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H. Sosa-Martínez and J. C. Gutiérrez-Vega Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2109

Optical forces on a Mie spheroidal particlearbitrarily oriented in a counterpropagating trap

Héctor Sosa-Martínez and Julio C. Gutiérrez-Vega*

Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey 64849, México*Corresponding author: juliocesar@itesm.mx

Received July 15, 2009; accepted September 15, 2009;posted September 22, 2009 (Doc. ID 114243); published October 19, 2009

The axial and transverse optical forces exerted by Gaussian beams on an arbitrarily oriented and homoge-neous spheroid are calculated and studied within the framework of the Mie theory. The results are applied tostudy the behavior of the forces in a counterpropagating optical trap. We calculate the trapping efficiencies fora wide range of physical parameters, including the beam waist separation distance, the equivalent spheroidradius, the spheroid eccentricity, and the refractive index ratio between the particle and the surroundingmedium. © 2009 Optical Society of America

OCIS codes: 290.0290, 140.7010, 290.5850.

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. INTRODUCTIONn recent years, the use of light to produce rotation, trans-ation, stretching, and other types of manipulation of mi-roparticles such as living cells and other biologicalamples has been achieved by means of a techniquenown as optical trapping [1–6]. The physical principlehat governs optical trapping is the transfer of momen-um from the photons of a light beam to a particle. As aesult of this interaction between the light and the par-icle, a net force will be experienced by the particle. Coun-erpropagating optical traps, and its variants, are moti-ated by the three dimensional control, such as rotation ofhe particle, that can be exerted in a noncontact manner.ince the advent of the multibeam optical technique, eventher noncontact approaches to manipulate small par-icles have been demonstrated with success, e.g., dielec-rophoresis [7,8].

There has been a significant theoretical treatment ofhe forces exerted over particles with simple shapes suchpheres and cylinders [9–11]. The relevance of the mea-urement and characterization of optical forces on non-pherical particles relies on the fact that the actual shapef real particles is not spherical. For instance, humanperm, red blood cells, and many other micro-organismells cannot be adequately modeled by spheres. Addition-lly, shape distortion due to the softness of biological par-icles plays a significant role under the action of radiationressure forces; a nonspherical model accounts better forhis and other effects. Maybe the simplest nonsphericalodel of a particle is given by the spheroidal geometry.ssentially, a spheroid is a sphere that has been uni-

ormly scaled along a particular diameter, i.e., the axis ofhe spheroid. If the scale factor is greater than unity, thepheroid is prolate; otherwise, it is oblate. The scatteringf light beams by spheroids have been studied by severaluthors using electromagnetic expansion methods12–17], the scattering T-matrix theory [18–21], and theeometrical optics framework [22,23].

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In this paper we calculate and study the behavior of thexial and transverse forces on spheroidal dielectric par-icles in a counterpropagating optical trap [24–28]. Wealculate the trapping efficiencies for a wide range ofhysical parameters, including the beam waist separationistance, the equivalent spheroid radius, the spheroid ec-entricity, and the refractive index ratio between the par-icle and the surrounding medium. The size of the particles assumed to be much greater than the wavelength of theptical fields such that the ray-optics approximation (i.e.,he Mie theory) can be applied [3,10]. In the course of ob-aining the forces, we first calculate the scattering andradient forces exerted by a single ray hitting the spher-id in an arbitrary direction. We have modeled the coun-erpropagating fields emerging from the fibers as stan-ard Gaussian beams propagating in a homogeneousedium. Unlike the spherical case, the spheroidal case

annot be reduced to a two-dimensional problem; i.e., theay trajectories inside the spheroid are not confinedithin a meridional plane. Additionally, the spheroidaleometry opens the possibility of having total internal re-ection at the boundary of the particle. This work consoli-ates and extends previous studies on the calculation oforces on spherical and spheroidal particles [10,14–20,23].

