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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 8 — Jul. 30, 2009
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Extreme axial optical force in a standing wave achieved by optimized object shape

J. Trojek, V. Karásek, and P. Zemánek  »View Author Affiliations


Optics Express, Vol. 17, Issue 13, pp. 10472-10488 (2009)
http://dx.doi.org/10.1364/OE.17.010472


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Abstract

Standing wave optical trapping offers many useful advantages in comparison to single beam trapping, especially for submicrometer size particles. It provides axial force stronger by several orders of magnitude, much higher axial trap stiffness, and spatial confinement of particles with higher refractive index. Mainly spherical particles are nowadays considered theoretically and trapped experimentally. In this paper we consider prolate objects of cylindrical symmetry with radius periodically modulated along the axial direction and we present a theoretical study of optimized objects shapes resulting in up to tenfold enhancement of the axial optical force in comparison with the original unmodulated object shape. We obtain analytical formulas for the axial optical force acting on low refractive index objects where the light scattering by the object is negligible. Numerical results based on the coupled dipole method are presented for objects with higher refractive indices and they support the previous simplified analytical conclusions.

© 2009 Optical Society of America

1. Introduction

Optical trapping of microobjects and nanoobjects is now a well established micromanipulation technique that revolutionized various branches of physics, biology and engineering [1

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

-3

3. A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis.” Electophoresis 29, 4813–4851 (2008). [CrossRef]

]. Up to now majority of efforts have been devoted to the manipulations with objects of spherical or near spherical shape. Thus, theoretical treatment of optical forces acting on spherical objects is most advanced. The theoretical description of optical forces is based on Lorenz-Mie scattering theory for a single sphere [4

4. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989). [CrossRef]

7

7. A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006). [CrossRef] [PubMed]

] or multiple Mie scattering theory for more simultaneously trapped spheres [8

8. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

, 9

9. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005). [CrossRef]

]. Except spherical objects, spheroidal or cylindrical ones are frequently considered theoretically, too [10

10. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A.M. Bránczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A: Pure Appl. Opt. 9, S196–S203 (2007). [CrossRef]

-12

12. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, “Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field,” J. Opt. Soc. Am. A 23, 2324–2330 (2006). [CrossRef]

]. Optical forces acting upon particles with more complex shapes must be calculated using numerical approaches - for example coupled dipole method (CDM) [13

13. P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 1011, 023106 (2007). [CrossRef]

, 14

14. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001). [CrossRef]

], finite element method (FEM) [15

15. D. A. White, “Vector finite element modeling of optical tweezers,” Comp. Phys. Commun. 128, 558–564 (2000). [CrossRef]

] or finite-difference time-domain method (FDTD) [16

16. R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express 13, 3707–3718 (2005). [CrossRef] [PubMed]

-19

19. D. C. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008). [CrossRef] [PubMed]

].

The most well-known optical micromanipulation technique, called optical tweezers, is based on a single tightly focused laser beam [20

20. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). [CrossRef] [PubMed]

]. However, another and even older configuration, called dual-beam trap, uses two counter-propagating beams. These beams are not so tightly focused and, therefore, lower optical intensity can be used for the particle spatial confinement. This configuration has found numerous applications in the advanced optical micromanipulation techniques such as optical fiber trapping schemes [21

21. A. Constable and J. Kim, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

, 22

22. S. D. Collins, R. J. Baskin, and D. G. Howitt, “Microinstrument gradient-force optical trap,” Appl. Opt. 38, 6068–6074 (1999). [CrossRef]

], optical stretcher [23

23. J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5154 (2000). [CrossRef] [PubMed]

, 24

24. J. Guck, S. Schinkinger, B. Lincoln, F. Wottawah, S. Ebert, M. Romeyke, D. Lenz, H. M. Erickson, R. Ananthakrishnan, D. Mitchell, J. Käs, S. Ulvick, and C. Bilby, “Optical deformability as an inherent cell marker for testing malignant Transformation and Metastatic Competence,” Biophys. J. 88, 3689–3698 (2005). [CrossRef] [PubMed]

], integrated opto-fluidic systems [25

25. S. J. Cran-McGreehin, T. F. Krauss, and K. Dholakia, “Integrated monolithic optical manipulation,” Lab Chip 6, 1122–1124 (2006). [CrossRef] [PubMed]

], Raman microspectroscopy of trapped objects [26

26. P. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fibre trap for Raman microspectroscopy of single cells,” Opt. Express 14, 5779–5791 (2006). [CrossRef] [PubMed]

