Analytical approach of ordinary frozen waves for optical trapping and micromanipulation.

Analytical approach of ordinary frozen waves for optical trapping and micromanipulation Leonardo André Ambrosio1,* and Michel Zamboni-Rached2 1

Department of Electrical and Computer Engineering, São Carlos School of Engineering, University of São Paulo, 400 Trabalhador São-carlense Ave., 13566-590 São Carlos, Brazil 2

Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Street, Toronto, ON, Canada *Corresponding author: [email protected] Received 7 October 2014; revised 11 February 2015; accepted 11 February 2015; posted 11 February 2015 (Doc. ID 224275); published 23 March 2015

The optical properties of frozen waves (FWs) are theoretically and numerically investigated using the generalized Lorenz-Mie theory (GLMT) together with integral localized approximation. These waves are constructed from a suitable superposition of equal-frequency ordinary Bessel beams and are capable of providing almost any desired longitudinal intensity profile along their optical axis, thus being of potential interest in applications in which intensity localization may be used advantageously, such as in optical trapping and micromanipulation systems. In addition, because FWs are composed of nondiffracting beams, they are also capable of overcoming the diffraction effects for longer distances when compared to conventional (ordinary) beams, e.g., Gaussian beams. Expressions for the beam-shape coefficients of FWs are provided, and the GLMT is used to reconstruct their intensity profiles and to predict their optical properties for possible biomedical optics purposes. © 2015 Optical Society of America OCIS codes: (050.1940) Diffraction; (140.3460) Lasers; (170.4520) Optical confinement and manipulation; (290.4020) Mie theory. http://dx.doi.org/10.1364/AO.54.002584

1. Introduction

The use of nondiffracting beams in optical trapping and micromanipulating systems is recent and relies on the possibility of simultaneously trapping several scatterers [1]. Single-beam experimental setups with Bessel beams (BBs) for manipulating microsized structures appeared in the literature during the first years of the past decade, proving to be an efficient tool for two-dimensional (2D) micromanipulation [2–6]. Particles embedded on a host medium can be mechanically displaced and rotated by exchanging (linear or angular) momentum with zero-order or higher-order nondiffracting beams, and several of them, depending on their refractive index relative to the external medium, can be simultaneously trapped 1559-128X/15/102584-10$15.00/0 © 2015 Optical Society of America 2584

APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

in one of the various bright or dark annular disks of intensity, aligned in the radial or in the longitudinal directions [1–7]. One of the advantages of using BBs in biomedical optics is the self-reconstruction property of such beams after passing through an obstacle due to the temporally constant lateral energy provided to the optical axis, which is intrinsically related to the multiple trap property. Simple optical elements or computer-generated holograms can be adapted in optical tweezers systems, for example, to generate arbitrary-order BBs and alternative schemes for three-dimensional (3D) micromanipulation [2,6,8]. In a recent work, we provided paraxial theoretical derivations of the beam-shape coefficients (BSCs) for a single ordinary (zero-order) BB using integral localized approximation (ILA) [9] in the framework of the generalized Lorenz-Mie theory (GLMT) [10–15]. Anam lytical expressions for the BSCs gm n;TM and gn;TE , which

ultimately describe the geometrical shape of a laser beam satisfying Maxwell’s equations, were introduced basically considering potential biomedical optics applications, viz., for optical force calculations over microsized scatterers. Because the GLMT applies to any arbitrary optical regime, our method may be expected to serve as a robust alternative to those already available in the literature [16,17]. Because of the nondiffracting nature of BBs, a type of wave has been proposed in which a suitable superposition of BBs with different longitudinal (or, equivalently, transverse) wave numbers can generate an almost unlimited number of longitudinal intensity profiles (LIPs), even in absorbing media [18–21]. The idea is to take advantage of the long-range propagation of BBs (when compared to conventional diffracting beams) to design a pre-chosen arbitrary LIP for specific purposes, such as in free-space optics, atom guides, optical microlithography, and optical or acoustical bistouries. These waves have been called frozen waves (FWs) because they provide localized wave fields with high transverse localization and static intensity envelopes whose longitudinal intensity patterns can be chosen a priori. Recently, FWs were experimentally produced for the first time by using computer-generated holograms and a spatial light modulator [22]. Because of their intrinsic nondiffracting nature and their high degree of freedom for the specification of predetermined LIPs, they surely must be viewed as promising alternative laser beams for optical trapping experiments, and, therefore, it would be of interest to delineate their basic properties, such as optical forces (or, equivalently, radiation pressure cross sections) and torques. This work is, therefore, devoted to the theoretical derivation of the BSCs that correctly describes paraxial FWs in the framework of the GLMT, thus allowing the study of their optical properties whenever they are used as laser beams for optical trapping and micromanipulation of particles, and is organized as follows: Section 2 presents the GLMT for (single) ordinary paraxial BBs using ILA, based on a previous work of the authors and assuming both linear and circular polarizations [9]. In Section 3, we outline the theory of FWs and use the results of Section 2 for evaluating the BSCs for this class of laser beams. In Section 4 we provide some examples of prechosen LIPs, all providing good agreement with previous works [18,19]. Then, in Section 5, we use the results and numerical values of Sections 3 and 4 to calculate longitudinal optical forces exerted by FWs over homogeneous spherical particles with arbitrary ratios between their radii and the wavelength and with arbitrary (both positive and negative) real refractive indices. 2. ILA Description of Ordinary Bessel Beams