. GEOMETRY OF THE PROBLEMe begin by considering the complex amplitude�r�exp�−i�t� of an optical field propagating in a homoge-eous medium essentially (but not necessarily paraxially)long the z axis of a fixed laboratory coordinate system

r = xx + yy + zz. �1�

The optical field illuminates a spheroidal particlehose center is located on the z axis at a distance z=z0, as

hown in Fig. 1. We introduce a second coordinate system

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2110 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 H. Sosa-Martínez and J. C. Gutiérrez-Vega

rcm = xcmxcm + ycmycm + zcmzcm, �2�

ith its origin 0cm fixed at the center of the spheroid andhose axes are parallel to the r frame. We shall refer to

his as the center-of-mass �cm� frame. We assumehroughout that the longitudinal displacement of the par-icle always occurs in the z direction. Thus z0 is taken toe positive.The spheroid is obtained by rotation of an ellipse about

n axis of symmetry. Let us denote by zsp the axis of revo-ution of the spheroid and by

rsp = xspxsp + yspysp + zspzsp �3�

he set of coordinates axes with origin at 0cm but fixed inhe particle. In general, the spheroid system rsp is arbi-rarily rotated with respect to the center-of-mass systemcm.

The physical size and ellipticity of the spheroid areully determined by its semimajor axis a and semiminorxis b, but, alternatively, the semifocal distance f�a2−b2 and the eccentricity e= f /a may be employed toharacterize the particle. Throughout the paper, the semi-xes a and b may change but the volume of the spheroid=4�ab2 /3 is kept constant in order to make meaningful

omparisons under different conditions. The values of and b are assumed to be much greater than the wave-ength of the optical field such that the ray-optics approxi-

ation can be applied. The refractive indices of the spher-id and the surrounding medium are denoted nsp and n0,espectively, and N=nsp /n0 is the effective refractive in-ex of the particle.

. Position of a Point C on the Surface of the Spheroidwo parameters are needed to specify the position of aoint C on the surface of the spheroid. In the coordinateystem of the particle, the position Csp of point C is deter-ined with the prolate spheroidal coordinates as follows:

Csp = �Cx

sp

Cysp

Czsp� = �

b sin � cos �

b sin � sin �

a cos �� , �4�

here �� �0,�� is the polar angle measured from theorth pole of the spheroid and �� �0,2�� is the azimuthalngle with respect to the axis zsp.The rotation of the spheroid with respect to its center ofass is described by two additional parameters. Among

arious possible choices, here we follow the definition

ig. 1. (Color online) Geometry and axes definitions for thepheroid.

iven in Fig. 1 in terms of the azimuthal �� �0,2�� andolar �� �0,�� angles of the spheroid axis zsp relative tohe origin 0cm of the center-of-mass frame. In this system,he position Ccm of the point C on the surface is given by

Ccm = �Cx

cm

Cycm

Czcm� = T��,���

Cxsp

Cysp

Czsp� , �5�

here

T��,�� = �cos � cos � − sin � cos � sin �

sin � cos � cos � sin � sin �

− sin � 0 cos �� �6�

s the rotation matrix.Finally, in the laboratory frame of reference, the posi-

ion C= �Cx ;Cy ;Cz� of the point C on the surface of thepheroid is written as

C = Ccm + �0

0

z0� = T��,���

b sin � cos �

b sin � sin �

a cos �� + �

0

0

z0� . �7�

. Unit Vector Normal to the Spheroidal Surfacet a given point C on the spheroid, the unit vector normal

o the surface is given by the normalized gradient of theurface equation �xsp /b�2+ �ysp /b�2+ �zsp /a�2=1 evaluatedt Csp= �Cx

sp ,Cysp ,Cz

sp�. We obtain

nsp =�Cx

sp/b2�xsp + �Cysp/b2�ysp + �Cz

sp/a2�zsp

��Cxsp/b2�2 + �Cy

sp/b2�2 + �Czsp/a2�2

. �8�

The transformation of coordinates

n = T��,��nsp, �9�

xpresses n in terms of the coordinate system of the labo-atory.