], multiple 3-D trapping using moderately focused beams [27

27. D. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75(9), 2960–2970 (2004). [CrossRef]

, 28

28. P. Rodrigo, L. Gammelgaard, P. Boggild, I. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express 13, 6899–6904 (2005). [CrossRef] [PubMed]

], or optical binding [29

29. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002). [CrossRef]

, 30

30. V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008). [CrossRef] [PubMed]

]. If both beams interfere, a standing wave is created along the beams propagation. Due to the steep axial intensity modulation optical trapping into the standing wave provides several advantages comparing to single beam trapping especially for sub-micrometer size objects. Maximal axial force and axial trap stiffness can be higher by several orders of magnitude and object confinement is possible with lower trapping power [31

31. P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998). [CrossRef]

-34

34. P. Zemánek, A. Jonáš, P. Jákl, M. Šerý, J. Ježek, and M. Liška, “Theoretical comparison of optical traps created by standing wave and single beam,” Opt. Commun. 220, 401–412 (2003). [CrossRef]

]. Additionally, in contrast to single beam trap, particles with higher refractive index can be trapped. Up to now standing wave optical trapping has been used for confinement of thousands of submicrometer objects [35

35. T. Čižmár, M. Šiler, and P. Zemánek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]

] and their optical delivery [36

36. T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). [CrossRef]

-38

38. D. M. Gherardi, A. E. Carruthers, T. Čižmár, E. M. Wright, and K. Dholakia, “A dual beam photonic crystal fibre trap for microscopic particles,” Appl. Phys. Lett. 93, 041110 (2008). [CrossRef]

]. Its variation using counter-propagating evanescent waves has been used for optical sorting of sub-micrometer objects [39

39. T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of sub-micron objects in a motional standing wave,” Phys. Rev. B 74, 035105 (2006). [CrossRef]

] and nanoparticle surface delivery [40

40. M. Šiler, T. Čižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84, 157–165 (2006). [CrossRef]

, 41

41. M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New. J. Phys. 10, 113010 (2008). [CrossRef]

].

2. Theory for weakly polarizable objects

We assume that the radial extent of the considered object is much smaller comparing to radial variation of illuminating beams. This allows us to take the illuminating field in the form of a standing wave generated by an interference of two counter-propagating plane waves of the same intensity I 0. The resulting optical intensity I of the standing wave is expressed as:

I(z)=2I0[1+cos(2kz)],
(1)

where k=2π/λ=k0n2 is the wavenumber in a surrounding medium (liquid) of refractive index n2, k0 is the wavenumber in vacuum, and λ is the wavelength in the medium.

To simplify the study while keeping the key features we assume that the refractive index of the object n 1 is slightly higher than the refractive index of the medium n 2. Thus the minimized influence of the object on the total field is not taken into account. Moreover, due to the counterpropagating geometry of the beams the radiation pressures (scattering forces) coming from both beams cancel each other and only the spatial variation of the standing wave optical intensity determines the final optical force. Therefore, the axial optical force Fz(r o) acting on a dielectric object placed at a position r o can be expressed as [33

33. P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A 19, 1025–1034 (2002). [CrossRef]

, 48

48. T. Tlusty, A. Meller, and R. Bar-Ziv, “Optical gradient forces of strongly localized fields,” Phys. Rev. Lett. 81, 1738–1741 (1998). [CrossRef]

]

Fz(ro)=α2n2cSI(r)nz(r)dS,
(2)

where S is the surface of the object, c is the speed of the light in vacuum, and nz is the axial component of the outer normal unit vector to the surface S at the position r. The object polarizability α is expressed as α=(n 1/n 2)2-1. In this article we will evaluate and analyze Eq. (2) for different objects shapes substituting the intensity from Eq. (1).

Fig. 1. Two different shapes of the prolate objects of cylindrical and mirror symmetry considered in the force optimization: overlapping spheres (case A) and sinusoidal chain (case B). R denotes the maximal radius from the optical axis z, D is the length of the period (distance between centres of the neighbouring overlapping spheres; D≤2R), A represents the amplitude of the sinusoidal modulation of the radius, and the bases on both sides can be shifted axially by a distance d. The shown objects are made of N=4 units, one of them is shaded.

