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Analyticalapproachofordinaryfrozenwavesforopticaltrappingandmicromanipulation

ArticleinAppliedOptics·April2015

ImpactFactor:1.78·DOI:10.1364/AO.54.002584

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LeonardoAndréAmbrosio

UniversityofSãoPaulo

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MichelZamboni-Rached

UniversityofCampinas

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Analytical approach of ordinary frozen waves foroptical trapping and micromanipulation

Leonardo André Ambrosio1,* and Michel Zamboni-Rached2

1Department of Electrical and Computer Engineering, São Carlos School of Engineering, University of São Paulo,400 Trabalhador São-carlense Ave., 13566-590 São Carlos, Brazil

2Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Street,Toronto, ON, Canada

*Corresponding author: leo@sc.usp.br

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 thegeneralized Lorenz-Mie theory (GLMT) together with integral localized approximation. These waves areconstructed from a suitable superposition of equal-frequency ordinary Bessel beams and are capable ofproviding almost any desired longitudinal intensity profile along their optical axis, thus being of potentialinterest in applications in which intensity localization may be used advantageously, such as in opticaltrapping and micromanipulation systems. In addition, because FWs are composed of nondiffractingbeams, they are also capable of overcoming the diffraction effects for longer distances when comparedto conventional (ordinary) beams, e.g., Gaussian beams. Expressions for the beam-shape coefficients ofFWs are provided, and the GLMT is used to reconstruct their intensity profiles and to predict theiroptical properties for possible biomedical optics purposes. © 2015 Optical Society of AmericaOCIS codes: (050.1940) Diffraction; (140.3460) Lasers; (170.4520) Optical confinement and manipu-

lation; (290.4020) Mie theory.http://dx.doi.org/10.1364/AO.54.002584

1. Introduction

The use of nondiffracting beams in optical trappingand micromanipulating systems is recent and relieson the possibility of simultaneously trapping severalscatterers [1]. Single-beam experimental setups withBessel beams (BBs) for manipulating microsizedstructures appeared in the literature during the firstyears of the past decade, proving to be an efficienttool for two-dimensional (2D) micromanipulation[2–6]. Particles embedded on a host medium canbemechanically displaced and rotated by exchanging(linear or angular) momentum with zero-order orhigher-order nondiffracting beams, and several ofthem, depending on their refractive index relative tothe external medium, can be simultaneously trapped

in one of the various bright or dark annular disks ofintensity, aligned in the radial or in the longitudinaldirections [1–7].

One of the advantages of using BBs in biomedicaloptics is the self-reconstruction property of suchbeams after passing through an obstacle due tothe temporally constant lateral energy providedto the optical axis, which is intrinsically related tothe multiple trap property. Simple optical elementsor computer-generated holograms can be adaptedin optical tweezers systems, for example, to generatearbitrary-order BBs and alternative schemes forthree-dimensional (3D) micromanipulation [2,6,8].

In a recent work, we provided paraxial theoreticalderivations of the beam-shape coefficients (BSCs) fora single ordinary (zero-order) BB using integral local-ized approximation (ILA) [9] in the framework of thegeneralized Lorenz-Mie theory (GLMT) [10–15]. Ana-lytical expressions for theBSCs gmn;TM and gmn;TE,which

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2584 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

ultimately describe the geometrical shape of a laserbeam satisfying Maxwell’s equations, were intro-duced basically considering potential biomedical op-tics applications, viz., for optical force calculationsover microsized scatterers. Because the GLMT ap-plies to any arbitrary optical regime, our methodmay be expected to serve as a robust alternative tothose already available in the literature [16,17].