. FORCE EXERTED BY A SINGLE LIGHTAY ON THE SPHEROID

n the ray-optics regime the field U�r� is treated as aundle of individual rays (light pencils) representing infi-itely localized conduits of power, each with appropriate

ntensity, direction, and state of polarization. Each rayropagates in straight lines in media of uniform refrac-ive index and can change direction when it reflects, re-racts and changes polarization at dielectric interfaces ac-ording to the usual Fresnel equations.

At a given observation point r, the power dP associatedith the corresponding ray is written as

dP = �I�r�dS, �10�

here �I�r� is the average intensity at r and dS is a sur-ace element whose normal unit vector points in directionf the ray. For nonparaxial electromagnetic fields, the in-ensity of the ray is given by the Poynting vector. In thease of paraxial beams, each linear component of the elec-ric field can be treated as a scalar wave U�r�U�r�exp�i��r��, the of which intensity is determined by

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H. Sosa-Martínez and J. C. Gutiérrez-Vega Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2111

he energy flux vector U2��t�+kz� evaluated at r. Theomentum per second transported by a ray travelingithin a medium with refractive index n0 is given by

dF =n0

cdP =

n0

c�I�r�dS, �11�

here c is the speed of light in vacuum. The net force ex-erimented by the particle is calculated by integratingq. (11) over the entire surface area that is illuminated by

he incident field.

. Force and Trapping Efficiency of a Single Rayonsider a single ray of power P traveling within the me-ium with refractive index n0 in the direction of the unitector ı0 as shown in Fig. 2. According to Eq. (11), the mo-entum per second carried by the ray is �n0P /c�ı0. The

ncident ray hits the spheroid at point C and is multiplyeflected and refracted at the particle surface, transfer-ing momentum between the ray and the particle. Theorce f exerted by the ray on the particle is the vector sub-raction of the incident ray force and the forces producedy the scattered rays, that is

f =n0P

cı0 − �n0P0

cr0 + �

m=1

� n0Pm

ctm , �12�

here P0, r0 and Pm, tm are the powers and the unit vec-ors in the direction of the first reflected �m=0� and theuccessive refracted �m1� rays, respectively.

The light ray travels in straight lines inside the par-icle and is bounced at its boundary according to the clas-ical laws of reflection and refraction. The incident ım, re-ected rm, transmitted tm, and normal nm unit vectorsre related as follows.For the initial ray reflection �m=0�,

r0 = ı0 + �2 cos 0�n0, �13�

ig. 2. (Color online) Geometry for calculating the force due tohe scattering of a single incident ray of power P by a dielectricpheroid, showing the reflected ray PR0 and infinite set of re-racted rays PT ��m−1R �T .

0 j=1 j m

t0 = ı1 =1

Nı0 − ��1 −

sin2 0

N2 −cos 0

N n0, �14�

here cos 0=−ı0 · n0 and N=nsp /n0 is the refractive indexatio between the particle and the surrounding medium.

For the successive ray reflections �m1�,

rm = ım+1 = ım − �2 cos m�nm, �15�

tm = Nım + ��1 − N2 sin2 m − N cos m�nm, �16�

here cos m= ım · nm.From Fig. 2 we see that the power carried by the scat-

ered rays take the values

P0 = PR0, �17�

Pm = PT0��j=1

m−1

Rj Tm, m 1, �18�

here Rm and Tm are the power reflectances and trans-ittances of the ray at the mth reflection point of the tra-

ectory.Replacing the powers in Eq. (12), we explicitly obtain

f = �n0P/c�q, �19�

here

q � ı0 − R0r0 − T0 �m=1

�

Tm��j=1

m−1

Rj�tm �20�

s a dimensionless vector quantity that will be called therapping efficiency of the ray and depends on the geomet-ic properties of the ray trajectory.