2.1. Cylindrical object

Let us start with the simplest case – a cylinder of length L and radius R oriented with its longitudinal axis parallel to the z axis. Since nz=0 over the cylinder coat, the surface integration in Eq. (2) reduces only to the area of cylinder bases placed at z=zA and z=zB:

FzbaseA=α2n2cI(zA)SA,FzbaseB=+α2n2cI(zB)SB,
(3)

where SA or SB denotes the area of the base at zA or zB, respectively. Therefore, the resulting optical force acting on the cylinder is equal to

Fzcyl(Z)=α2n2c2I0[cos(2kzA)+cos(2kzB)]πR2
=2F0(kR)2sin(kL)sin(2kZ),withforceunitF0=αn2cπk2I0,
(4)

and Z=(zA+zB)/2, L=zB-zA representing the cylinder centre and cylinder length, respectively. The force acting on the cylinder depends on its axial position just through the term sin(2kZ) and it enables to define the force amplitude -2F0k2R2 sin(kL) corresponding to sin(2kZ)=1. This amplitude depends periodically on the cylinder length L and rises quadratically with radius R. If the force amplitude is negative (i.e. term sin(kL) is positive), the axial equilibrium position of the cylinder centre overlaps with the intensity maximum of the standing wave (Z=Mλ/2, M is integer). Vice versa, positive force amplitudes localize the cylinder centre in the standing wave intensity minimum. If sin(kL)=0 (i.e. the cylinder length L is an integer multiple of λ/2) the axial force amplitude equals to zero for any cylinder position Z and consequently the cylinder would move freely along z axis.

2.2. Spherically modulated object

In this case we deal with an object composed of N identical units in the form of a sphere of radius R cut symmetrically at both ends by a plane perpendicular to the z axis. The shape of the whole object is shown in Fig. 1-A, the shaded region denotes the single unit. The whole object is composed of N adjacent units, centres of which are regularly spaced by D. We further consider the option that both outer units (spheres) can be cropped by a plane perpendicular to the z axis at the distance d from the outer unit centre. Therefore, the object ends with two flat circular bases and positions of them can be parametrically changed. The adopted approximation, described in Eq. (2), allows to express the total optical force acting upon such object as the sum of the forces acting upon its two plane bases and the object coat composed of coats of N units.

Let us first consider axial optical force acting upon the coat of the single unit – a cropped sphere. Using Eq. (2) and procedure described in Appendix 4.1 one obtains:

Fzcoatsph(z1)=F0G(kD)sin(2kz1),withtermG(kD)=sin(kD)kDcos(kD),
(5)

where z 1 is the axial position of the unit (sphere) centre. The force depends on the position z1 through sin(2kz 1) and on the geometric term G(kD). Figure 2 shows its dependence on the shape parameter D with periodic extremes at D=M λ/2, where M is an integer. In order to obtain the force amplitude we further set sin(2kz 1)=1 yielding to

Fzcoatexsph=(1)MF0πM.
(6)

Without cropping (D=2R) we obtain the known force acting on a sphere placed into the standing wave [33

33. P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A 19, 1025–1034 (2002). [CrossRef]

].

FzcoatNsph(Z)=F0T(kD,N)sin(2kZ),withtermT(kD,N)=G(kD)sin(NkD)sin(kD).
(7)

The term T plays the same role as G in Eq. (5) or sin(kL) in Eq. (4) and determines whether the equilibrium position of the object centre is placed in the standing wave intensity maximum or minimum. There are many local extremes of the function T(kD,N) but only the dominant ones satisfy condition D=/2 (see Fig. 3). This condition gives the extreme amplitude of the force done by Eq. (7):

FzcoatexNsph=F0T(kMλ2,N)=(1)MNF0πMN,
(8)

with restriction M=1, 2, …M̄≤4R/λ coming from condition D≤2R (see Fig. 1). Equation (8) shows that the absolute value of the force amplitude increases with the number N of overlapping spheres and, surprisingly, with the period D between the units centres - represented here by parameter M. Moreover, if the product MN is kept constant, the strength of the final force does not depend on the particular shape of the object. Therefore, N overlapping spheres with D=λ/2 experience the same extreme force as the single cropped sphere with D=Nλ/2. This force contribution from the object coat does not depend explicitly on the sphere radius R but we must keep in mind restriction D≤2R.

Fig. 2. Dependence of the term G on period D/λ given by Eq. (5).
Fig. 3. Dependence of the term T on period D/λ given by Eq. (7) for N overlapping spheres.

More complex geometry is discussed in the Appendix 4.2 with the outer spheres cropped symmetrically with respect to centre of the object but asymmetrically with respect to the outer spheres. The force acting upon the bases is derived there, too.