Because of the nondiffracting nature of BBs, a typeof wave has been proposed in which a suitable super-position of BBs with different longitudinal (or, equiv-alently, transverse) wave numbers can generate analmost unlimited number of longitudinal intensityprofiles (LIPs), even in absorbing media [18–21].The idea is to take advantage of the long-rangepropagation of BBs (when compared to conventionaldiffracting beams) to design a pre-chosen arbitraryLIP for specific purposes, such as in free-space optics,atom guides, optical microlithography, and optical oracoustical bistouries. These waves have been calledfrozen waves (FWs) because they provide localizedwave fields with high transverse localization andstatic intensity envelopes whose longitudinal inten-sity patterns can be chosen a priori. Recently, FWswere experimentally produced for the first time byusing computer-generated holograms and a spatiallight modulator [22].

Because of their intrinsic nondiffracting natureand their high degree of freedom for the specificationof predetermined LIPs, they surely must be viewedas promising alternative laser beams for optical trap-ping experiments, and, therefore, it would be of inter-est to delineate their basic properties, such as opticalforces (or, equivalently, radiation pressure cross sec-tions) and torques.

This work is, therefore, devoted to the theoreticalderivation of the BSCs that correctly describes para-xial FWs in the framework of the GLMT, thusallowing the study of their optical properties when-ever they are used as laser beams for optical trappingand micromanipulation of particles, and is organizedas follows: Section 2 presents the GLMT for (single)ordinary paraxial BBs using ILA, based on a pre-vious work of the authors and assuming both linearand circular polarizations [9]. In Section 3, we out-line the theory of FWs and use the results of Section 2for evaluating the BSCs for this class of laser beams.In Section 4 we provide some examples of prechosenLIPs, all providing good agreement with previousworks [18,19]. Then, in Section 5, we use the resultsand numerical values of Sections 3 and 4 to calculatelongitudinal optical forces exerted by FWs overhomogeneous spherical particles with arbitrary ra-tios between their radii and the wavelength and witharbitrary (both positive and negative) real refractiveindices.

2. ILA Description of Ordinary Bessel Beams

ILA is an alternative to the more time-consumingquadrature schemes [16] and finite series ap-proaches [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 forthe scattering of arbitrary waves by a spherical par-ticle. It avoids the numerical difficulties arising inthe calculation of the BSCs when quadratures areemployed by eliminating the oscillatory behavior ofthe integrands, while still maintaining a flexiblecharacter (only the kernel is to be modified in bothquadratures and in ILA when the nature of the inci-dent 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 behaviorof Bessel functions of order (n� 1∕2), n representingthe corresponding order, was adopted from the plane-wave situation (Lorenz-Mie theory) and successfullyapplied for Gaussian beams, laser sheets, top-hatbeams, and, more recently, zero-order paraxial BBs.(For a review on the GLMT, its different techniquesfor determining the BSCs, and its applications, see[12,13,27] and references therein.)

To start, let us now briefly resume ILA by consid-ering a linearly polarized (along u or v) paraxial BBpropagating along positive w [9,14,15]. From sym-metry considerations of the BSCs, circular polariza-tion can be implemented [28–30]. Notice that theorigin Ouvw is displaced from Oxyz (which will beconsidered as the reference for the center of thespherical particle) by �x0; y0; z0� and that the axesu; v;w are parallel to the axes x; y; z, respectively.Thus, a paraxial ordinary BB with linear polariza-tion in x or y is described, using cylindrical coordi-nates relative to the xyz system, as [9]

E�ρ;ϕ; z� ��xy

�E0J0�χ�e−ikz�z−z0�; (1)

where χ � kρ����������������������������������������������������������ρ2 � ρ20 − 2ρρ0 cos�ϕ − ϕ0�

q, E0 is the

field amplitude, J0�·� is the zero-order Bessel func-tion, kρ (kz) is the transverse (longitudinal) wavenumber, ρ � �x2 � y2�1∕2, ρ0 � �x20 � y20�1∕2, andϕ0 � arctan �y0∕x0�. A factor exp�iωt� is omittedfor simplicity. For a BB, kρ and kz are intrinsicallyrelated to the axicon angle θa, that is, kρ � k sin θaand kz � k cos θa (k � nref2π∕λ, nref being the re-fractive index and λ the wavelength in vacuum),thus forming the well-known k cone in space. Forthe paraxial regime, one must ensure β �sin θa∕�1� cos θa� ≪ 1 [5]. Contrary to [9], we keepthe factor exp�ikzz0�, because the longitudinal dis-placement of the beam relative to the center ofthe scatterer will now be important in the caseof FWs. The magnetic field H can be readily ob-tained from Maxwell’s equations by assuming loss-less media with no sources.