ILA is an alternative to the more time-consuming quadrature schemes [16] and finite series approaches [17] for the computation of the BSCs of

arbitrary laser beams in the framework of the GLMT, which is an extension of the Lorenz-Mie theory for the scattering of arbitrary waves by a spherical particle. It avoids the numerical difficulties arising in the calculation of the BSCs when quadratures are employed by eliminating the oscillatory behavior of the integrands, while still maintaining a flexible character (only the kernel is to be modified in both quadratures and in ILA when the nature of the incident beam is changed, in contrast with finite series) [23–25]. The principle of localization of van de Hulst [26], which is deduced from the asymptotic behavior of Bessel functions of order (n 1∕2), n representing the corresponding order, was adopted from the planewave situation (Lorenz-Mie theory) and successfully applied for Gaussian beams, laser sheets, top-hat beams, and, more recently, zero-order paraxial BBs. (For a review on the GLMT, its different techniques for determining the BSCs, and its applications, see [12,13,27] and references therein.) To start, let us now briefly resume ILA by considering a linearly polarized (along u or v) paraxial BB propagating along positive w [9,14,15]. From symmetry considerations of the BSCs, circular polarization can be implemented [28–30]. Notice that the origin Ouvw is displaced from Oxyz (which will be considered as the reference for the center of the spherical particle) by x0 ; y0 ; z0 and that the axes u; v; w are parallel to the axes x; y; z, respectively. Thus, a paraxial ordinary BB with linear polarization in x or y is described, using cylindrical coordinates relative to the xyz system, as [9] Eρ; ϕ; z

xˆ E0 J 0 χe−ikz z−z0 ; yˆ

(1)

q where χ kρ ρ2 ρ20 − 2ρρ0 cosϕ − ϕ0 , E0 is the field amplitude, J 0 · is the zero-order Bessel function, kρ (kz ) is the transverse (longitudinal) wave number, ρ x2 y2 1∕2 , ρ0 x20 y20 1∕2 , and ϕ0 arctan y0 ∕x0 . A factor expiωt is omitted for simplicity. For a BB, kρ and kz are intrinsically related to the axicon angle θa , that is, kρ k sin θa and kz k cos θa (k nref 2π∕λ, nref being the refractive index and λ the wavelength in vacuum), thus forming the well-known k cone in space. For the paraxial regime, one must ensure β sin θa ∕1 cos θa ≪ 1 [5]. Contrary to [9], we keep the factor expikz z0 , because the longitudinal displacement of the beam relative to the center of the scatterer will now be important in the case of FWs. The magnetic field H can be readily obtained from Maxwell’s equations by assuming lossless media with no sources. ILA can be applied to Eq. (1) by (i) using a spherical coordinate system r; θ; φ centered at Oxyz ; (ii) finding the corresponding radial fields Er and H r ; (iii) applyˆ that performs the ing to Er and H r an operator G transformations z → 0, kr → n 0.5 and introducing prefactors Zm n , n, and m being integers related 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

2585

to the associated Legendre functions Pm n ·; and (iv) performing an integration over angular coordinate ϕ. Details are found in [9]. Thus, using the procedure above and introducing the factor expikz z0 , m the BSCs gm n;TM and gn;TE for arbitrary ϕ0 and z0 can be easily obtained, after some simple algebra, as g0n;TMfxg i y

gm≠0 n;TMfxg y

2nn 1 cos ϕ0 J 1 ϖJ 1 ξ expikz z0 ; sin ϕ0 2n 1 (2)

1 1 −2i jmj−1 J jmj−1 ϖJ jmj−1 ξ 2 2n 1 ∓i × cosjmj − 1ϕ0 ∓i sinjmj − 1ϕ0 1 J jmj1 ϖJ jmj1 ξ i

× cosjmj 1ϕ0 ∓i sinjmj 1ϕ0 × expikz z0 ;

g0n;TEfxg i y

gm≠0 n;TEfxg y

(3)