The total axial optical force is equal to the sum of the forces acting upon the object coat done by Eqs. (7,22) and upon the bases described in Eq. (23):

FztotalNsph(Z)=FzcoatNsph(Z)+ΔFzcoatNsph(Z)+FzbasesNsph(Z)
=F0{[sin(kD)kDcos(kD)]sin(NkD)sin(kD)
[2kdcos(kL)sin(kL)+sin(NkD)kDcos(NkD)]
+2[(kR)2(kd)2]sin(kL)}sin(2kZ).
(9)

In Appendix 4.2 we derive combinations of parameters

D=M̄λ/2,whereM̄=max(M)4R/λ,andd={+λ8foroddM̄,λ8forevenM̄,
(10)

giving extreme amplitude of the total force in Eq. (9) for fixed R and N>1

FztotalexNsph(N,R)=(1)M̄NF0{πM̄(N1)+1π28+8π2(Rλ)2},
(11)

which increases linearly with the number of spheres N and becomes always stronger than the force (4) acting upon a cylinder with the same radius R and length L that is given by the last term in curly brackets in Eq. (11). The first term in curly brackets expresses the influence of the object coat shape on the optical force.

Fig. 4. Amplitude of the axial optical force FztotalNsph (see Eq. (9)) as a function of the distance D between the centres of neighbouring overlapping spheres and the sphere radius R for optimized displacements of bases d/8 (upper graph) and d=-λ/8 (lower graph) resulting in extreme forces. Number of units N=4 is the same for both graphs, associated movie reveals the results for other N (Media 1). Refractive index of the environment n 2=1.33, refractive index of the object n 1=1.35, and F 0 is normalized to 1 pN. The marked points indicate local extremes of the force calculated numerically.

Figures 4 and 5 present the amplitude of the total force FztotalNsph from Eq. (9) (i.e. for sin(2kZ)=1) as a function of the distance between the sphere centres D and the sphere radius R for displacements of the bases d=±λ/8, d=0, and d=R. Here the white areas correspond to forbidden combinations of parameters due to the relevant inequalities noted in the figures. Movies (Media 1,Media 2) associated with these figures reveal the profiles of the force amplitude for other values of N. Note that the condition D=Mλ/2 provides extremal amplitude exactly only for the force acting on the object coat with d=D/2 (see Eq. (7)) and, therefore, the interaction coming from the object bases and their displacements d more or less disturbs this condition. However, Eq. (11) shows that larger N or M (corresponding to larger D) strengthen the force contribution coming from the object coat which consequently determines the final behaviour of such object in the considered spatially periodic optical pattern. On the other hand, larger sphere radius R increases the forces acting upon the bases in Eq. (23) and the shape of the coat becomes less important for the total amplitude of the force – a limit case of this behaviour is the cylinder treated in Sec. 2.1. The figures illustrate the main results of this section: for a given number of spheres N and their radius R the amplitude of the total axial optical force acting upon such object is higher for D close to D=/2 and extreme for conditions stated in Eq. (10).

Fig. 5. Amplitude of the axial optical force FztotalNsph (see Eq. (9)) as a function of the distance D between the centres of neighbouring overlapping spheres and the sphere radius R for the bases displacements d=0 (upper graph) and d=R (lower graph). Number of units N=4 is the same for both graphs, associated movie reveals the results for other N (Media 2). Refractive index of the environment n 2=1.33, refractive index of the object n 1=1.35, F 0=1 pN. The marked points indicate local extremes of the force calculated numerically.

2.3. Sinusoidally modulated object

In this section we will consider an object with sinusoidally modulated radius along z axis that forms a sinusoidal chain as depicted in Fig. 1B. The amplitude of this modulation is denoted A and other parameters are the same as for the spherically modulated object. In Appendix 4.3 integration over the object coat is performed and the axial optical force acting upon it can be expressed as

Fzcoatsin=F0(T1+T2)sin(2kZ),
(12)

where the dimensionless terms T 1 and T 2 are equal to

T1=(kA)2D̄cos(kL)sin(2kd/D̄)sin(kL)cos(2kdD̄)(D+1)(D1),
(13)
T2=(kA)(kRkA)2D̄cos(kL)sin(kdD̄)sin(kL)cos(kdD̄)(D̄+12)(D̄12),
(14)

with modulation period in wavelength units D̄≡D/λ. The sum of both terms T 1+T 2 is a nontrivial function of the dimensionless parameter D̄ with many local extremes. However, as Fig. 6 demonstrates, there are always two dominant extremes at singular points D̄1=1 and D̄2=1/2. This property enables to express the extreme total optical forces analytically.

Fig. 6. Dependence of the terms T 1 and T 2 (expressed by Eq. (13) and Eq. (14), respectively) on D̄=D/λ for bases displacement d=0, object radius R=λ and sinusoidal modulation of the radius A=0.3λ

Fzbasessin=2F0(kB)2sin(kL)sin(2kZ),
(15)

where the bases radii are denoted by Br(zA)=r(zB).

