## 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 [1-3]. 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—7] or multiple Mie scattering theory for more simultaneously trapped spheres [8, 9]. Except spherical objects, spheroidal or cylindrical ones are frequently considered theoretically, too [10-12]. Optical forces acting upon particles with more complex shapes must be calculated using numerical approaches - for example coupled dipole method (CDM) [13, 14], finite element method (FEM) [15] or finite-difference time-domain method (FDTD) [16-19].

The most well-known optical micromanipulation technique, called optical tweezers, is based on a single tightly focused laser beam [20]. 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, 22], optical stretcher [23, 24], integrated opto-fluidic systems [25], Raman microspectroscopy of trapped objects [26], multiple 3-D trapping using moderately focused beams [27, 28], or optical binding [29, 30]. 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 [31-34]. 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] and their optical delivery [36-38]. Its variation using counter-propagating evanescent waves has been used for optical sorting of sub-micrometer objects [39] and nanoparticle surface delivery [40, 41].

The optical forces produced by the single focused laser beam on spherical objects are typically in the range from pN to hundreds of pN. Utilization of spatially structured light (for example standing waves, or interference pattern of two and more interfering beams) can increase this force by about one order. In this paper we made the next step and considered structured shape of the object placed into spatially structured light with the hope to further increase the acting force keeping the laser power fixed. In this first analysis we consider a prolate object of cylindrical symmetry with periodic modulation of its radius along object axial axis (see Fig. 1). We choose two different types of the modulation - spherical and sinusoidal. The choice of the spherical modulation relates to the fact that majority of the laboratory particles are shaped close to sphere and can be sticked together thermally or by proper functionalization of their surfaces. Therefore structure shown in Fig. 1(a) can be reached and this model also includes the case of touching spheres. The second considered shape expects sinusoidal modulation of the object radius because such shape can be manufactured by the photopolymerization if the monomer is illuminated by the standing wave (similarly as in Ref. [42] but with both counter-propagating interfering beams). Generally even more complex object shapes can be manufactured by the photopolymerization if a single focused laser beam is used [43-45].

Such artificial object shapes would be interesting handles for more complex probes with a particle or tip attached to the end. Due to the stronger confinement and non-contact manipulation they offer an interesting alternative to atomic force microscope and its applications [46,47]. Such a probe with a tip can be used for mechanical disposal of the adhered cells, inducing mechanical stress at selected placed of the cell, and with more complex setup could also measure interaction forces between the probe and an object. The other issue is the orientation of the trapping beams making the spatially structured trapping field. One might consider a standing wave (presented here) with a tip attached perpendicularly to the beams axis. The beams can be placed either above the studied object and in parallel with the surface or perpendicularly to the surface next to the object. In both cases the beams are not effected by the object and the mechanical interaction is mediated by the attached tip. Moreover, phase change in one beam would provide a fine axial positioning of the structured handle with the tip even without mechanical movement of the sample [36]. The other alternative is to use two or more co-propagating but tilted beams making interference fringes. If the shape of the object-handle is properly designed with respect to these fringes, one might expect similar enhancement of the interaction forces as in the case of standing wave illumination. In the following part of the paper we will deal with the structured trapping field in the form of the standing wave.

## 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:

where *k*=2*π/λ*=*k _{0}n_{2}* 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 *F _{z}*(

**r**

_{o}) acting on a dielectric object placed at a position

**r**

_{o}can be expressed as [33, 48]

where *S* is the surface of the object, *c* is the speed of the light in vacuum, and *n _{z}* 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).

#### 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*=*z _{A}* and

*z*=

*z*:

_{B}where *S _{A}* or

*S*denotes the area of the base at

_{B}*z*or

_{A}*z*, respectively. Therefore, the resulting optical force acting on the cylinder is equal to

_{B}$$=-2{F}_{0}{\left(kR\right)}^{2}\mathrm{sin}\left(kL\right)\mathrm{sin}\left(2kZ\right),\phantom{\rule[-0ex]{.7em}{0ex}}\mathrm{with}\mathrm{force}\mathrm{unit}\phantom{\rule[-0ex]{.4em}{0ex}}{F}_{0}=\alpha \genfrac{}{}{0.1ex}{}{{n}_{2}}{c}\genfrac{}{}{0.1ex}{}{\pi}{{k}^{2}}{I}_{0},$$

and *Z*=(*z _{A}*+

*z*)/2,

_{B}*L*=

*z*representing the cylinder centre and cylinder length, respectively. The force acting on the cylinder depends on its axial position just through the term sin(2

