices outside the nontrapping re- of axial forces acting on particles of radii a р 0.25␭ and gions, where the effect of the GSW is not minimized. relative refractive indices m р 1.21 placed into the GSW у ␭ ␣ ϭ 2 Ϫ of beam waist w0 0.75 . If the term (m 1) is replaced by the term 3(m2 Ϫ 1)/(m2 ϩ 2) taken from the APPENDIX A: TIME-AVERAGED CHANGE Rayleigh scattering theory, the relative error of the ESA IN FIELD ENERGY method with respect to the GLMT decreases below 16%. If object and immersion media are linear (i.e., D ϭ ⑀E, The ESA model straightforwardly explains why spheres B ϭ ␮H) and both media are nonmagnetic (␮ ϭ ␮ ), Eq. ϳ ␭ 0 of radius smaller than 0.3 are trapped in the antinodes (1) can be rewritten in the form and spheres of radius larger than ϳ0.3␭, but smaller than ϳ ␭ 0.55 are confined in the nodes. 1 We found that the SWT, in contrast to the SBT, permits ⌬ ϭ ͵ ͑ Ϫ ͒ W ED0 DE0 dV confinement of particles of high refractive indices even in 2 V moderately focused beams and could therefore be em- 1 ployed for easier optical confinement of such particles in ϩ ͵ ͑ ϩ ͒͑ Ϫ ͒ E E0 D D0 dV. (A1) air. This conclusion, however, is valid only for particle 2 V 1032 J. Opt. Soc. Am. A/Vol. 19, No. 5/May 2002 Zemanek et al.

͐ Ϫ We get for the first integral in Eq. (A1) V(ED0 DE0) If the radius R of the surface goes to infinity, the area in- ϭϪ͐ ⑀ Ϫ ⑀ 2 ␾ Ϫ V ( 2 1)EE0dV. The integration is performed creases as R but the product T(D D0) for electro- 1 Ϫ3 over only the volume of the inserted dielectric V1 , be- static fields decreases as R . However, this relation- cause outside the inserted object the integrand vanishes ship is not valid for the radiation fields. To eliminate this ⑀ Ϫ ⑀ ϭ 35 as ( 1 1)EE0 0. integral, we apply Wolf’s arguments even though they If we employ the Maxwell equations and introduce po- cannot be regarded as the rigorous ones. We assume -t, the second integral in that the radiation field does not exist at all times but inץ/Aץ tentials A, ␾ as E ϭϪٌ␾ Ϫ Eq. (A1) can be rewritten as stead is produced by some source at time t0 . At any time Ͼ t t0 , the field fills the region of space reaching to c(t 1 Ϫ t ) from the source (c is the velocity of the light). If ͵ ͑ ϩ ͒͑ Ϫ ͒ 0 E E0 D D0 dV the integration surface is far enough from the source, the 2 V Ϫ field does not reach it within t t0 , and the integral can ץ ץ 1 A0 A be omitted. Therefore we can conclude that the time- ͒ ϭ ͵ ͩ Ϫٌ␾ Ϫ Ϫ ٌ␾ Ϫ ͪ ͑ Ϫ 0 D D0 dV, averaged energy change caused by the insertion of the di- tץ tץ V 2 electric object into the electric field is described by ϩ ץ 1 ͑A0 A͒ ͒ ϭ ͵ ͫ Ϫٌ͑␾ ϩ ␾ ͒ Ϫ ͬ͑ Ϫ 0 D D0 dV, t 1ץ V 2 ͗⌬ ͘ ϭϪ ͵ ͑⑀ Ϫ ⑀ ͒͗ ͘ WE T 2 1 EE0 TdV.(A7) V 2 ץ 1 1 AT i ͒ ϭϪ ͵ ٌ␾ ͑ Ϫ ͒ Ϫ ͵ ͑ Ϫ T D D0 dV D D0 dV, tץ V 2 V 2 Because of the time averaging, the equation takes the (A2) same form as in the electrostatic case. ␾ ϭ ␾ ϩ ␾ ϭ where new total potentials T 0 and AT A0 ϩ A were defined. Since we assume harmonic fields, the second term in Eq. (A2) can be rewritten as APPENDIX B: INTEGRATION IN THE GAUSSIAN STANDING WAVE IN ͒ ␻ ͑ץ 1 AT0 sin t SPHERICAL COORDINATES Ϫ ͵ ͑D Ϫ D ͒ sin ␻t dV ™ 0 0 ץ 2 V t 1. a w0 Let us express the principal terms in Eq. (9) in spherical 1 ϭϪ ͵ ␻ ͑ Ϫ ͒ ␻ ␻ coordinates and simplify them as AT0 D D0 0 cos t sin t dV.(A3) 2 V r 2 r 2 r r sin ␹ cos ␾ r2 sin2 ␹ Therefore the time-averaged value over the field period is B ϭ ͩ s ϩ s ϩ ͪ 2 2 2 2 2 2 2 equal to zero: wi/r wi/r wi/r wi/r 2 ץ 1 AT rs Ϫ ͳ ͵ ͑D Ϫ D ͒dV ʹ ϭ 0. (A4) Ӎ 2 ,(B1) 0 2 t wץ V 2 T i/r The first term in Eq. (A2) can be rewritten with use of in- Ϯ ϩ ␹ 2 tegration by parts, as zs z0 r cos w 2 ϭ w 2ͫ 1 ϩ ͩ ͪ ͬ i/r 0 z 1 1 R Ϫ ͵ ٌ␾ ͑D Ϫ D ͒dV ϭϪ ͵ ٌ͓ ␾ ͑D Ϫ D ͔͒dV Ϯ 2 T 0 T 0 zs z0 2 V 2 V Ӎ 2ͫ ϩ ͩ ͪ ͬ w0 1 ,(B2) zR 1 ͒ ϩ ͵ ␾ ٌ͑ Ϫ T D D0 dV. 2 V k 1 k z Ϯ z ϩ r cos ␹ ϭϯ s 0 (A5) 2 ϩ Ϯ ϩ ␰ 2 2 2 Ri/r 2 zR ͓1 ͑zs z0 r cos ͒ /zR ͔ The second integral in Eq. (A5) is equal to zero if we make z Ϯ z use of the fact that the sources of the field are unchanged Ӎ ϯ s 0 2 ,(B3) during the placement of the dielectric particles, i.e., ٌD zRwi/r ϭ ٌ ϭ ␳ ␳ D0 , where is the free-charge density. With the Gauss theorem, the first integral in Eq. (A5) can be 2 ϩ Ϫ krB 1 1 zs z0 zs z0 rewritten as the integration of the normal component of ͩ Ϫ ͪ Ӎ Ϫ 2ͩ ϩ ͪ rs .(B4) ␾ Ϫ 2 R R z w 2 z w 2 the product T(D D0) over a surface embedding the ob- i r R i R r ject: Integration of noninterference terms in Eqs. (11) and (12) 1 1 Ϫ ͵ ٌ͓ ␾ ͑ Ϫ ͔͒ ϭϪ ͵ ͓ ␾ ͑ Ϫ ͔͒ is equal to zero, or these terms can be taken in front of the T D D0 dV T D D0 ndS. 2 V 2 S integral. The rewritten interference term, with the sub- ␰ ϭ ␹ ␾ ϭ ␾ (A6) stitution cos and s (ra , zs), gives for the force Zemanek et al. Vol. 19, No. 5/May 2002/J. Opt. Soc. Am. A 1033