ILA can be applied to Eq. (1) by (i) using a sphericalcoordinate system �r; θ;φ� centered atOxyz; (ii) findingthe corresponding radial fields Er and Hr; (iii) apply-ing to Er and Hr an operator G that performs thetransformations z → 0, kr → �n� 0.5� and introduc-ing prefactors Zm

n , n, and m being integers related

1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2585

to the associated Legendre functions Pmn �·�; and

(iv) performing an integration over angular coordi-nate ϕ. Details are found in [9]. Thus, using the pro-cedure above and introducing the factor exp�ikzz0�,the BSCs gmn;TM and gmn;TE for arbitrary ϕ0 andz0 can be easily obtained, after some simplealgebra, as

g0n;TMfxyg � i2n�n� 1��2n� 1� J1�ϖ�J1�ξ�

�cos ϕ0

sin ϕ0

�exp�ikzz0�;

(2)

gm≠0n;TMfxyg �

12

�−2i

2n� 1

�jmj−1�� 1

∓i

�Jjmj−1�ϖ�Jjmj−1�ξ�

× �cos�jmj − 1�ϕ0∓i sin�jmj − 1�ϕ0�

��1

�i

�Jjmj�1�ϖ�Jjmj�1�ξ�

× �cos�jmj � 1�ϕ0∓i sin�jmj � 1�ϕ0��

× exp�ikzz0�; (3)

g0n;TEfxyg � i2n�n� 1��2n� 1� J1�ϖ�J1�ξ�

�sin ϕ0

− cos ϕ0

�exp�ikzz0�;

(4)

gm≠0n;TEfxyg �

12

�−2i

2n� 1

�jmj−1��∓i

−1

�Jjmj−1�ϖ�Jjmj−1�ξ�

× �cos�jmj − 1�ϕ0∓i sin�jmj − 1�ϕ0�

���i

−1

�Jjmj�1�ϖ�Jjmj�1�ξ�

× �cos�jmj � 1�ϕ0∓i sin�jmj � 1�ϕ0��

× exp�ikzz0�; (5)

where ϖ � sin θa�n� 1∕2�, ξ � ρ0k sin θa, and, as inEq. (1), the terms inside the slashes are related to thecorresponding x or y polarization. Notice that thefactor exp�ikzz0� now appears explicitly. The set ofEqs. (2)–(5) may be further simplified if we considerparticular polarizations and displacements such asϕ0 � 0 or π∕2 for x or y polarization [9]. For theon-axis case (ρ0 � 0), when the longitudinal Poyntingvector does not depend on ϕ (axisymmetric case),the GLMT requires nonzero BSCs to occur only form � �1 or m � −1, with the additional constraintg1n;TM � g−1n;TM � ig1n;TE � −ig−1n;TE. These BSCs reduceto the plane wave case when θa � 0, as expected,because J0�ϖ� → 1 [28]. Numerical values for gmn;TM,together with some examples of ordinary BBdescriptions, can be found elsewhere and will notbe reproduced here [9].

3. Extension to Frozen Waves

Initially, the method of the FWs was developed forthe ideal case where infinity-energy BBs, all withthe same frequency but different longitudinal wavenumbers, were superposed in order to achieve someprechosen LIP. This, however, would require thatour BBs be generated by infinite apertures, thusmaking the method unrealizable in practice[18,19]. But situations may exist in which trun-cated versions of FWs (using transmitters with fi-nite apertures) can furnish the same LIP as itsideal counterpart, with low error [20–22]. This ispossible because of a careful compromise betweenthe radius R of the aperture (which, by the way,limits the maximum distance L up to which theBBs can regenerate themselves) and the numberof lateral lobes of each BB for a radial distanceR (thus imposing the self-reconstruction capacityof this BB [31,32]).

