Optical alignment and confinement of an ellipsoidalnanorod in optical tweezers: a theoretical study

Jan Trojek, Lukáš Chvátal, and Pavel Zemánek*

Institute of Scientific Instruments of the ASCR, v.v.i., Academy of Sciences of the Czech Republic,Královopolská 147, 612 64 Brno, Czech Republic

*Corresponding author: zemanek@isibrno.cz

Received December 6, 2011; revised February 19, 2012; accepted March 9, 2012;posted March 9, 2012 (Doc. ID 159516); published June 6, 2012

Within the Rayleigh approximation, we investigate the behavior of an individual ellipsoidal metal nanorod that isoptically confined in three dimensions using a single focused laser beam. We focus on the description of the op-tical torque and optical force acting upon the nanorod placed into a linearly polarized Gaussian beam (scalardescription of the electric field) or a strongly focused beam (vector field description). The study comprisesthe influence of the trapping laser wavelength, the angular aperture of focusing optics, the orientation of the el-lipsoidal nanorod, and the aspect ratio of its principal axes. The results reveal a significantly different behavior ofthe nanorod if the trapping wavelength is longer or shorter than the wavelength corresponding to the longitudinalplasmon resonance mode. Published experimental observations are compared with our theoretical predictionswith satisfactory results. © 2012 Optical Society of America

OCIS codes: 140.7010, 350.4855.

1. INTRODUCTIONOptical trapping of micrometer- and nanometer-scale parti-cles by lasers has found many useful applications over thepast two decades [1–3]. While optical trapping of microparti-cles is nowadays a very common tool in physical, biological,and chemical sciences, experimental extension to nanoparti-cles is not straightforward: the gradient force responsible forthe trapping falls down with the particle volume and is easilyovercome by thermal forces arising from molecular fluctua-tions of the surrounding medium.

Metallic nanoparticles can resolve the difficulties with thestability of optical trapping due to their larger refractive indexrelative to the surrounding medium, which leads to strongeroptical trapping compared to dielectric nanoparticles at wave-lengths longer than the localized surface plasmon resonancewavelength [4]. At the plasmon resonance wavelength, collec-tive oscillations of conduction electrons lead to enhanced lightabsorption and scattering, and the optical trapping forcechanges significantly near this wavelength. Spherical metallicnanoparticles are localized at the beam axis for trapping wave-lengths longer than the plasmon resonance wavelength butrepelled from the axis for shorter trapping wavelengths [5,6].The plasmon resonant wavelength can be tuned over a widespectral range by varying the size and shape of themetallic par-ticles [7]. Elongated nanoparticles tend to orient with respectto the polarization or propagation axis of the incident trappingbeam [8–10]. By monitoring the polarized light emission froman individual gold nanorod (NR), it is possible to determine itsorientation as a function of time [11]. The large signal-to-noiseratio, photochemical stability, nonblinking and nonbleaching,fast time response, and small size of theseNRsmake them idealprobes for orientation sensing in material science and molecu-lar biology [12–14].GoldNRsare easily takenupbymammaliancells and are frequently used in biological experiments [15].

Inorganic nanotubes have been successfully employed forsensing of single DNA molecule translocation in nanofluidicdevices [16].