Equations (12)–(19) constitute a complete formulationor calculating the force exerted by a single ray on a di-lectric particle independently of its shape. The sphericalarticle has been of relevant interest for many years. Inhis case, the incident ray, the trajectory inside thephere, and the scattered rays are always confined in aeridional plane that crosses the center of the sphere; in

his way the three-dimensional problem turns into a two-imensional one. Additionally, for a spherical particle thempact angles m, the reflectances Rm, and the transmit-ances Tm are always the same for all ray reflections. As aonsequence, once a ray enters the sphere, it never expe-iences total internal reflection in the subsequent reflec-ions. All these facts together simplify Eqs. (12)–(19),aking it possible to get closed-form expressions for the

orce exerted by a single ray on a spherical particle [3,10].None of the simplifications mentioned above for the

ase of a spherical particle can be applied in the case of apheroidal particle. The most important consequences ofhe lack of spherical symmetry are the existence of new,ifferent types of ray trajectories inside the particle andhe possibility of having total internal reflection at itsoundary. These situations are illustrated in Fig. 3 for thepecial case of meridional ray trajectories. Figures 3(a)nd 3(b) show rotational trajectories without and with to-al internal reflection, respectively. In both situations thentrance point is the same, but the angle of incidence is

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2112 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 H. Sosa-Martínez and J. C. Gutiérrez-Vega

lightly different. Despite this small difference, we canee that the scattered rays behave quite differently. In theotational trajectories, the ray lines avoid the interior of apheroidal caustic, touching its boundary between everywo consecutive reflections at the particle boundary. Onhe other hand, Figs. 3(c) and 3(d) show librational trajec-ories without and with total internal reflection, respec-ively. In the librational case, the ray trajectories are con-ned inside two caustics in the form of confocalyperboloids.Equation (19) applies to both linearly and circularly po-

arized incident beams. At each reflection point, the trans-erse electric (TE) and transverse magnetic (TM) compo-ents of the beam ray must be treated separately toalculate the correct values of the power reflectance andransmittance. For a circularly polarized incident beam,he ray can be considered to be composed of an equalumber of photons in both polarizations; therefore the re-ectance and transmittance may be replaced by its aver-ge value [9,10]. We thus have

Rm = ��RmTE + Rm

TM�/2 m � c

1 m � c� , �21�

Tm = �1 − Rm m � c

0 m � c� , �22�

here c=arcsin�n0 /nsp� is the critical angle of total inter-al reflection and

ig. 3. (Color online) (a), (b) Rotational trajectories in a meridi-nal plane without and with total internal reflection. (c), (d) Li-rational trajectories without and with total internal reflection.

RmTE = �N�ım · nm� − �tm · nm�

N�ım · nm� + �tm · nm��2

, �23�

RmTM = � �ım · nm� − N�tm · nm�

�ım · nm� + N�tm · nm��2

�24�

re the usual power reflectances for the TE and TM po-arizations evaluated at the m1 impact point.

. Decomposition into a Scattering and Two Gradientorceshe expression for the force f exerted by a single ray [Eq.

19)] has been written in a general vector form. It is con-enient to decompose the force into three orthogonal com-onents:

1. The scattering force f�s�, which is parallel to the in-ident ray ı0,

f�s� =n0P

cq�s� =

n0P

c�q · ı0�ı0. �25�

2. The transverse gradient force f��g�, which is perpen-

icular to the plane of incidence of the input ray (i.e., di-ected along of the TE component of the incident ray),

u� �n0 ı0

n0 ı0, �26�

f��g� =

n0P

cq�

�g� =n0P

c�q · u��u�. �27�

3. The parallel gradient force f��g� which is parallel to

he plane of incidence of the input ray (i.e., directed alongf the TM component of the ray),

u� ��n0 ı0� ı0

�n0 ı0� ı0, �28�

f��g� =

n0P

cq�

�g� =n0P

c�q · u��u�. �29�

The three vectors �u� , u� , ı0� define a local Cartesian co-rdinate system mounted on the light ray. The net gradi-nt force exerted on the particle directed perpendicular tohe incident ray is thus given by f�g�=f�

�g�+f��g�.