Regardless of how FWs are generated, as a firstapproximation we can assume that a longitudinal(ρ � 0) intensity profile jΨ�ρ � 0; z�j2 � jF�z�j2 hasbeen previously specified inside the chosen interval0 ≤ z ≤ L (or, as assumed hereafter, −L∕2 ≤ z ≤ L∕2)and that the associated FW propagating along �z isgiven by the following scalar solution to the waveequation [19]:

Ψ�ρ; z� �XNq�−N

AqJ0�kρqρ�exp�−ikzqz�: (6)

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

kzq � Q� 2πL

q�0 ≤ Q� 2π

Lq ≤

nrefω

c

�

kρq ���

nrefω

c

�2− k2zq

�1∕2

Aq � 1L

ZL∕2

−L∕2F�z�exp�i2πqz∕L�dz; (7)

where ω is the angular frequency, and Q > 0 is aconstant value [limited according to Eq. (7)] chosenaccording to the given experimental situation andthe desired degree of transverse field localization.Notice that the constraints on Q ensure propagat-ing waves (no evanescent waves) in the positive zdirection. The integrand for Aq differs from pre-vious work in the sign of the exponential becauseof our temporal convention exp�iωt� [19]. Now, ifone displaces the beam by a rectangular distance�x0; y0; z0�, in analogy with the previous section,Eq. (6) can be recast in the form

Ψ�ρ;ϕ; z� �XNq�−N

AqJ0�χq�e−ikzq�z−z0�; (8)

2586 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

where χq � kρq�ρ2 � ρ20 − 2ρρ0 cos�ϕ − ϕ0��1∕2. Onecan assume Eq. (8) to represent our new transverse(x or y) electric field component, the determinationof the BSCs being similar to that of a single BB.In fact, because FWs are linear superposition ofequal-frequency BBs, gm�FW�

n;TM and gm�FW�n;TE will also in-

corporate this linear condition. From Eqs. (2)–(5),one finds

gm�0�FW�n;TMfxyg � i

2n�n�1�2n�1

×XNq�−N

AqJ1�ϖq�J1�ξq��cosϕ0

sinϕ0

�exp�ikzqz0�;

(9)

gm≠0�FW�n;TMfxyg

� 12

�−2i

2n� 1

�jmj−1 XNq�−N

Aq

��1

∓i

�Jjmj−1�ϖq�Jjmj−1�ξq�

× �cos�jmj − 1�ϕ0∓i sin�jmj − 1�ϕ0�

��1

�i

�Jjmj�1�ϖq�Jjmj�1�ξq�

× �cos�jmj � 1�ϕ0∓i sin�jmj � 1�ϕ0��exp�ikzqz0�;

(10)

gm�0�FW�n;TEfxyg � i

2n�n� 1�2n� 1

×XNq�−N

AqJ1�ϖq�J1�ξq�� sin ϕ0

− cos ϕ0

�

× exp�ikzqz0�; (11)

gm≠0�FW�n;TEfxyg � 1

2

�−2i

2n� 1

�jmj−1 XNq�−N

Aq

��

∓i

−1

�Jjmj−1�ϖq�Jjmj−1�ξq�

�cos�jmj − 1�ϕ0∓i sin�jmj − 1�ϕ0�

���i

−1

�Jjmj�1�ϖq�Jjmj�1�ξq�

× �cos�jmj � 1�ϕ0∓i sin�jmj � 1�ϕ0��

× exp�ikzqz0�; (12)

where ϖq � sin θaq�n� 1∕2�, ξq � ρ0k sin θaq, and

sin θaq ��1 −

�kzqk

�2�1∕2

: (13)

Obviously, only an approximation of the desiredLIP is observed from Eqs. (9)–(12) because thesums are necessarily truncated (q ≤ Nmax). We em-phasize that all the copropagating BBs that com-pose our FW have the same frequency (the wavenumber k is fixed) and are capable of maintainingtheir nondiffracting properties up to a longitudinaldistance Z ≈ R∕ tan θa when generated by a finiteaperture of radius R. According to Eq. (13), there-fore, L < Zmin, where Zmin is the field depth of theBB with the smallest longitudinal wave num-ber kz�q�−N� � Q − 2πN∕L.