2nn 1 sin ϕ0 J 1 ϖJ 1 ξ expikz z0 ; − cos ϕ0 2n 1 (4)

× cosjmj 1ϕ0 ∓i sinjmj 1ϕ0 (5)

where ϖ sin θa n 1∕2, ξ ρ0 k sin θa , and, as in Eq. (1), the terms inside the slashes are related to the corresponding x or y polarization. Notice that the factor expikz z0 now appears explicitly. The set of Eqs. (2)–(5) may be further simplified if we consider particular polarizations and displacements such as ϕ0 0 or π∕2 for x or y polarization [9]. For the on-axis case (ρ0 0), when the longitudinal Poynting vector does not depend on ϕ (axisymmetric case), the GLMT requires nonzero BSCs to occur only for m 1 or m −1, with the additional constraint 1 −1 g1n;TM g−1 n;TM ign;TE −ign;TE . These BSCs reduce to the plane wave case when θa 0, as expected, because J 0 ϖ → 1 [28]. Numerical values for gm n;TM, together with some examples of ordinary BB descriptions, can be found elsewhere and will not be reproduced here [9]. 2586


Ψρ; z

N X

Aq J 0 kρq ρexp−ikzq z:

(6)

In Eq. (6), kρq (kzq ) are the transverse (longitudinal) wave numbers of each BB with complex amplitude Aq :

× expikz z0 ;

Initially, the method of the FWs was developed for the ideal case where infinity-energy BBs, all with the same frequency but different longitudinal wave numbers, were superposed in order to achieve some prechosen LIP. This, however, would require that our BBs be generated by infinite apertures, thus making the method unrealizable in practice [18,19]. But situations may exist in which truncated versions of FWs (using transmitters with finite apertures) can furnish the same LIP as its ideal counterpart, with low error [20–22]. This is possible because of a careful compromise between the radius R of the aperture (which, by the way, limits the maximum distance L up to which the BBs can regenerate themselves) and the number of lateral lobes of each BB for a radial distance R (thus imposing the self-reconstruction capacity of this BB [31,32]). Regardless of how FWs are generated, as a first approximation we can assume that a longitudinal (ρ 0) intensity profile jΨρ 0; zj2 jFzj2 has been previously specified inside the chosen interval 0 ≤ z ≤ L (or, as assumed hereafter, −L∕2 ≤ z ≤ L∕2) and that the associated FW propagating along z is given by the following scalar solution to the wave equation [19]:

q−N

∓i 1 −2i jmj−1 J jmj−1 ϖJ jmj−1 ξ 2 2n 1 −1 × cosjmj − 1ϕ0 ∓i sinjmj − 1ϕ0 i J jmj1 ϖJ jmj1 ξ −1

3. Extension to Frozen Waves

2π 2π n ω kzq Q q 0 ≤ Q q ≤ ref L L c 2 1∕2 nref ω − k2zq kρq c Z 1 L∕2 Aq Fzexpi2πqz∕Ldz; L −L∕2

(7)

where ω is the angular frequency, and Q > 0 is a constant value [limited according to Eq. (7)] chosen according to the given experimental situation and the desired degree of transverse field localization. Notice that the constraints on Q ensure propagating waves (no evanescent waves) in the positive z direction. The integrand for Aq differs from previous work in the sign of the exponential because of our temporal convention expiωt [19]. Now, if one displaces the beam by a rectangular distance x0 ; y0 ; z0 , in analogy with the previous section, Eq. (6) can be recast in the form Ψρ; ϕ; z

N X q−N

Aq J 0 χ q e−ikzq z−z0 ;

(8)

where χ q kρq ρ2 ρ20 − 2ρρ0 cosϕ − ϕ0 1∕2 . One can assume Eq. (8) to represent our new transverse (x or y) electric field component, the determination of the BSCs being similar to that of a single BB. In fact, because FWs are linear superposition of mFW equal-frequency BBs, gmFW will also inn;TM and gn;TE corporate this linear condition. From Eqs. (2)–(5), one finds 2nn 1 2n 1 N X cos ϕ0 Aq J 1 ϖ q J 1 ξq expikzq z0 ; × sin ϕ0

gm0FW n;TMfxg i y

q−N

(9) m≠0FW gn;TMf x g y

N 1 1 −2i jmj−1 X J jmj−1 ϖ q J jmj−1 ξq Aq 2 2n 1 ∓i q−N

Obviously, only an approximation of the desired LIP is observed from Eqs. (9)–(12) because the sums are necessarily truncated (q ≤ N max ). We emphasize that all the copropagating BBs that compose our FW have the same frequency (the wave number k is fixed) and are capable of maintaining their nondiffracting properties up to a longitudinal distance Z ≈ R∕ tan θa when generated by a finite aperture of radius R. According to Eq. (13), therefore, L < Zmin , where Zmin is the field depth of the BB with the smallest longitudinal wave number kzq−N Q − 2πN∕L. Let us again look at the on-axis case but now for a FW described in the GLMT by means of Eqs. (9)–(12). As said, the only nonzero coefficients are those with m 1. This is easily seen by imposing ρ0 0. Because ξq ρ0 k sin θaq 0 and J v 0 0 for v ≠ 0 (v integer), Eqs. (9)–(12) for x-polarized BBs simplify, for example, to