Optical control of the orientation and position of NRs en-hances the possibilities of their applications. Holographic op-tical tweezers have been employed for trapping, deposition,and photochemical transformation of single wall carbon nano-tubes [17]. NRs have been also utilized as light driven nanoro-tors [9,18]. Rotation of the beam polarization inducedcontrolled NR rotation. Such a principle can be applied to con-trol and manipulate important elements in nanoelectronics,such as nanotubes and nanowires, measure the local viscoe-lastic properties of biological and rheological systems, or per-form torsional analysis experiments with biomolecules suchas DNA [19]. Tiny volumes of NRs compared to conventionaldielectric particles pave the way to study the mechanical het-erogeneity of liquid environments on a nanoscale level. Therotational and translational motions of NRs can be detectedusing holographic video microscopy that also provides realtime feedback for their three-dimensional (3D) micromanipu-lation [20]. The first quantitative measurements of the opticaltorque exerted on a single gold NR in a polarized 3D opticaltrap were presented by Ruijgrok et al. [21]. They determinedthe torque both by observing the time-averaged orientationdistribution and by measuring the dynamics of the rotationalBrownian fluctuations. Stable optical trapping of a single goldNR immersed in water was demonstrated for an objective lenswith high NA and trapping wavelengths longer than the long-itudinal plasmon resonances of the NRs [22,23]. The stiffnessof a 3D optical trap was measured in the case of a single op-tically trapped gold NR [23]. It was shown in [23] that the op-tical forces correlate with NR polarizability, which depends onboth the particle volume and its aspect ratio.

In this paper we present an in-depth theoretical study of NRbehavior in a single focused Gaussian beam in both scalar and

1224 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Trojek et al.

1084-7529/12/071224-13$15.00/0 © 2012 Optical Society of America

vector descriptions. We focus on the orientation and lateraland longitudinal confinement of a single gold, silver, and silicaNR, and we show that the usually used scalar field description(paraxial Gaussian beam) cannot explain optical trapping ob-served in the experiments mentioned above. However, moreexact vector field description provides an optical potentialwell deep enough for stable confinement of NRs.

The paper is organized into logical steps leading to the de-termination of the stable orientations and positions of an el-lipsoidal NR trapped by optical tweezers. Section 2 introducesthree different descriptions of the incident focused laser beamand compares their spatial intensity distributions. Section 3presents theoretical descriptions of optical force and torqueacting upon the NR if the Rayleigh approximation and scalaror vector descriptions of the incident beam are applied.Section 4 analyzes step by step the NR behavior if the NR isilluminated by a linearly polarized focused beam. Again boththe scalar and vector descriptions of the incident fields arecompared. In Subsection 4.A the torques acting upon theNR are analyzed and the stable NR alignment is determined.In Subsection 4.B we assume the NR is localized on the opticalaxis and we look for its longitudinal stable position. Conse-quently, knowing the stable NR orientation and longitudinalposition, in Subsection 4.C we analyze the stability of the lat-eral trapping if the NR is placed on the optical axis. Since, incontrast to spherical objects, the stability of the NR confine-ment is influenced by the NR orientation, we analyze inSubsection 4.D the stability of the NR optical trapping depend-ing on the NR orientation. Since we consider metal NRs thatabsorb energy from the trapping beam and thus heat up,Section 5 analyzes the effects of temperature increase of theNR trapped by the laser beam. Section 6 compares presentedtheoretical results with experimental data published by otherresearch groups.

2. DESCRIPTION OF THE INCIDENT LASERBEAMCorrect description of the laser beam focused by a highnumerical aperture (NA) objective is a challenging topic[24–27], especially if optical aberrations are taken into ac-count [28]. In this study we neglect the optical aberrationsand focus on the two most frequently used descriptions:scalar Gaussian beam description (SD) [29] and vector de-scription (VD) based on the Debye approximation of the planewave diffraction on a circular aperture [25,26].

A. Scalar Beam DescriptionLet us consider a laser beam propagating in the medium ofrefractive index nm along z-axis and polarized along x-axis.The beam is focused by a lens with an NA and an angularaperture Θ related as

NA � nm sin Θ: (1)

The electric field of a moderately focused beam can be de-scribed in the paraxial approximation as the scalar Gaussianbeam [27,29]:

Ex�x; y; z� � E0w0

w�z� exp�−�x2 � y2�

w2�z�

�

× exp�ikz� ik

x2 � y2

2R�z� − iζ�z��; (2)

Ey�x; y; z� � 0; Ez�x; y; z� � 0; (3)

where E0 is the electric field intensity at the beam focusx � y � z � 0, k � 2π∕λ is the wavenumber, and λ is thebeam wavelength in the surrounding medium of refractive in-dex nm. The following quantities denote the well-knownparameters of the Gaussian beam: the beam width w�z� �w0