. Behavior of the Forceso get a feeling for the magnitudes of the forces, we cal-ulate the force exerted by a single circularly polarizeday on the spheroid as a function of the angle of incidence0. The location of the impact point C runs along the me-idian �=0 from the north pole ��=0� to the equatoriallane ��=� /2� for the two situations shown in Fig. 4.In Fig. 5 we show the trapping efficiencies q�s��� ,0�

nd q��g��� ,0� corresponding to the meridional ray inci-

ence depicted in Fig. 4(a) for three spheroids with eccen-ricities e=0,0.3,0.6 and refractive index ratio N=1.2.

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H. Sosa-Martínez and J. C. Gutiérrez-Vega Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2113

ote that for meridional incidence the transverse gradi-nt force q�

�g� vanishes. Red lines in subplots Figs. 4(b),(d), and 4(f) are contour lines of zero gradient force. Wehus see that a ray hitting the particle with normal inci-ence �0=0� will not generate gradient force indepen-ently of the impact point or the ellipticity of the spher-id. For a spherical particle �e=0� the forces depends only

ig. 5. Trapping efficiencies q�s��� ,0� and q��g��� ,0� correspond-

ng to the situation depicted in Fig. 4(a) for spheroids of eccen-ricities e=0,0.3,0.6 and refractive index ratio N=1.2. Note thatach subplot has its own vertical scale.

ig. 4. Plane of incidence of the input ray (a) equal to a meridi-nal plane and (b) orthogonal to the meridional plane.

n the incident angle 0 and exhibit maximum values asigh as approximately 0.5 for rays at angles 0�70°. Theffect of increasing the ellipticity of the particle on theorce distributions is illustrated in Figs. 5(c), 5(d), 6(e),nd 5(f). We see a notable increment of q�s� and q�

�g� for raympacts close to the poles of the spheroid (i.e., ��45°) andncidence angles 0�70°. The scattering force is clearly

uch more affected than the gradient force by ellipticityhat assumes values even larger than the unit. The incre-ent of the force is provoked by the reshaping of the par-

icle and the consequent presence of the total internal re-ection inside the spheroid.Figure 6 shows the three components of the force for

he situation depicted in Fig. 4(b), where now the plane ofncidence is orthogonal to the situation in Fig. 4(a); thats, the plane of incidence is tangent to the hyperboloidalurface at point C. The trapping efficiencies were calcu-ated for spheroids with e=0.3 and e=0.6 and relative re-ractive index N=1.2. From Figs. 6(e) and 6(f) we see thathe magnitude of the perpendicular gradient force f�

�g� isuch smaller than the parallel gradient force f�

�g� for allngles of f impact. For all force components the effect ofhe total internal reflection is more notable, as the inci-ent point approaches the pole and the angle of incidences greater than 70°.

ig. 6. Scattering and gradient trapping efficiencies correspond-ng to the situation depicted in Fig. 4(b) for spheroids of eccen-ricities e=0.3 and e=0.6 and refractive index ratio N=1.2. Notehat each subplot has its own vertical scale.

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2114 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 H. Sosa-Martínez and J. C. Gutiérrez-Vega

. FORCE EXERTED BY THE OPTICAL FIELD. Force Due to a Single Beame now turn our attention to the force exerted on the

pheroid by an optical field. The net force F is calculatedy summing the contributions of the individual rays thatre hitting the surface area of the particle. From Eqs. (11)nd (19) we have

F =� n0

cdP � �

jfj =

n0

c �j

Pjqj, �30�

here Pj and qj are the power and trapping efficiency ofhe jth ray. To define a global trapping efficiency Q it isonvenient to normalize the forces with respect to the to-al beam power PT; thus

F = �n0PT

c Q = �n0PT

c �j

Pj

PTqj. �31�

As optical field, let us consider a monochromatic Gauss-an beam of total power PT, wavenumber k=n0� /c, andalf-waist size w0 traveling along the propagation coordi-ate z. The beam waist is located at z=0 in the coordinateystem of the laboratory. At a given point r, the differen-ial power of the Gaussian beam flowing through a smallegion of the spheroidal surface is given by