× cosjmj − 1ϕ0 ∓i sinjmj − 1ϕ0 1 J jmj1 ϖ q J jmj1 ξq i

× cosjmj 1ϕ0 ∓i sinjmj 1ϕ0 expikzq z0 ; (10) 2nn 1 2n 1 N X sin ϕ0 Aq J 1 ϖ q J 1 ξq × − cos ϕ0 q−N

i gm0FW n;TEfxg y

× expikzq z0 ; m≠0FW gn;TEf x g y

(11)

N 1 −2i jmj−1 X Aq 2 2n 1 q−N ∓i × J jmj−1 ϖ q J jmj−1 ξq −1

cosjmj − 1ϕ0 ∓i sinjmj − 1ϕ0 i J jmj1 ϖ q J jmj1 ξq −1

× cosjmj 1ϕ0 ∓i sinjmj 1ϕ0 × expikzq z0 ;

(12)

where ϖ q sin θaq n 1∕2, ξq ρ0 k sin θaq , and

sin θaq

kzq 1− k

2 1∕2

:

(13)

8 m≠1FW m≠1FW g gn;TEf 0 x x > > yg > n;TMfyg < P 12 N gm1FW q−N Aq J 0 ϖ q expikzq z0 ; 14 n;TMfxyg > > P > : gm1FW ∓i N q−N Aq J 0 ϖ q expikzq z0 n;TMfxg 2 y

and, as expected, Eq. (14) represents a summation of the BSCs given by Eq. (5), the contribution of each ordinary BB to the LIP being weighted by complex amplitudes Aq. Normalization factors can be introduced into Eqs. (9)–(12) and Eq. (14) to ensure that BSCs for FWs reduce to the plane-wave case when θa → 0 and z0 0. These factors would explicitly take into account the fact that all BBs are generated by the same (and not by distinct) power sources. Finally, the correct description of the prechosen LIP, jFzj2 , using Eq. (6), depends, as we will see in the next section, on the number (2N 1) of BBs or, equivalently, on N N max , the choice of which being limited to real values of the transverse wave numbers kρq (kzq < k). This, according to Eq. (7), also depends on the longitudinal distance L up to which the desired LIP is to be reproduced and on the parameter Q. For example, for Q Q0 nrel ω∕c 0.99nrel ω∕c, N max 0.01L∕λm , where λm is the wavelength in the propagating medium. Thus, for a FW generated by zero-order BBs with λm λ∕nrel 1064 nm∕1.33 (nrel 1.33 being approximately the refractive index of water, and λ 1064 nm the wavelength usually adopted in optical tweezers systems), N max ∼ 18 for L 2 mm (thus, the LIP of interest is within jzj < 1 mm). This ensures propagating waves only, and sin θa is real. For the above values, one can verify that the aperture radius R necessary to generate such LIP must satisfy R ≥ 3.76 × 10−4 m. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

2587

4. FWs Generated by the Generalized Lorenz-Mie Theory

The representation of a laser beam in the framework of the GLMT allows one to calculate optical forces and torques, radiation pressure cross sections, scattering amplitudes, incident and scattered electromagnetic fields, etc., only by the knowledge of the Mie scattering coefficients and the BSCs gm n;TM and gm . In this section, we shall consider a few examn;TE ples of applications of the analytical solutions shown in Section 3 for reproducing the intensity profile of some FWs using the GLMT formulation. We assume that all BSCs have been normalized so that the plane-wave case can be recovered. A.