��������������������������1� �z∕zR�2

p, the wave front radius R�z� � z�1� �zR∕z�2�,

the Guoy phase ζ�z� � arctan�z∕zR�, the Rayleigh rangezR � kw2

0∕2, and the beam waist radius w0. Let us recallthe beam waist is equal to the radial distance from the opticalaxis at the focal plane where the electric field intensity fulfills

jEx�w0; 0; 0�j � jEx�0; w0; 0�j � jEx�0; 0; 0�j∕e: (4)

This paraxial scalar (zeroth-order) Gaussian beam approxima-tion can be enhanced to higher orders by taking more terms inthe Taylor expansion of the vector potential A when theHelmholtz equation for A is solved [27]. Such field descriptionis generally vectorial. The fifth-order Gaussian beam approx-imation (G5) [27] is among the most frequently used descrip-tions related to the theoretical studies of optical forces actingupon particles [30]. Therefore we mention it here to link it toother descriptions used in this paper. The G5 approximationgives the same field distribution on the optical axis as SD; ithas only a transversal electric field component in yz plane andan extra longitudinal field component in xz plane:

E�0; y; z� � �EG5x �0; y; z�; 0; 0�; (5)

E�x; 0; z� � �EG5x �x; 0; z�; 0; EG5

z �x; 0; z��: (6)

B. Vector Beam DescriptionThe following Debye approximation of the diffraction of aconvergent spherical wave on a circular aperture [25,26] isconsidered as highly appropriate for the description of astrongly focused beam without aberrations:

Ex�x; y; z� � −i2k�I0 � I2

x2 − y2

x2 � y2

�; (7)

Ey�x; y; z� � −ikI2xy

x2 � y2; (8)

Ez�x; y; z� � −kI1x�����������������

x2 � y2p ; (9)

where

I0�r; z;Θ� �Z Θ

0A�α� sin α�1� cos α�J0�kr sin α�

exp�ikz cos α�dα; (10)

I1�r; z;Θ� �Z Θ

0A�α� sin2 α J1�kr sin α� exp�ikz cos α�dα;

(11)

I2�r; z;Θ� �Z Θ

0A�α� sin α�1 − cos α�J2�kr sin α�

exp�ikz cos α�dα; (12)

Trojek et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1225

where r ������������������x2 � y2

pdenotes the radial distance from the

optical axis z, Jn denotes the Bessel function of the first kindand nth order, and the angular amplitude distribution in theaplanatic projection follows [26]

A�α� � A0������������cos α

p: (13)

The amplitude A0 can be related to the electric field intensityat the beam focus Ex�0; 0; 0� using Eqs. (7)–(10):

Ex�0; 0; 0� �k15

A0�8 − 3 cos5∕2 Θ − 5 cos3∕2 Θ�: (14)

Equation (14) thus enables a direct comparison of the vectorand the scalar beam descriptions if we set the electric fieldintensity at the beam focus identical in both cases, i.e.,E0 � Ex�0; 0; 0�. Furthermore, one can immediately concludethat the vector beam has only Ex-component in yz plane. Thisproperty enables a direct comparison of lateral profiles ofboth fields assuming that the effective Gaussian beam waistw0 for the SD model corresponds to the radial position wherethe vector beam satisfies jEx�0; w0; 0�j � jEx�0; 0; 0�j∕e.Figure 1 compares the beam intensity profiles in all consid-ered approximations SD, G5, and VD for different angularapertures Θ. The top left plot of jE�x; 0; 0�j reveals noticeabledifferences between all models along the direction of thebeam polarization. The bottom left plot demonstrates negligi-ble differences in the higher values of the electric field inten-sity profiles jE�0; y; 0�j between all three compared modelsalong y-axis. The plots of the longitudinal field intensity pro-files jE�0; 0; z�j in the right column prove stronger longitudinalfield gradients (Dz) obtained from VD.