Let us again look at the on-axis case but now for aFW described in the GLMT by means of Eqs. (9)–(12).As said, the only nonzero coefficients are those withm � �1. This is easily seen by imposing ρ0 � 0. Be-cause ξq � ρ0k sin θaq � 0 and Jv�0� � 0 for v ≠ 0 (vinteger), Eqs. (9)–(12) for x-polarized BBs simplify,for example, to

8>>><>>>:

gm≠1�FW�n;TMfxyg � gm≠1�FW�

n;TEfxyg � 0

gm��1�FW�n;TMfxyg � 1

2

PNq�−N AqJ0�ϖq�exp�ikzqz0�

gm��1�FW�n;TMfxyg � ∓i

2

PNq�−N AqJ0�ϖq�exp�ikzqz0�

; �14�

and, as expected, Eq. (14) represents a summationof the BSCs given by Eq. (5), the contribution ofeach ordinary BB to the LIP being weighted bycomplex amplitudes Aq. Normalization factors canbe introduced into Eqs. (9)–(12) and Eq. (14) to en-sure that BSCs for FWs reduce to the plane-wavecase when θa → 0 and z0 � 0. These factors wouldexplicitly take into account the fact that all BBsare generated by the same (and not by distinct)power sources.

Finally, the correct description of the prechosenLIP, jF�z�j2, using Eq. (6), depends, as we will seein the next section, on the number (2N � 1) of BBsor, equivalently, on N � Nmax, the choice of whichbeing limited to real values of the transverse wavenumbers kρq (kzq < k). This, according to Eq. (7), alsodepends on the longitudinal distance L up to whichthe desired LIP is to be reproduced and on theparameter Q. For example, for Q � Q0nrelω∕c �0.99nrelω∕c, Nmax � 0.01�L∕λm�, where λm is thewavelength in the propagating medium. Thus, fora FW generated by zero-order BBs with λm �λ∕nrel � �1064 nm�∕1.33 (nrel � 1.33 being approxi-mately the refractive index of water, and λ �1064 nm the wavelength usually adopted in opticaltweezers systems), Nmax ∼ 18 for L � 2 mm (thus,the LIP of interest is within jzj < 1 mm). This en-sures propagating waves only, and sin θa is real.For the above values, one can verify that the apertureradius R necessary to generate such LIP mustsatisfy R ≥ 3.76 × 10−4 m.

1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2587

4. FWs Generated by the Generalized Lorenz-MieTheory

The representation of a laser beam in the frameworkof the GLMT allows one to calculate optical forcesand torques, radiation pressure cross sections, scat-tering amplitudes, incident and scattered electro-magnetic fields, etc., only by the knowledge of theMie scattering coefficients and the BSCs gmn;TM andgmn;TE. In this section, we shall consider a few exam-ples of applications of the analytical solutions shownin Section 3 for reproducing the intensity profile ofsome FWs using the GLMT formulation. We assumethat all BSCs have been normalized so that theplane-wave case can be recovered.