The total axial optical force is done as

Fztotalsin(Z)=Fzcoatsin(Z)+Fzbasessin(Z)=F0[2(kB)2sin(kL)T1T2]sin(2kZ).
(16)

Due to the mirror symmetry of this object the position term sin(2kZ) can be again separated and the position independent amplitude of the total force can be expressed analytically. Extreme value of the total force in Eq. (16) at extreme points D̄1 and D̄2 is reached for d=(2Q+1)λ/8, where Q is an integer. Because of restrictive conditions on bases displacement d given by Eq. (29) only one solution d=λ/8 is allowed and the extreme amplitude of the total force has the following forms for the two cases D 1=λ, D 2=λ/2:

Fz1totalmaxsin=F0(kR)2{[23234+(34N)π16]v2+[2432]v2},
(17)
Fz2totalmaxsin=(1)NF0(kR)2{[56(2N1)π4]v2+[(2N1)π22]v+2},
(18)

where we used the dimensionless coefficient v≡2A/R∊〈0;1〉 expressing deviation of the object shape from a cylinder (v=0). We see that the amplitudes of both forces in Eq. (17) and Eq. (18) grow with the area of the widest profile (πR 2) and number of units N if the parameter v is fixed. However, only the force amplitude in Eq. (18) changes its sign with consequent numbers of N. If N=1, both forces amplitudes reach extreme value at v=0 and they are equal. It gives the shape of a cylinder of length L=λ/4. If N≥2, the force amplitude of Eq. (18) is extreme for

v=123π(2N1)10'
(19)

which is always larger than the extreme force amplitude from Eq. (17) occurring at v=1. Both extreme values of the forces occur for v close to 1. Interestingly, if the parameter v grows from 0 to 1 and the force amplitude increases, the volume of the object decreases

V2=R2λ8[2v(2v)+π(2N1)(22v+34v2)].
(20)
Fig. 7. Amplitude of the total axial optical force Fztotalsin (done by Eq. (16)) acting upon the sinusoidal chain as a function of modulation period D and coefficient v=2A/R with fixed R=0.6λ (upper graph) or radius R with fixed v=0.5 (lower graph) for the same displacements of bases d=λ/8, number of units N=4, and F 0=1 pN. Associated movie shows the development for different number of units N (Media 3).

Figure 7 and associated movie (Media 3) demonstrate how the amplitude of the total optical force acting upon sinusoidal chain varies with D, R, and v. They illustratively demonstrate the analytical conclusions in Eqs. (17,18,19) presented above, too.

It is worth to stress that the extreme amplitude of the force acting upon the sinusoidal chain expressed by Eq. (18) with v=1 is always stronger than the corresponding extreme force amplitude acting upon the overlapping cropped spheres expressed by Eq. (11), for the same parameters D=λ/2 and d=λ/8 (see Fig. 8). Figure 8 demonstrates that the transfer of momentum from the light to the sinusoidal chain is more efficient than to the spherical chain, because the light intensity is also sinusoidally modulated (see Eq. (1)).

Fig. 8. Ratio of the maximal optical forces acting on the sinusoidal chain (see Eq. (18)) and on the overlapping cropped spheres (see Eq. (11)) as a function of the radius R for the following fixed parameters: D=λ/2, d=λ/8, and v=1. Note that for large radius a limit of the ratio rises linearly with increasing number of units N.

3. Calculation by coupled dipole method

Figure 9 compares the axial optical force acting upon five overlapping spheres calculated analytically and by CDM. Two different object refractive indices were considered n 1=1.35 and n 1=1.41 (e.g. silica). Figure 9 demonstrates that the resulting force amplitudes obtained from CDM are in very good agreement with the analytical ones if the object refractive index is very close to the host medium refractive index (n 2=1.33; mn 1/n 2=1.015). It also shows significant deviations for the object made of silica.

The influence of the refractive index of several overlapping spheres on the optical force is shown in Fig. 10 in more details. It again reveals very good coincidence between analytical approximation and numerical computation by CDM for low refractive indices but significant deviations for higher refractive indices of the object. We can immediately see that the value of D giving extreme force amplitude decreases uniformly with the object refractive index. The optimized shape of the object again ensures extreme force which increases with number of units N (see associated movie (Media 4)). By proper design of the object composed of several units one can reach the optical force several times stronger comparing to the single unit.