3. OPTICAL FORCES AND TORQUESIn this study, we focus on finding the general rules for the be-havior of NR in a focused laser beam and therefore we lookfor an analytical description of the force interaction betweenthe NR and the laser beam. We adopt the simple Rayleigh ap-proximation, which is only valid for an NR much smaller com-pared to the trapping wavelength, so that the electromagneticfield inside the NR can be considered constant. In the case oflarger NRs, mainly numerical methods based on the transitionmatrix (T-matrix) approach [31–34] are used to investigate op-tical trapping in the focal region of a high NA optical system.Radiation torque and trapping force on highly elongated linearchains formed from nanospheres were also studied by themeans of the T-matrix approach [35]. It was found that theshortest chains were stably oriented transversely to the opti-cal axis, whereas the longest ones aligned themselves alongthe optical axis. Very good agreement between the T-matrixand the finite difference time domain (FDTD) methods hasbeen reported by Simpson et al. [34] for different symmetrygroups of nanoparticles. Also the coupled dipole method(CDM) was used to compute optical torque acting upon a mi-cropropeller illuminated by a circularly polarized plane wave[36]. In contrast to the above numerical models, the rigorousdiffraction theory can be used only in some special cases ofparticle geometry, e.g., for elliptically shaped dielectric nano-cylinders [37,38].

A. Rayleigh ApproximationAssuming an NR is much smaller compared to the wavelength(NR length l ≪ λ), the components of the optical force F ξ(with ξ � x; y; z) and the optical torque Mξ acting on suchan NR are given by the following formulas [10,34,39,40]:

F ξ �12Rfp� · ∂ξEg; (15)

Mξ �12Rfp� × Egξ; (16)

where p � αE is the dipole moment of the NR induced by thebeam and α denotes the polarizability tensor.

B. Polarizability Tensor of an Ellipsoidal NanorodLet us consider an ellipsoidal NR with semiaxes a, b, c andvolume V � 4πabc∕3. The polarizability αj along the j-axisof the ellipsoid is given by [41,42]

αj �α0j

1 −ik3α0j6πϵ

; α0j �ϵ0n2

mV

Lj � 1m2−1

; m � np

nm; (17)

L1 �Z

∞

0

abcds

2�s� a2�3∕2�s� b2�1∕2�s� c2�1∕2 ; (18)

where np is the refractive index of the NR, nm is the refractiveindex of the surrounding medium, and the same formula[Eq. (18)] is valid forL2 andL3 with cyclical exchanges of indexand axis.

In the case of an ellipsoidal NR with one long axis lL andtwo identical short axes lS , the polarizability tensor has thefollowing form in its principal axes coordinate system:

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

x [nm]

|E/E

0|

0 500 1000 1500 2000 25000

0.2

0.4

0.6

0.8

1

z [nm]

|E/E

0|

0 200 400 600 8000

0.2

1/e

0.6

0.8

1

y [nm]

|E/E

0|

0 500 1000 1500 2000 2500−1

0

1

2

3

z [nm]

−D

z |E/E

0| [1

/µm

]

Θ = 30°Θ = 70°

VDSDG5

w0

Fig. 1. (Color online) Comparison of three different descriptions ofthe incident laser beam. Normalized profiles of the absolute values ofthe electric field intensity in the incident beam along x (top left), y(bottom left), and z (top right) axis are shown for the vector descrip-tion [VD—solid, Eqs. (7)–(9)], the scalar description [SD—dot,Eq. (2)], and the fifth-order description [G5—dash, Eq. (27)]. Beamwaist w0 (corresponding to the radial distance along y-axis wherejEj � jE0j∕e, which is denoted by the dash-dotted line) is identicalfor all considered approximations, and two different angular aper-tures of the VD beam,Θ � 30° (green) andΘ � 70° (red), are studied.The electric field is normalized to its absolute value at the beam focusjE0j. The negative gradient (−Dz) of the longitudinal field is plottedalong the optical axis z (bottom right). The following quantitiesand values are used in the calculations: vacuum laser wavelengthλ0 � 500 nm, nm � 1.33.