␣ 2 ϭ 2 ϩ 2 1/2 n2 w0 W wi͑zs͒wr͑zs͒/͓wi ͑zs͒ wr ͑zs͔͒ , F ͑r , z ͒ ϭ I 4␲␳ a2 exp͑Ϫr 2/w 2 z s n 2 c 00 w w s i i r X ϭϪkW/ͱC, 1 Ϫ 2 2͒ ͵ ͑ ␰ ϩ ␾ ͒␰ ␰ ϭ ͱ rs /wr cos 2ka s d dr, Y a C/W, Ϫ1 ϭ Ϫ 2 C 1 iZW , n2 w0 ϭ ␣ I ␲␳ exp͑Ϫr 2/w 2 00 2 s i ϭ 2 2 c k wiwr ZW W /R0 , Ϫ 2 2 2 ϭ ͑ ϩ ͒ ͑ 2͒ ϩ ͑ ϩ ͒ ͑ 2͒ rs /wr ͒͑2ak cos 2ak 1/R0 zs z0 / zRwi zs z0 / zRwr . Ϫ ␾ If the error function is transformed into the Dawson inte- sin 2ak͒sin s .(B5) gral by using Equation (16) can be obtained in similar manner. 2i ͑ ͒ ϭ ͑ 2͒ ͑ ͒ ™ ¶ erf ix exp x daw x , 2. a zR , a w0 ͱ␲ ϭ If the sphere is placed on axis (rs 0) and is so large that the lateral variation of the intensity cannot be ne- x ϭ 2 Ϫ 2 Ӷ daw͑x͒ ͵ exp͑t x ͒dt, glected, but the condition a zR is satisﬁed, the approxi- 0 mations in Eqs. (B2) and (B3) can be used again, but instead of relations (B1) and (B4) the following expressions Eq. (19) can be obtained. Moreover, the expression for the have to be employed: energy change needs integration by parts of the noninterference and the interference terms, and Eq. (18) can be r 2 r2 sin2 ␹ B ϭ obtained in this way. 2 2 2 2 ,(B6) wt/r wi/r

2 ϩ Ϫ krB 1 1 zs z0 zs z0 ACKNOWLEDGMENTS ͩ Ϫ ͪ Ӎ Ϫr2 sin2 ␹ͩ ϩ ͪ . 2 2 The authors thank J. Komrska for fruitful discussion and 2 Ri Rr zRwi zRwr (B7) comments and the Grant Agency of the Czech Republic for ﬁnancial support (grants 202/99/0959, 101/98/P106 and The axial force can be obtained by using Eq. (12). It 101/00/0974). can be shown that the integral of the noninterference terms is equal to zero, and the interference term gives, Author Pavel Zema´nek may be reached by e-mail at with the substitution ␰ ϭ cos ␹, [email protected].

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