A. First Example

In this first example, let us suppose that a constantLIP is desired over the range −Zmax ≤ z ≤ �Zmax bysuperposing 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 � 0.95nrelω∕c�Q0 � 0.95�, which impliesNmax � 15 (that is, our FW will be generated by2Nmax � 1 � 31 BBs) [19]. All BSCs with jmj ≠ 1are zero. According to the previous section, determin-ing g1�FW�

n;TM automatically allows us to find g1�FW�n;TE ,

g−1�FW�n;TM , and g−1�FW�

n;TE . BSCs g1�FW�n;TM (up to n � 400)

can be viewed in Fig. 1. Notice that, as n increases,the amplitude of the BSCs and, consequently, theircontribution to the generated field, decreases. Itshould be emphasized that we have chosen tonormalize the gm�FW�

n;TM s over g1�FW;OA�n;TM so that their

maximum possible value (on-axis case, OA) is 0.50(as seen in Fig. 1). This ensures that the BSCs forordinary FWs will reduce to those expected for planewaves in the limit θaq→0 [27].

The intensity profile associated with this FW canbe evaluated in the GLMT by using expressions forthe electromagnetic fields available in the literature[27]. Figure 2 shows 2D and 3D plots of both theexpected FW generated by using the method avail-able 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 thelateral lobules of the 31 BBs, constantly feeds thelongitudinal 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. 2could, in principle, be significantly improved by usinglower values for Q0 (or, for instance, by choosing alarger L). As long as decreasing the value of Q0 in-creasesNmax, the LIP is expected to get closer to thatof the ideal case (constant). Although this would bedesired for practical purposes, one may not accom-plish it without having to introduce nonparaxialBBs [the condition sin θaq∕�1� cos θaq� ≪ 1 is nolonger valid for all BBs, specially for those withq < 0]. This extension to nonparaxial BBs is cur-rently under investigation and will be consideredin a future work.

B. Second Example

We now consider a LIP in the range −Zmax ≤ z ≤�Zmax that exhibits an exponential growth of theform exp�5z∕L�. Again we impose on-axis BBs withλ � 1064 nm in water. The following parametershave been chosen: L � 2 × 10−3 m, Zmax � 0.10L,Q � 0.99nrelω∕c�Q0 � 0.99), and Nmax � 25 [i.e., 51BBs 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, con-sequently, their contribution to the generated field,increases. The imaginary values (red squares) havebeen multiplied by 10 for visualization purposes.Figure 4 shows the 3D and 2D patterns forjΨ�ρ; z�j2, revealing the exponential growth of the in-tensity along the optical axis as a consequence of aconstant lateral energy feeding.

The results presented in Figs. 2 and 4 of this sec-tion can be compared with their long-range versions[18–21]. Whereas it is relatively easy to fulfill theparaxial requirement of small axicon angles forlong-range FWs, designing FWs for optical trappingand micromanipulation using a scalar theory maypossess serious limitations, mainly because L mustbe of the order of millimeters, which significantlylimits the number of BBs with small axicon angleswhen their wavelengths are of the order of amicrometer.

5. Applications in Optical Trapping andMicromanipulation

We saw in the previous sections that paraxial FWscan be accurately described through knowledge ofthe BSCs and their introduction in the GLMTformulation. The general off-axis BSCs given byEqs. (9)–(12) or their on-axis simplified versions,Eq. (14), may be readily implemented not only inany GLMT algorithm but also in similar methodsby introducing some prefactors [27,33,34].

All above considerations lead us to considerFWs as potential laser beams for biomedical optics

Fig. 1. BSCs g1n;TM (real part as blue triangles, imaginary part asred squares) using ILA for an on-axis FW that provides an idealconstant LIP. As expected, g1n;TM → 0 as n increases, their contri-bution to jF�z�j2 being insignificant for n > 400.

2588 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

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

could also serve, for instance, as alternative laserbeams for optical bistouries.

According to Zamboni–Rached et al., practical FWscould be generated by an array of annular disks, usinglight space modulators or even by means of computerholograms [18–22]. They would take advantage of thenondiffracting and self-reconstruction propertiesof BBs with an additional degree of freedom associ-ated with its longitudinal field localization potential-ities, thus providing an effective 3D trap for opticaltweezers systems.As the advantages ofmultiple trap-ping andmanipulation of particles usingBBs arewellknown, it would certainly be desired, even if onlytheoretically and numerically, to study the trappingproperties of FWs such as optical forces or, equiva-lently, radiation pressure cross sections.