1226 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Trojek et al.

α �������α0L � iα00L 0 0

0 α0S � iα00S 00 0 α0S � iα00S

������; (19)

with real components α0L, α00L, α0S , α00S calculated from Eqs. (17)and (18).

We assume the NR is initially oriented with its long axis Lparallel to the polarization of the beam, i.e., along the x-axis.The general orientation of the NR in the coordinate system x,y, z connected to the incident beam is described using the azi-muthal angle ψ and the polar angle θ (see Fig. 2). The principalaxes of the NR correspond to the axes of the rotated system ofcoordinates x00y00z00, where the NR long axis is oriented alongthe x00-axis. Consequently, the transformation matrix betweenthe coordinate systems xyz↣x00y00z00 has the following form:

R �������cos θ 0 − sin θ0 1 0

sin θ 0 cos θ

������

������cos ψ sin ψ 0− sin ψ cos ψ 0

0 0 1

������; (20)

where both angles increase in accordance with the righthand rule.

The polarizability tensor of the rotated NR in the system ofcoordinates xyz is then done by the matrix product

A � R−1α R; (21)

and the NR dipole moment p excited by the total electric fieldE at the NR position r � �x; y; z� is

p � A�ψ ; θ�E�x; y; z�: (22)

C. Optical Force and Torque in a Paraxial GaussianBeamIn order to obtain the simplest analytical expressions for theoptical force and the optical torque acting upon an ellipsoidalNR, we first consider an incident Gaussian beam given by thescalar description in Eq. (2) and use it in Eqs. (15) and (16):

Fx�x; y; z;ψ ; θ� � −E20

z2Rkx

2�z2R � z2�2 exp�−kzR�x2 � y2�

z2R � z2

�

�a0�ψ ; θ�zR − a00�ψ ; θ�z�; (23)

Fy�x; y; z;ψ ; θ� � −E20

z2Rky

2�z2R � z2�2 exp�−kzR�x2 � y2�

z2R � z2

�

�a0�ψ ; θ�zR − a00�ψ ; θ�z�; (24)

Fz�x; y; z;ψ ; θ� � −E20

z2R4�z2R � z2�3 exp

�−kzR�x2 � y2�

z2R � z2

�

× f−2ka00�ψ ; θ�z4 � 2a0�ψ ; θ�z3

� �kx2 � ky2 − 4kz2R � 2zR�a00�ψ ; θ�z2

− �2kzR�x2 � y2� − 2z2R�a0�ψ ; θ�z− �kz2R�x2 � y2� � 2kz4R − 2z3R�a00�ψ ; θ�g;

(25)

Mx�x; y; z;ψ ; θ� � 0; (26)

My�x; y; z;ψ ; θ� � −E20

z2R4�z2R � z2� exp

�−kzR�x2 � y2�

z2R � z2

�

× �α0L − α0S� sin 2θ cos ψ ; (27)

Mz�x; y; z;ψ ; θ� � −E20

z2R4�z2R � z2� exp

�−kzR�x2 � y2�

z2R � z2

�

× �α0L − α0S� sin 2ψ cos2 θ; (28)

where

a0�ψ ; θ� � α0S � �α0L − α0S�cos2 θ cos2 ψ ; (29)

a00�ψ ; θ� � α00S � �α00L − α00S�cos2 θ cos2 ψ : (30)

The torque expression (28) for x � y � z � 0 and θ � 0 ex-actly agrees with the analogical one from the appendix in[10], where the authors considered an incident plane waveand only a two-dimensional rotation of the NR. More general3D rotation of a spheroid placed at the focus of a Gaussianbeam was published in [43]. Although they chose anotherway to describe the NR spatial orientation, their torque vectoralso matches with Eq. (28) for x � y � z � 0.