_{B}-z_{A}*kZ*) and it enables to define the force amplitude -2

*F*sin(

_{0}k^{2}R^{2}*kL*) corresponding to sin(2

*kZ*)=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:

where *z*
_{1} is the axial position of the unit (sphere) centre. The force depends on the position z1 through sin(2*kz*
_{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(2*kz*
_{1})=1 yielding to

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

In the next step we derive the force acting on the coat of object composed of *N* identical (*d=D/2) cropped spheres*. The spheres are mutually separated by *D* and therefore the centre of *n*-th cropped sphere is located at *z _{n}*=

*z*

_{1}+(

*n*-1)

*D*with

*n*=1…

*N*. The total axial component of the optical force acting on the coat of such structure is equal to the sum of contributions of each unit in the form of Eq. (5). By definition of the centre of the object

*Z*=(

*z*

_{1}+

*z*)/2 the total axial optical force acting upon the

_{N}*object coat*has the form:

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*=*Mλ*/2 (see Fig. 3). This condition gives the extreme amplitude of the force done by Eq. (7):

with restriction *M*=1, 2, …*M*̄≤4*R*/λ coming from condition *D*≤2*R* (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*.

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):

$$=-{F}_{0}\{\left[\mathrm{sin}\left(kD\right)-kD\mathrm{cos}\left(kD\right)\right]\genfrac{}{}{0.1ex}{}{\mathrm{sin}\left(NkD\right)}{\mathrm{sin}\left(kD\right)}$$

$$-\left[2kd\mathrm{cos}\left(kL\right)-\mathrm{sin}\left(kL\right)+\mathrm{sin}\left(NkD\right)-kD\mathrm{cos}\left(NkD\right)\right]$$

$$+2\left[{\left(kR\right)}^{2}-{\left(kd\right)}^{2}\right]\mathrm{sin}\left(kL\right)\}\mathrm{sin}\left(2kZ\right).$$

In Appendix 4.2 we derive combinations of parameters

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

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.

Figures 4 and 5 present the amplitude of the total force ${F}_{{z}_{\mathit{total}}}^{\mathit{Nsph}}$ from Eq. (9) (i.e. for sin(2*kZ*)=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*=*M*λ/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*=*Mλ*/2 and extreme for conditions stated in Eq. (10).

#### 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

where the dimensionless terms *T*
_{1} and *T*
_{2} are equal to

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

Following our previous results expressed by Eq. (4)
*the force on the object bases* takes analogous form

where the bases radii are denoted by *B*≡*r*(*z _{A}*)=

*r*(

*z*).

_{B}*The total axial optical force* is done as

Due to the mirror symmetry of this object the position term sin(2*kZ*) 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 *D*̄_{1} and *D*̄_{2} is reached for *d*=(2*Q*+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:

where we used the dimensionless coefficient *v*≡2*A/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

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

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)).

## 3. Calculation by coupled dipole method

The analytical results of the previous sections enable us to study the problem of optimized shape in a great detail. Even though the results are limited by the assumption of small relative refractive index between studied objects and medium there could a possibility to generalize the behaviour also to systems where the scattering from the studied objects plays substantial role. In order to prove this assumption, a more rigorous method must be used to express the optical forces using proper light scattering theory. We used our numerical code based on coupled dipole method (CDM) which was previously successfully employed in the calculation of the scattered light and subsequently the optical forces acting upon several objects of various sizes [30,49,50]. In CDM the shape of the object is approximated by induced elementary dipoles spread in a cubic lattice. We used the cubic lattice parameter of the order of 0.01*λ* to have quite smooth surface of the object. This distance between dipoles amply satisfies the upper limit *λ*/20 given by CDM [51].

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; *m*≡*n*
_{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.