First Example

In this first example, let us suppose that a constant LIP is desired over the range −Zmax ≤ z ≤ Zmax by superposing on-axis BBs with λ 1064 nm in water (nrel 1.33). For such, we chose L 10−3 m, Zmax 0.1L, and Q 0.95nrel ω∕cQ0 0.95, which implies N max 15 (that is, our FW will be generated by 2N max 1 31 BBs) [19]. All BSCs with jmj ≠ 1 are zero. According to the previous section, determin1FW ing g1FW n;TM automatically allows us to find gn;TE ,

−1FW −1FW , and gn;TE . BSCs g1FW gn;TM n;TM (up to n 400) can be viewed in Fig. 1. Notice that, as n increases, the amplitude of the BSCs and, consequently, their contribution to the generated field, decreases. It should be emphasized that we have chosen to 1FW;OA normalize the gmFW so that their n;TM s over gn;TM maximum possible value (on-axis case, OA) is 0.50 (as seen in Fig. 1). This ensures that the BSCs for ordinary FWs will reduce to those expected for plane waves in the limit θaq →0 [27]. The intensity profile associated with this FW can be evaluated in the GLMT by using expressions for the electromagnetic fields available in the literature [27]. Figure 2 shows 2D and 3D plots of both the expected FW generated by using the method available in [19] [Figs. 2(a) and 2(b)] and that obtained

by the GLMT with BSCs up to n 400 [Figs. 2(c) and 2(d)]. Good agreement is achieved, as expected. One clearly sees that lateral energy, provided by the lateral lobules of the 31 BBs, constantly feeds the longitudinal axis close to the region of interest −0.1L ≤ z ≤ 0.1L in order to reproduce the LIP. Notice that the constant LIP as shown in Fig. 2 could, in principle, be significantly improved by using lower values for Q0 (or, for instance, by choosing a larger L). As long as decreasing the value of Q0 increases N max, the LIP is expected to get closer to that of the ideal case (constant). Although this would be desired for practical purposes, one may not accomplish it without having to introduce nonparaxial BBs [the condition sin θaq ∕1 cos θaq ≪ 1 is no longer valid for all BBs, specially for those with q < 0]. This extension to nonparaxial BBs is currently under investigation and will be considered in a future work. B. Second Example

We now consider a LIP in the range −Zmax ≤ z ≤ Zmax that exhibits an exponential growth of the form exp5z∕L. Again we impose on-axis BBs with λ 1064 nm in water. The following parameters have been chosen: L 2 × 10−3 m, Zmax 0.10L, Q 0.99nrel ω∕cQ0 0.99), and N max 25 [i.e., 51 BBs for the superpositions in Eqs. (9)–(12)]. Figure 3 is equivalent to Fig. 1. Notice again that, as n increases, the amplitude of the BSCs and, consequently, their contribution to the generated field, increases. The imaginary values (red squares) have been multiplied by 10 for visualization purposes. Figure 4 shows the 3D and 2D patterns for jΨρ; zj2 , revealing the exponential growth of the intensity along the optical axis as a consequence of a constant lateral energy feeding. The results presented in Figs. 2 and 4 of this section can be compared with their long-range versions [18–21]. Whereas it is relatively easy to fulfill the paraxial requirement of small axicon angles for long-range FWs, designing FWs for optical trapping and micromanipulation using a scalar theory may possess serious limitations, mainly because L must be of the order of millimeters, which significantly limits the number of BBs with small axicon angles when their wavelengths are of the order of a micrometer. 5. Applications in Optical Trapping and Micromanipulation

Fig. 1. BSCs g1n;TM (real part as blue triangles, imaginary part as red squares) using ILA for an on-axis FW that provides an ideal constant LIP. As expected, g1n;TM → 0 as n increases, their contribution to jFzj2 being insignificant for n > 400. 2588


We saw in the previous sections that paraxial FWs can be accurately described through knowledge of the BSCs and their introduction in the GLMT formulation. The general off-axis BSCs given by Eqs. (9)–(12) or their on-axis simplified versions, Eq. (14), may be readily implemented not only in any GLMT algorithm but also in similar methods by introducing some prefactors [27,33,34]. All above considerations lead us to consider FWs as potential laser beams for biomedical optics

Fig. 2. (a) 3D and (b) 2D views of jΨρ; zj2 generated by the technique developed by Zamboni–Rached et al. [18], using L 10−3 m, Q0 0.95, N max 15, Zmax 0.1L. All on-axis BBs have λ 1064 nm and are supposed to generate a constant LIP (along ρ 0) within the range −Zmax ≤ z ≤ Zmax . (c) and (d) Same as (a) and (b) but using the method developed in Section 3.

purposes, e.g., the optical micromanipulation or trapping of viruses, bacteria, biological organelles, etc., using laser beams [3,35–37]. In addition, due to the spatial localization of their optical field, FWs

Fig. 3. BSCs g1n;TM (real part as blue triangles and imaginary part in red squares) necessary to generate the growing exponential LIP. In comparison with Fig. 1, one sees that a higher number of BSCs are necessary to adequately represent the desired jΨρ; zj2 in the framework of the GLMT. The imaginary values are scaled by a factor of 10.