The analytical expressions developed in the pre-vious sections were numerically implemented on aGLMT Fortran code, the evaluation of the BSCsbeing based on ILA. Longitudinal and transverse ra-diation pressure cross sections over homogeneousspherical particles can then be calculated based onthe available literature [38,39]. These cross sectionsare linear with respect to the longitudinal and

Fig. 3. BSCs g1n;TM (real part as blue triangles and imaginary partin red squares) necessary to generate the growing exponential LIP.In comparison with Fig. 1, one sees that a higher number of BSCsare necessary to adequately represent the desired jΨ�ρ; z�j2 in theframework of the GLMT. The imaginary values are scaled by afactor of 10.

Fig. 2. (a) 3D and (b) 2D views of jΨ�ρ; z�j2 generated by the technique developed by Zamboni–Rached et al. [18], using L � 10−3 m,Q0 � 0.95, Nmax � 15, Zmax � 0.1L. All on-axis BBs have λ � 1064 nm and are supposed to generate a constant LIP (along ρ � 0) withinthe range −Zmax ≤ z ≤ �Zmax. (c) and (d) Same as (a) and (b) but using the method developed in Section 3.

1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2589

transverse optical forces exerted over these scatter-ers, respectively, so that the results for the formercan be automatically extended to the latter [27].

Let us refer to the case of the FW with constantLIP (first example, Section 4) and suppose that it im-pinges on a spherical dielectric particle with radiusa � 17.5 μm. The longitudinal radiation pressurecross section (Cpr;z) for this scatterer is sketched inFig. 5 for four positive relative refractive indices(nrel � 0.95, 1.005, 1.01, and 1.20) between the par-ticle and the water. It is clear that, depending onnrel, this type of FW could provide a 3D trap becausepossible points of stable equilibrium can be observedfor z0 ≈ −37 μm (the two zero force points repre-sented by points P and Q for nrel � 1.005 and 1.01,respectively). As z0 changes, scattering forces tendto push the particle back to its stable equilibriumpoint. Finally, Fig. 6 shows plots of Cpr;z for negativerefractive index particles (simultaneous negativepermeability μ and permittivity ε with μ � −1.0),and, as noticed by Ambrosio and Hernández-Figueroa [40–45], achieving a 3D trap with nrel < 0may be much more challenging.

When we replace the previous FW with the grow-ing exponential one, the results for Cpr;z again revealpossible positions of stable equilibrium for a 3D trap.These points occur at z0 ≈ −105 μm and are denoted,

respectively, as P0 and Q0 in Fig. 7. Figure 8 corre-sponds to Fig. 6 when nrel < 0, and again we observerestrictions in achieving an efficient 3D trap for thevalues chosen.

Fig. 5. Longitudinal radiation pressure cross section Cpr;z for theconstant LIP of the previous sections over a dielectric sphericalparticle with radius a � 17.5 μm. Four different nrel�nwater �1.33� are shown, and the center of the scatterer is fixed atz � 0. Possible points of stable equilibrium for nrel � 1.005 and1.01 are indicated as P and Q, respectively. For visualization pur-poses, slopes nrel � 1.005 and 1.01 are scaled by factors of 10 and 5,respectively.

Fig. 4. jΨ�ρ; z�j2 for the growing exponential profile.