Fig. 9. Comparison of the amplitudes of the axial optical forces acting upon five overlapping spheres made of refractive indices n 1=1.35 and n1=1.41 (silica). The forces are calculated analytically (dashed line) and numerically by CDM (solid line) as a function of sphere diameter 2R. The following parameters were used: refractive index of the host medium n 2=1.33, number of units N=5, sphere period D=0.7λ, bases displacement d=0, F 0=1pN.

4. Conclusion

Up to now mainly optical forces acting upon spherical or cylindrical objects have been studied. Obtained optical forces are generally very weak and any method is very useful which increases them keeping the same incident laser power. This study shows a novel way that combines two possible optical methods how to address such a force increase - utilization of structured light illumination of the trapping beam combined with the spatially structured shape of the object. We considered a prolate object with radially modulated shape placed into a spatially periodic light pattern (optical standing wave). First we assumed that the refractive index of the object is close to the refractive index of the surrounding medium. It allowed us to express the optical forces analytically for objects composed of several units in the form of overlapping spheres or sinusoidal chain. We found analytical conditions giving maximal amplitude of the optical force. This force amplitude increases linearly with the number of units and with the squared radius of the bases. Proper distance between the centres of the units ensures rapid increase of the force. To extend our study to objects of higher refractive index we used coupled dipole methods to calculate numerically the optical forces. We obtained very good coincidence with the analytical results for low refractive indices. The expected significant deviations from the analytical results were obtained for higher refractive indices. Especially, the distance D between the units (axial periodicity of the object), providing the extreme force amplitude, decreases with increasing refractive index of the object. However, if this condition is fulfilled, obtained optical force can be many times stronger.

As we stressed in the introduction, such artificial object shapes would be interesting handles for more complex probes with a particle or tip attached to the end and they could offer an interesting contactless alternative to atomic force microscope and its applications.

Appendix

4.1. Single cropped sphere

Let us first assume a sphere of radius R centred at [0,0, z1]. If the spherical polar coordinate ϑ denotes an angle between ẑ and n(r), then nz=cosϑ=(z-z 1)/R. Differentiation of the last relation gives -sinϑdϑ=dz/R and the polar surface element dS of the sphere can be expressed in the Cartesian coordinates using dS=|2πR 2 sinϑdϑ|=2πRdz. Therefore, Eq. (2) can be rewritten to the following form:

Fz(z1)=α2n2c2πzazbI(z)(zz1)dz,
(21)

where za and zb denote the axial positions of the planes cropping the sphere perpendicular to the optical axis. If Eq. (1) is substituted to Eq. (21) and the following substitution is used za=z1-D/2, zb=z 1+D/2, integration of Eq. (21) over one cropped sphere gives Eq. (5).

Fig. 10. Comparison of the axial optical force amplitudes as a function of period D of overlapping spheres (Fig. 1A) and object refractive index n 1. Lower graph shows analytical results from Eq. (9) and middle graph shows results by numerical CDM, both for the sphere radius R=0.6λ, bases displacement d=0, refractive index of the host medium n 2=1.33, and F 0=1 pN. Number of units N=4 is the same for all graphs and it varies in the associated movie (Media 4). The profiles of the force amplitudes at n 1=1.35 are in a very good agreement (see upper graph). However with increasing the object refractive index n1 locations of the axial force extremes with respect to D and force magnitudes change. This is because of the optical field scattered by the object that cannot be neglected anymore and that modifies the incident (standing wave) field.

4.2. Overlapping cropped spheres

We consider here more general case where the outer spheres are not cropped symmetrically with respect to their centres i.e. the condition d=D/2 is not required any more. The object bases placed originally at zA=z 1-D/2 and zB=zN +D/2 are now located in more general positions zA=z 1-d and zB=zN+d. This symmetrical modification at both ends keeps the object centre at Z=(zA+zB)/2. The final force acting on the coat given by Eq. (7) has to be corrected with the term ΔFzcoatNsph evaluated in Appendix 4.1 from Eq. (21) with integration limits zAz1D2 and zN+D2zB::

ΔFzcoatNsph(Z)=F0[2kdcos(kL)sin(kL)+sin(NkD)kDcos(NkD)]sin(2kZ),
(22)

where L=zB-zA=(N-1)D+2d denotes the length of the whole object and the following restrictions are valid: d∊(0;R〉 for N=1 and d∈(-D/2;R〉 for N>1. This extension makes the analytical results less straightforward and it will be studied below within the concept of the total force acting on such object. Let us note that it is not possible to separate the phase term sin(2kZ) in Eq. (22) and define the position independent force amplitude if the displacements of both object ends are asymmetric.