could also serve, for instance, as alternative laser beams for optical bistouries. According to Zamboni–Rached et al., practical FWs could be generated by an array of annular disks, using light space modulators or even by means of computer holograms [18–22]. They would take advantage of the nondiffracting and self-reconstruction properties of BBs with an additional degree of freedom associated with its longitudinal field localization potentialities, thus providing an effective 3D trap for optical tweezers systems. As the advantages of multiple trapping and manipulation of particles using BBs are well known, it would certainly be desired, even if only theoretically and numerically, to study the trapping properties of FWs such as optical forces or, equivalently, radiation pressure cross sections. The analytical expressions developed in the previous sections were numerically implemented on a GLMT Fortran code, the evaluation of the BSCs being based on ILA. Longitudinal and transverse radiation pressure cross sections over homogeneous spherical particles can then be calculated based on the available literature [38,39]. These cross sections are linear with respect to the longitudinal and 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

2589

Fig. 4. jΨρ; zj2 for the growing exponential profile.

transverse optical forces exerted over these scatterers, respectively, so that the results for the former can be automatically extended to the latter [27]. Let us refer to the case of the FW with constant LIP (first example, Section 4) and suppose that it impinges on a spherical dielectric particle with radius a 17.5 μm. The longitudinal radiation pressure cross section (Cpr;z ) for this scatterer is sketched in Fig. 5 for four positive relative refractive indices (nrel 0.95, 1.005, 1.01, and 1.20) between the particle and the water. It is clear that, depending on nrel , this type of FW could provide a 3D trap because possible points of stable equilibrium can be observed for z0 ≈ −37 μm (the two zero force points represented by points P and Q for nrel 1.005 and 1.01, respectively). As z0 changes, scattering forces tend to push the particle back to its stable equilibrium point. Finally, Fig. 6 shows plots of Cpr;z for negative refractive index particles (simultaneous negative permeability μ and permittivity ε with μ −1.0), and, as noticed by Ambrosio and HernándezFigueroa [40–45], achieving a 3D trap with nrel < 0 may be much more challenging. When we replace the previous FW with the growing exponential one, the results for Cpr;z again reveal possible positions of stable equilibrium for a 3D trap. These points occur at z0 ≈ −105 μm and are denoted, 2590


respectively, as P0 and Q0 in Fig. 7. Figure 8 corresponds to Fig. 6 when nrel < 0, and again we observe restrictions in achieving an efficient 3D trap for the values chosen.

Fig. 5. Longitudinal radiation pressure cross section Cpr;z for the constant LIP of the previous sections over a dielectric spherical particle with radius a 17.5 μm. Four different nrel nwater 1.33 are shown, and the center of the scatterer is fixed at z 0. Possible points of stable equilibrium for nrel 1.005 and 1.01 are indicated as P and Q, respectively. For visualization purposes, slopes nrel 1.005 and 1.01 are scaled by factors of 10 and 5, respectively.

framework of the GLMT and using the geometrical and electromagnetic parameters considered. The flexibility in designing virtually any desired jFzj, however, can be used advantageously to specify any arbitrary-shape intensity pattern along ρ 0. As an example, consider a LIP given by the two inverted parabola 8 z−l1 z−l2 for l1 ≤ z ≤ l2 < −4 l1 −l2 2 ; p z−l z−l 3 4 Fz −4 2 for l3 ≤ z ≤ l4 l3 −l4 2 : 0 elsewhere; Fig. 6. Cpr;z for the constant LIP of Fig. 2 assuming a 17.5 μm and considering negative values for the refractive index of the particle. In contrast with Fig. 5, Cpr;z is always repulsive (pushing the scatterer away from the laser source) and no 3D trap would be possible for such parameters.

Fig. 7. Cpr;z for the exponential LIP of the previous sections over a dielectric spherical particle with radius a 17.5 μm. Four different nrel nwater 1.33 are shown, and the center of the scatterer is fixed at z 0. Possible points of stable equilibrium for nrel 1.005 and 1.01 are indicated as P0 and Q0 , respectively. For visualization purposes, the slopes nrel 1.005 and 0.950 are scaled by factors of 10.

Incidentally, neither a constant nor an exponential LIP is capable of providing multiple axial traps, at least with the results predicted by ILA in the

Fig. 8. Similar to Fig. 6 but for a growing exponential LIP.