## 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, z_{1}]. If the spherical polar coordinate ϑ denotes an angle between *z*̂ and * n(r)*, then

*n*=cos

_{z}*ϑ*=(

*z*-

*z*

_{1})/

*R*. Differentiation of the last relation gives -sinϑdϑ=

*dz/R*and the polar surface element d

*S*of the sphere can be expressed in the Cartesian coordinates using d

*S*=|2

*πR*

^{2}sinϑdϑ|=2

*πR*dz. Therefore, Eq. (2) can be rewritten to the following form:

where *z _{a}* and

*z*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

_{b}*z*-

_{a}=z_{1}*D*/2,

*z*=

_{b}*z*

_{1}+

*D*/2, integration of Eq. (21) over one cropped sphere gives Eq. (5).

#### 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 *z _{A}*=

*z*

_{1}-

*D*/2 and

*z*=

_{B}*z*+

_{N}*D*/2 are now located in more general positions

*z*=

_{A}*z*

_{1}-

*d*and

*z*=

_{B}*z*+

_{N}*d*. This symmetrical modification at both ends keeps the object centre at

*Z*=(

*z*+

_{A}*z*)/2. The final force acting on the coat given by Eq. (7) has to be corrected with the term $\mathrm{\Delta}{F}_{{z}_{\mathit{coat}}}^{\mathit{Nsph}}$ evaluated in Appendix 4.1 from Eq. (21) with integration limits ${\int}_{{z}_{A}}^{{z}_{1}-D\u20442}$ and ${\int}_{{z}_{N}+D\u20442}^{{z}_{B}}:$:

_{B}where *L=z _{B}-z_{A}*=(

*N*-1)

*D*+2

*d*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(2

*kZ*) 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 *z _{A}* and

*z*are obtained in the same way as in Eq. (3). In the studied case of overlapping cropped spheres we have

_{B}*S*=

_{A}=S_{B}*π*(

*R*

^{2}-

*d*

^{2}) and the resulting optical force acting upon both bases together is equal to

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):

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

It can be shown that the dominant extremes of the total force ${F}_{{z}_{\mathit{total}}}^{\mathit{Nsph}}$ 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*=(2*Q*+1)*λ*/8,

*Q*=0,1, …*Q*̄≤4*R/λ*-1/2 for *N*=1;

*Q*=-*M*, …-1,0,1, … *Q*̄ 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*=*Mλ*/2 and d=(2*Q*
*+*1)λ/8, we obtain

$$\left\{{\left(-1\right)}^{Q-M}\left[\genfrac{}{}{0.1ex}{}{{\pi}^{2}}{8}{\left(2Q+1\right)}^{2}-2{\left(kR\right)}^{2}-1\right]+\pi M\left(N-1\right)\right\}\mathrm{sin}\left(2kZ\right).$$

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 *d*=±*R* 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=M*̄≤ 4*R/λ* (see Eq. (8)). Since both terms in the curly brackets must be positive, the term (-1)^{Q-M} must be negative. Thus if *M*̄ is even, we take *Q*=-1, whereas if *M*̄ 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*=(2*P*)λ/8; *P*=1,2,3, …, *M*=1,2, …*M*̄≤*P*

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

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:

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*=2*R* (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 *z _{A}* to

*z*. Assume an arbitrary point of coordinate

_{B}*z*on the surface of the object. If the function

*r(*

*z*) is descending at this point, the axial component

*n*of the outer normal unit vector to the surface is positive, and vice versa. The triangle similarity validates

_{z}*n*: 1=-

_{z}*dr*: dl, where dl is the length of the curve

*r*(

*z*) along axis element d

*z*. The polar surface element d

*S*around axis element d

*z*is given by d

*S*=2

*πrdl*and the axial optical force in Eq. (2) then takes the following form:

where *r*′=dr/d*z* and *Z*=(*z _{A}*+

*z*)/2.

_{B}In the case of sinusoidally modulated coat the following relation is valid for the object radius

with *R*≥2*A, 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*=(*z _{A}*+

*z*)/2. The object having the mirror symmetry is expressed by relation

_{B}*r*(

*z*̃)=

*r*(-

*z*̃). Substituting

*z*and

*I*(

*z*) into Eq. (28) we get

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

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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