The axial optical forces acting upon both planar circular bases located at zA and zB are obtained in the same way as in Eq. (3). In the studied case of overlapping cropped spheres we have SA=SB=π(R 2-d 2) and the resulting optical force acting upon both bases together is equal to

FzbasesNsph(Z)=2F0[(kR)2(kd)2]sin(kL)sin(2kZ).
(23)

This contribution to the optical force can be neglected if d=R (no planar edges of the object) or sin(kL)=0.

The total axial optical force is equal to the sum of the forces acting upon the object coat done by Eqs. (7,22) and upon the bases described in Eq. (23):

FztotalNsph(Z)=FzcoatNsph(Z)+ΔFzcoatNsph(Z)+FzbasesNsph(Z).
(24)

Its final form is presented in Eq. (9).

It can be shown that the dominant extremes of the total force FztotalNsph lie close to the condition D=Mλ/2. Its substitution into Eq. (9) enables to find analytically all extreme points with respect to parameter d. Solving the relevant equation leads to two solutions:

A1) d=R,

A2) d=(2Q+1)λ/8,

Q=0,1, …Q̄≤4R/λ-1/2 for N=1;

Q=-M, …-1,0,1, … Q̄ for N>1;

where the confinement of Q follows from restrictive conditions d ∈ (0;R〉 for N=1 and d ∈ (-D/2;R〉 for N>1 mentioned in Section 2.2.

Let us first analyse the solution A2. If we evaluate the total force (9) in extreme points D=/2 and d=(2Q +1)λ/8, we obtain

FztotalexNsph(M,Q)=(1)MNF0×
{(1)QM[π28(2Q+1)22(kR)21]+πM(N1)}sin(2kZ).
(25)

We want to find the values of parameters M and Q (and consequently D and d) providing extreme amplitude of the total force in Eq. (25) if the number of spheres N and their radius R are fixed. The second term in curly brackets is always positive (or zero for N=1) and grows up linearly with both parameters M and N. Therefore, the first term in the curly brackets must be positive to get the extreme value of the force amplitude. Its value is close to zero for dR and the biggest for d=0. Its magnitude is significant due to quadratic dependence on the sphere radius R. Consequently the first term in the square brackets must be as small as possible leading to Q=0,-1 corresponding to displacements d=±λ/8. Hence for given N, R we take the greatest available integer M=M̄≤ 4R/λ (see Eq. (8)). Since both terms in the curly brackets must be positive, the term (-1)Q-M must be negative. Thus if M̄ is even, we take Q=-1, whereas if M̄ is odd, the choice Q=0 leads to the extreme value of the force in Eq. (25) for given parameters N and R expressed by Eq. (11).

Solution A1 describes the case when the outer spheres are not cropped and, therefore, the force acting upon plane bases done by Eq. (23) is equal to zero. The total force is done as the sum of Eq. (7) and Eq. (22) and it increases only linearly with parameter R. Hence, the extreme values of this force must be lower comparing to those from the solution A2 for the same parameters M, N and R. However, this total force has many local extremes with respect to the sphere radius R compared to only one in Eq. (25). Solution of the relevant equation leads to the condition

A1′) d=R=(2P)λ/8; P=1,2,3, …, M=1,2, …M̄≤P

for extreme points. The amplitude of the total force in Eq. (9) consequently takes the form

FztotalexNsphA1=(1)MNF0{(1)PMπP+πM(N1)}.
(26)

Consequent analysis are similar to those of solution A2. The highest value of this force amplitude can be achieved by taking the greatest available values of parameters M and P but only with even difference P-M. So that the choice M=P gives the extreme optical force amplitude in the following form:

FztotalmaxNsphA1=F0(1)PNπPN.
(27)

Due to these conditions the object shape corresponding to A1 giving the extreme amplitude of the force is composed of N identical touching spheres forming linear chain. Therefore, this result is a special case of Eq. (8) for D=2R (i.e. M=P).

4.3. Optical force upon objects with rotational symmetry

Let us consider an object with rotational symmetry around the optical axis z, which profile is modulated by a function r(z) along this axis from zA to zB. Assume an arbitrary point of coordinate z on the surface of the object. If the function r( z) is descending at this point, the axial component nz of the outer normal unit vector to the surface is positive, and vice versa. The triangle similarity validates nz : 1=-dr : dl, where dl is the length of the curve r(z) along axis element dz. The polar surface element dS around axis element dz is given by dS=2πrdl and the axial optical force in Eq. (2) then takes the following form:

Fz(Z)=2πα2n2czAzBI(z)r(z)r(z)dz.
(28)

where r′=dr/dz and Z=(zA+zB)/2.