(15)

where l1 1.5L∕10 − Δz, l2 1.5L∕10 Δz, l3 −1.5L∕10 − Δz, and l4 −1.5L∕10 Δz, with Δz L∕70. When Eq. (15) is inserted in Eq. (7) and the first n 300jmj 1 BSCs [Eqs. (9)–(12)] are evaluated, the (normalized) resulting jΨρ; zj becomes that shown in Fig. 9 for L 10−3 m, Zmax 0.4L, and Q 0.988nrel ω∕c (N max 15). This is an adaptation of Fig. 2 from [19] and, in comparison with the previous examples, a computationally time-consuming pattern in the GLMT due to the fact that the BSCs (jmj 1) decay very slowly with increasing n. Lateral energy feeding the LIP is immediately observed in this figure (notice that Fig. 11 represents jΨρ; zj and not jΨρ; zj2 as in Figs. 3 and 6). Cpr;z curves can be appreciated in Figs. 10 and 11, again for the same nrel as before. Notice that two points of stable equilibrium along z are readily observed in Fig. 10, which means that this FW allows for multiple axial trapping and, therefore, simultaneous micromanipulation of organic particles. The results presented in this paper are limited by the discrete superposition of zero-order BBs, all with the same frequency, and this does not give us any control over the transverse intensity profile as the FW propagates along z. However, this difficulty can be overcome by using higher-order BBs, which, as a consequence, will shift jFzj along ρ and create an transverse annular ring of intensity whose intensity profile along z can also be designed at will [21].

Fig. 9. 3D plot of jΨρ; zj for the two inverted parabola given by Eq. (15). Multiple trapping along the optical axis can be obtained using FWs. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

2591

Fig. 10. Cpr;z for the two inverted parabola configuration given by Eq. (15) on a dielectric spherical particle with radius a 17.5 μm. Two possible points of stable equilibrium or zero force points (in the vicinity of the two peaks of intensity) are observed for both nrel 1.005 (P1 and P2 ) and 1.010 (Q1 and Q2 ), meaning that multiple trapping along the optical axis is indeed possible.

can be constructed from a continuous supply of lateral energy, a behavior typical of nondiffracting beams. Furthermore, they can be modeled in order to provide virtually any prechosen pattern, simultaneously preserving the nondiffracting character of its constituents (BBs) and enabling an additional degree of freedom due do the axial control over the intensity. Because of this, they certainly deserve a more profound analysis of their capabilities in both long- and short-range applications (including practical implementations), with possible restrictions of the scalar solution provided in this paper depending on the LIP imposed for a specific situation. In a case where there are axicon angles (for a given BB composing the FW) for which the paraxial approximation must be discarded, we may still find the BSCs using ILA but now at a higher computational cost. In fact, this happens because the azimuth symmetry is broken in the nonparaxial regime, and the fields of vector BBs are described in terms of not only a zero-order but also of higher-order Bessel functions. Because of their propagating properties, FWs could be advantageously used as laser beams for optical trapping and micromanipulation, and we have also calculated the longitudinal radiation pressure cross sections for some specific LIPs and scatterers, including negative refractive index spherical particles. These cross sections are directly related to the scattering forces. The results showed that single FW laser beams could allow a 3D manipulation of organic and biological particles, with simultaneous traps along the optical axis, when desired. An experimental implementation of such beams may confirm our predictions.

Fig. 11. Cpr;z for the two inverted parabola configuration and assuming a 17.5 μm for negative refractive indices.

The authors wish to thank FAPESP (contract nos. 2014/04867-1 and 2013/26437-6) for supporting this work.

Concerning the experimental generation of the FW beams, we can cite the use of computer-generated holograms (CGH) optically reconstructed by spatial light modulators (SLMs), as it was made in [22,46]. More specifically, once the analytical solution of a FW beam is known, we can use the information about its amplitudes and phases at the initial plane to define the complex transmittance hologram function and perform the amplitude CGH. Distinct values of L, Q0 , and N max can be adopted to create some particular FW. The efficient generation of the desired beam, however, depends on the resolution of the SLM. If the amplitudes and phases of the signal used to create the CGH don’t undergo significant changes within a spatial interval of the order of the SLM spatial resolution, then the generation process occurs without further problems. 6. Conclusions

The BSCs that correctly describe FWs in the framework of the GLMT have been derived under ILA. Because these waves are composed of a suitable superposition of equal-frequency BBs, their LIPs 2592


References 1. V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). 2. V. Garces-Chavez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004). 3. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). 4. A. N. Rubinov, A. A. Afanas’ev, I. E. Ermolaev, Y. A. Kurochkin, and S. Y. Mikhnevich, “Localization of spherical particles under the action of gradient forces in the field of a zero-order Bessel beam. Rayleigh-Gans approximation,” J. Appl. Spectrosc. 70, 565–572 (2003). 5. G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13987 (2007). 6. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003). 7. T. Cižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for biophotonics workstation,” Opt. Express 16, 14024–14035 (2008). 8. T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001).