Pj = I�r�cos j�S =2PT

�w2 exp�−2r2

w2 cos j�S, �32�

here the beam width reads as w2�z�=w02�1+z2 /zR

2 �, withR=kw0

2 /2 being the Rayleigh range of the beam and �She differential surface on the illuminated part of the par-icle, and 0 is the angle between the ray direction andormal of the spheroid at the incident point. On the otherand, the energy flux vector of a paraxial field U�r�U�r�exp�i��r�� points along the vector �t�+kz, wheret stands for the transverse gradient. On replacing theorresponding values of a Gaussian beam, the direction ofhe rays at the observation point r is given by the unitector

ı =rr + R�z�z

�r2 + R2�z�, �33�

here R�z�=z�1+zR2 /z2� is the radius of curvature of the

pherical wavefronts of the Gaussian beam.On substituting Eqs. (32) and (33) into Eq. (31) the glo-

al trapping efficiency turns out to be

Q = �j

I�rj�cos j

PT qj �34�

=2

��j

exp�− 2rj2/wj

2�

wj2 cos j�Sjqj, �35�

here qj is given by Eq. (20). It is convenient to expresshe force exerted by the beam in terms of the Cartesianomponents of the laboratory system shown in Fig. 1, thats

Qx = Q · x, Qy = Q · y, Qz = Q · z. �36�

o find the forces we need only to multiply these efficien-ies by �n0PT /c�.

. Force Due to Two Counterpropagating Beamsdual beam trap consists of two counterpropagating

eams. The geometry of the problem and the definition ofhe relevant parameters are depicted in Fig. 7. For thisrap the total axial force component Fz and the totalransverse force Fxy are

Fz = Fz1 − Fz2 =n0

c�P1

TQz1 − P2TQz2�, �37�

Fxy = Fxy1 + Fxy2 =n0

c�P1

TQxy1 + P2TQxy2�, �38�

here the subscripts 1 and 2 represent beams 1 and 2, re-pectively. If the beam powers are similar, the longitudi-al forces tend to cancel out with the consequence that Fz

s negligible with respect to the net transverse force Fxy.

. NUMERICAL CALCULATIONSn this section numerical results are presented for a dual-eam optical trap. Qx, Qy, and Qz are evaluated for aange of values of the normalized parameters, includinghe beam waist separation distance S=s /w0, the equiva-ent spheroid radius R0=r0 /w0, and the effective refrac-ive index N=nsp /n0. The semiaxes a and b of the spher-id may change, but the volume of the spheroid V4�ab2 /3 is kept constant and equal to an equivalentphere of volume V=4�r0

3 /3.The trapping efficiency factors are calculated for vari-

us values of the eccentricity e of the spheroid and for dif-erent orientations of the particle characterized by thengles � and � shown in Fig. 1. In all cases, the reflec-ance R and transmittance T at the surface of the particlere each calculated as the average of that correspondingo TE polarization and that corresponding to TM polariza-ion, respectively, for each illuminated element of thepheroid. The laser beam wavelength in vacuum is fixedt �0=1.064 �m, and the laser beam waist is w0=5 �m.he particle is suspended in water, n0=1.33. The wave-

ength is assumed to be the same for both counterpropa-ating beams.

ig. 7. (Color online) Geometry of counterpropagating dualaussian beams illuminating a spheroid.

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H. Sosa-Martínez and J. C. Gutiérrez-Vega Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2115

. Transverse Trapping Forceigure 8 shows the dependence of the transverse trappingfficiency Qx on the transverse offset D=d /w0 for differ-nt values of beam waist spacing, spheroid size, and effec-ive refractive index, respectively. In each case Qx is cal-ulated at Z0=z0 /w0=S /2 where there is no axial force onhe particle. When not indicated in the legends of the fig-res, the parameters values S=30, R0=1, and N=1.2 aresed. Also, the spheroid has an eccentricity e=0.7, andhe particle is oriented with the angles �=0 and �=� /2.ote that when Qz2 and Qx2 are evaluated for beam 2, z0ust be replaced by s−z0, where s is the spacing between