In the case of sinusoidally modulated coat the following relation is valid for the object radius

r(z)=RA+Acos[2πD(zz1)],
(29)

with R≥2A, d ∈ (0;D/2〉 for N=1 or d ∈ (-D/2;D/2〉 for N>1.

4.4. Optical force upon objects with mirror symmetry

In this article, we only deal with objects having mirror symmetry (see Fig. 1) because in such case the amplitude of the optical force depends harmonically on the position of object centre Z=(zA+zB)/2. The object having the mirror symmetry is expressed by relation r(z̃)=r(-z̃). Substituting z and I(z) into Eq. (28) we get

Fz=2F0k2L2L2[1+cos(2kz˜)cos(2kZ)sin(2kz˜)sin(2kZ)]r(z˜)r(z˜)dz˜,
(30)

where we used substitution z=Z+z̃. The product of even function r(z̃) and odd function r′(z̃) is again odd function, and the integral of odd function over a symmetric interval vanishes. Hence, Eq. (30) can be simplified into the form

Fz=2F0k2sin(2kZ)L2L2sin(2kz˜)r(z˜)r(z˜)dz˜
(31)

with the modulation term sin(2kZ).

Acknowledgment

The authors appreciate valuable comments of Dr. A. Jonáš and acknowledge support from MEYS CR (LC06007, OC08034) projects, CSF (GA202/09/0348), and ISI IRP (AV0Z20650511).

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OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(070.6110) Fourier optics and signal processing : Spatial filtering
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: April 3, 2009
Revised Manuscript: May 2, 2009
Manuscript Accepted: May 14, 2009
Published: June 8, 2009

Virtual Issues
Vol. 4, Iss. 8 Virtual Journal for Biomedical Optics

Citation
J. Trojek, V. Karásek, and P. Zemanek, "Extreme axial optical force in a standing wave achieved by optimized object shape," Opt. Express 17, 10472-10488 (2009)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-13-10472


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References

  1. K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004). [CrossRef]
  2. K. Dholakia, P. Reece, and M. Gu, "Optical micromanipulation," Chem. Soc. Rev. 35, 42-55 (2008). [CrossRef]
  3. A. Jonáš and P. Zemánek, "Light at work: The use of optical forces for particle manipulation, sorting, and analysis." Electophoresis 29, 4813-4851 (2008). [CrossRef]
  4. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989). [CrossRef]
  5. K. F. Ren, G. Gréhan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt 35, 2702-2710 (1996). [CrossRef] [PubMed]
  6. A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc.R. Soc. Lond. A 459, 3021-3041 (2003). [CrossRef]
  7. A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, "Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric," Opt. Express 14, 13101-13106 (2006). [CrossRef] [PubMed]
  8. Y.-L. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995). [CrossRef] [PubMed]
  9. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, "Photonic clusters formed by dielectric microspheres: Numerical simulations," Phys. Rev. B 72, 085130 (2005). [CrossRef]
  10. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A.M. Bránczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical tweezers computational toolbox," J. Opt. A: Pure Appl. Opt. 9, S196-S203 (2007). [CrossRef]
  11. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, "Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam," Phys. Rev. E 75, 026613 (2007). [CrossRef]
  12. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006). [CrossRef]
  13. P. C. Chaumet and C. Billaudeau, "Coupled dipole method to compute optical torque: Application to a micropropeller," J. Appl. Phys. 1011, 023106 (2007). [CrossRef]
  14. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, "Radiation forces in the discrete-dipole approximation," J. Opt. Soc. Am. A 18, 1944-1953 (2001). [CrossRef]
  15. D. A. White, "Vector finite element modeling of optical tweezers," Comp. Phys. Commun. 128, 558-564 (2000). [CrossRef]
  16. R. C. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13, 3707-3718 (2005). [CrossRef] [PubMed]
  17. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, "Radiation pressure and the distribution of electromagnetic force in dielctric media," Opt. Express 13, 2321-2336 (2005). [CrossRef] [PubMed]
  18. W. L. Collet, C. A. Ventrice, and S. M. Mahajan, "Electromagnetic wave technique to determine radiation torque on micromachines driven by light," Appl. Phys. Lett. 82, 2730-2732 (2003). [CrossRef]
  19. D. C. Benito, S. H. Simpson, and S. Hanna, "FDTD simulations of forces on particles during holographic assembly," Opt. Express 16, 2942-2957 (2008). [CrossRef] [PubMed]
  20. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986). [CrossRef] [PubMed]
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