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces,” Biomed. Opt. Express 2, 1893–1906 (2011). 10. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982). 11. B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988). 12. J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110, 800–807 (2009). 13. G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009). 14. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994). 15. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994). 16. G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gm n in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988). 17. G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. 16, 83–93 (1985). 18. M. Zamboni-Rached, “Stationary optical wavefields with arbitrary longitudinal shape, by superposing equal frequency Bessel beams: frozen waves,” Opt. Express 12, 4001–4006 (2004). 19. M. Zamboni-Rached, E. Recami, and H. E. HernándezFigueroa, “Theory of ‘frozen waves’: modeling the shape of stationary wave fields,” J. Opt. Soc. Am. A 22, 2465–2475 (2005). 20. M. Zamboni-Rached, “Diffraction-attenuation resistant beams in absorbing media,” Opt. Express 14, 1804–1809 (2006). 21. M. Zamboni-Rached, L. A. Ambrosio, and H. E. HernándezFigueroa, “Diffraction-attenuation resistant beams: their higher-order versions and finite-aperture generations,” Appl. Opt. 49, 5861–5869 (2010). 22. T. A. Vieira, M. R. R. Gesualdi, and M. Zamboni-Rached, “Frozen waves: experimental generation,” Opt. Lett. 37, 2034–2036 (2012). 23. G. Gouesbet, G. Gréhan, and B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989). 24. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990). 25. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218–4225 (1998). 26. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957). 27. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer-Verlag, 2011). 28. K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 11, 1812–1817 (1994).

29. K. R. Fen, “Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques,” Ph.D. thesis (Université de Rouen, 1995). 30. G. Gouesbet, “Partial wave expansions and properties of axisymmetric light beams,” Appl. Opt. 35, 1543–1555 (1996). 31. J. Durnin, J. J. Miceli, and J. H. Eberli, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). 32. J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). 33. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988). 34. 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). 35. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987). 36. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. USA 94, 4853–4860 (1997). 37. D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Letters 209, 171–176 (2004). 38. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994). 39. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169–179 (1998). 40. L. A. Ambrosio and H. E. Hernández-Figueroa, “Doublenegative optical trapping,” presented at Biomedical Optics (BIOMED/Digital Holography and Three-Dimensional Imaging) on CD-ROM’10/OSA, Miami, Florida, 11–14 April 2010. 41. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double-negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17, 21918–21924 (2009). 42. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express 1, 1284–1301 (2010). 43. L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010). 44. L. A. Ambrosio and H. E. Hernández-Figueroa, “Radiation pressure cross-sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt. 50, 4489–4498 (2011). 45. L. A. Ambrosio and H. E. Hernández-Figueroa, “Spin angular momentum transfer from TEM00 focused Gaussian beams to negative refractive index spherical particles,” Biomed. Opt. Express 2, 2354–2363 (2011). 46. T. A. Vieira, M. Zamboni-Rached, and M. R. R. Gesualdi, “Modeling the spatial shape of nondiffracting beams: experimental generation of Frozen waves via holographic method,” Opt. Commun. 315, 374–380 (2014).

1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

2593

Analytical approximations for spiral waves.

An extended analytical approach for diffuse optical imaging.

Combined optical micromanipulation and interferometric topography (COMMIT).

Experimental and analytical comparative study of optical coefficient of fresh and frozen rat tissues.

Analytical expressions for the angle of arrival fluctuations for optical waves' propagation through moderate-to-strong non-Kolmogorov refractive turbulence.

Producing acoustic frozen waves: simulated experiments.

Optical trapping of silver nanoplatelets.

Optical trapping and manipulation of nanostructures.

Optical trapping in plant cells.

Bubble wrap for optical trapping and cell culturing.

Focus issue introduction: optical cooling and trapping.

Speckle optical tweezers: micromanipulation with random light fields.

Introduction: Optical trapping and applications feature issue.

Dynamic interfacial trapping of flexural waves in structured plates.

Optical Trapping of Gold Nanoparticles in Air.

2D Magneto-optical trapping of diatomic molecules.

Micromanipulation of mitotic chromosomes in PTK2 cells using laser-induced optical forces ("optical tweezers").

Optical trapping of the anisotropic crystal nanorod.

Cleaved fiber optic double nanohole optical tweezers for trapping nanoparticles.

Plasmonic optical trapping in biologically relevant media.

Optical trapping: Motor molecules in motion.

Dark-hollow optical beams with a controllable shape for optical trapping in air.

Optical rogue waves for the inhomogeneous generalized nonlinear Schrödinger equation.

Magnetic trapping of molecules via optical loading and magnetic slowing.