2590 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

Incidentally, neither a constant nor an exponentialLIP is capable of providing multiple axial traps, atleast with the results predicted by ILA in the

framework of the GLMT and using the geometricalandelectromagnetic parameters considered. The flex-ibility in designing virtually any desired jF�z�j,however, can be used advantageously to specify anyarbitrary-shape intensity pattern along ρ � 0. Asan example, consider a LIP given by the two invertedparabola

F�z� �8<:−4 �z−l1��z−l2�

�l1−l2�2 ; for l1 ≤ z ≤ l2

−4���2

p �z−l3��z−l4��l3−l4�2 for l3 ≤ z ≤ l4

0 elsewhere;

(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 firstn � 300�jmj � 1� BSCs [Eqs. (9)–(12)] are evaluated,the (normalized) resulting jΨ�ρ; z�j becomes thatshown in Fig. 9 for L � 10−3 m, Zmax � 0.4L, andQ �0.988nrelω∕c (Nmax � 15). This is an adaptation ofFig. 2 from [19] and, in comparison with the previousexamples, a computationally time-consumingpatternin the GLMT due to the fact that the BSCs (jmj � 1)decay very slowly with increasing n. Lateral energyfeeding the LIP is immediately observed in this figure(notice that Fig. 11 represents jΨ�ρ; z�j and notjΨ�ρ; z�j2 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 twopoints of stable equilibrium along z are readily ob-served in Fig. 10, which means that this FW allowsfor multiple axial trapping and, therefore, simultane-ous micromanipulation of organic particles.

The results presented in this paper are limited bythe discrete superposition of zero-order BBs, all withthe same frequency, and this does not give us anycontrol over the transverse intensity profile as theFW propagates along �z. However, this difficultycan be overcome by using higher-order BBs, which,as a consequence, will shift jF�z�j along ρ and createan transverse annular ring of intensity whose inten-sity profile along z can also be designed at will [21].

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

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

Fig. 9. 3D plot of jΨ�ρ; z�j for the two inverted parabola given byEq. (15). Multiple trapping along the optical axis can be obtainedusing FWs.

Fig. 6. Cpr;z for the constant LIP of Fig. 2 assuming a � 17.5 μmand considering negative values for the refractive index of theparticle. In contrast with Fig. 5, Cpr;z is always repulsive (pushingthe scatterer away from the laser source) and no 3D trap would bepossible for such parameters.

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Concerning the experimental generation of the FWbeams, we can cite the use of computer-generatedholograms (CGH) optically reconstructed by spatiallight modulators (SLMs), as it was made in [22,46].More specifically, once the analytical solution of a FWbeam is known, we can use the information about itsamplitudes and phases at the initial plane to definethe complex transmittance hologram function andperform the amplitude CGH.

Distinct values of L, Q0, and Nmax can be adoptedto create some particular FW. The efficient genera-tion of the desired beam, however, depends on theresolution of the SLM. If the amplitudes and phasesof the signal used to create the CGH don’t undergosignificant changes within a spatial interval of theorder of the SLM spatial resolution, then the gener-ation process occurs without further problems.

6. Conclusions

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

can be constructed from a continuous supply of lat-eral energy, a behavior typical of nondiffractingbeams. Furthermore, they can be modeled in orderto provide virtually any prechosen pattern, simulta-neously preserving the nondiffracting character of itsconstituents (BBs) and enabling an additional degreeof freedom due do the axial control over the intensity.

Because of this, they certainly deserve a more pro-found analysis of their capabilities in both long- andshort-range applications (including practical imple-mentations), with possible restrictions of the scalarsolution provided in this paper depending on theLIP imposed for a specific situation. In a case wherethere are axicon angles (for a given BB composing theFW) for which the paraxial approximation must bediscarded, we may still find the BSCs using ILA butnow at a higher computational cost. In fact, thishappens because the azimuth symmetry is brokenin the nonparaxial regime, and the fields of vectorBBs are described in terms of not only a zero-orderbut also of higher-order Bessel functions.Because of their propagating properties, FWs could

be advantageously used as laser beams for opticaltrapping and micromanipulation, and we have alsocalculated the longitudinal radiation pressure crosssections for some specific LIPs and scatterers, includ-ing negative refractive index spherical particles.These cross sections are directly related to the scat-tering forces. The results showed that single FW laserbeams could allow a 3D manipulation of organic andbiological particles, with simultaneous traps alongthe optical axis, when desired. An experimentalimplementation of such beams may confirm ourpredictions.

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

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