he waists of the two beams.When R0 and N are fixed, the total force exerted over

he particle decreases with increasing S because lesseam power illuminates the sphere; therefore the slopend the maximum value of Qx also decreases. However,here is a point close to D=1.7 where the force is equal forhe four values of S evaluated. When R0 is increased, theaximum value of Qx increases, and it happens for larger

alues of the transverse offset of the particle. When N isncreased the transverse force also increases as a result ofhe greater change in the momentum between the beamnd the particle.The dependence of Qx on the angle � between the z axis

f the spheroid coordinate system and the axis of propa-ation of the laser beam for different values of the eccen-ricity of the spheroid is shown in Fig. 9. The parameters=15, R0=1, N=1.2, D=1.0, and Z0=0 are used, and �0. When e=0 the particle has the shape of a sphere; thusx does not vary when the particle is rotated, as for the

ther cases in which as the eccentricity increases the ef-

ig. 8. (Color online) Variation of the transverse trapping effi-iency Qx as a function of the transverse offset D with (a) theeam waist separation S, (b) the spheroid size R0, and (c) the ef-ective index of refraction N as a parameter. Parameters nothown in the legends are S=30, R0=1, and N=1.2. Also, thepheroid has an eccentricity e=0.7.

ect on the trapping efficiency increases. It is worth notic-ng that Qx is smaller when the long axis of the spheroids oriented in the line of the propagation of the beam.

In Fig. 10, Qx is plotted versus the eccentricity e of thepheroid for several values of the angle �. For small val-es of � the trapping efficiency decreases with increasingccentricity, and for large values of �, Qx increases. Thisan be understood from the fact that the surface area ofhe spheroid that the beam touches changes even whenhe volume remains equal for every spheroid; so for smallalues of e the particle looks thinner, and the force is re-uced, and for large values of e the opposite occurs.

. Axial Trapping Forcehe axial trapping efficiency Qz as a function of thehifted axial offset

Z0� = Z0 − S/2 �39�

s shown for different values of beam waist spacing,pheroid radius, and effective refractive index in Fig. 11.n each case Qz is calculated at D=0, where the particle isentered on the beam axis. When not indicated in the leg-nds of the figures, the parameters values S=30, R0=1,nd N=1.2 are used. Also, the spheroid has an eccentric-ty e=0.7, and the particle is oriented with the angles �0 and �=� /2. Figure 11(a) illustrates that the axial ef-ciency Qz exhibits a region of instability for short waistpace distances such as S=10,20, but for S=30,40 theres a restoring force in the vicinity of Z0�=0 correspondingo a point of stable equilibrium. An opposite effect occursith the equivalent radius of the particle; in this case thexial trapping force passes from a stable to a unstablequilibrium as the size of the particle is increased [Fig.

ig. 9. (Color online) Variation of the transverse trapping effi-iency Qx as a function of the angle � between the z axis of thepheroid coordinate system and the axis of propagation of the la-er beam for different values of the eccentricity of the spheroid.arameters not shown in the legends are S=15, R0=1, and=1.2.

ig. 10. (Color online) Variation of the transverse trapping effi-iency Qx as a function of the eccentricity of the spheroid. Param-ters not shown in the legends are S=15, R =1, and N=1.2.

0

1d

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R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

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2116 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 H. Sosa-Martínez and J. C. Gutiérrez-Vega

1(b)]. When N is increased, the slope of Qz increases, asoes the slope of Qx, creating a more stable trap.

. CONCLUSIONSstudy of the transverse and axial forces exerted by two

ounterpropagating Gaussian beams in a dual-beam trapn a spheroidal particle has been presented. The size ofhe particle was assumed to be much larger than the lightavelength such that the ray-optics approximation coulde employed. The analysis can be straightforwardly ex-ended to optical beams with different transverse struc-ure. We showed the behavior of the trapping efficienciesnder the variation of meaningful physical parameters,

ncluding the beam waist separation distance, the equiva-ent spheroid radius, the spheroid eccentricity, and the re-ractive index ratio between the particle and the sur-ounding medium.

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