Since in the text below we consider the particles areoptically trapped on the optical axis, let us simplify theexpressions accordingly:

Fx�0; 0; z;ψ ; θ� � 0;

Fy�0; 0; z;ψ ; θ� � 0;

Mx�0; 0; z;ψ ; θ� � 0; (31)

Fz�0; 0; z;ψ ; θ� � E20

z2R2�z2R � z2�2 �a

00�ψ ; θ�k�z2R � z2�

− a0�ψ ; θ�z − a00�ψ ; θ�zR�; (32)

Fig. 2. (Color online) Orientation of the nanorod (NR) in the initialcoordinate system xyz and the rotated coordinate system x00y00z00. TheNR long axis is parallel to x00-axis. The axis x0 denotes the projection ofthe axis x00 to the plane xy.

Trojek et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1227

My�0; 0; z;ψ ; θ� � −E20

z2R4�z2R � z2� �α

0L − α0S� sin 2θ cos ψ ;

(33)

Mz�0; 0; z;ψ ; θ� � −E20

z2R4�z2R � z2� �α

0L − α0S� sin 2ψ cos2 θ:

(34)

Physical consequences coming from these equations arediscussed in Section 4.

D. Optical Force and Torque in a Tightly Focused BeamWe use the vector description of the beam presented inSubsection 2.B and insert it into Eqs. (15) and (16). If weagain assume the NR is placed on the optical axis, only thex-component of the electric field E is nonzero, and we obtainfor the components of the force and torque

Fx�0; 0; z;ψ ; θ� � 0;

Fy�0; 0; z;ψ ; θ� � 0;

Mx�0; 0; z;ψ ; θ� � 0; (35)

Fz�0; 0; z;ψ ; θ� �12Rf�a0�ψ ; θ� � ia00�ψ ; θ���

× Ex�0; 0; z��∂zEx�0; 0; z�g; (36)

My�0;0;z;ψ ;θ��−14

��Ex�0;0;z���2�α0L −α0S�sin 2θ cos ψ ; (37)

Mz�0;0;z;ψ ;θ��−14

��Ex�0;0;z���2�α0L −α0S�sin 2ψ cos2 θ: (38)

4. NANOROD OPTICAL ALIGNMENT ANDCONFINEMENTA. Stable Alignment of the NanorodIf we compare Eqs. (33) and (34) with Eqs. (37) and (38), oneimmediately sees that the torque components have the samedependence on the NR orientation (angles θ and ψ) and shape(α0L − α0S). Therefore, there would be no difference betweenthe NR rotation if the NR were placed on the optical axisand the scalar Gaussian beam or the focused vector beam de-scription were used. Stable alignment of the NR in the beam isdone by the torque equal to zero and the negative derivative ofthe torque components with respect to the related angles. Aswe can see, such stable orientations do not depend on the NRposition along z-axis and, therefore, they can be investigatedindependently of the stable longitudinal position of thetrapped NR. A careful inspection of the torque equations re-veals two stable orientations of the NR with respect to thedirection of the incident electric field (light polarization);these orientations depend on the sign of the polarizability dif-ference (α0L − α0S). Table 1 summarizes the results, and it also

expresses the forms of parameters a0, a00 from Eqs. (29) and(30) for the two cases of the stable NP orientation. The spe-cific form of a0, a00 then significantly influences the opticalforces acting upon the NR.

The sign of the polarizability difference (α0L − α0S) dependson the wavelength used for the NR illumination. As Fig. 3 de-monstrates for NRs made of three materials, there exist spec-tral regions of different stable NR orientations with respect tothe polarization of the incident beam. In the long wavelengthregion α0L > α0S , the NR is aligned parallel to the electric field Eof the incident beam. Conversely, in the short wavelength re-gion α0S > α0L, the NR aligns perpendicular to the electric fieldE of the incident beam. Surprisingly, the dielectric silica NRcan also switch from the parallel to the perpendicular orienta-tion although there is no plasmon resonance in this material.In the investigated range of wavelengths, gold and silica en-able one NR reorientation; however, silver reveals a morecomplex NR behavior with three NR reorientations.

The NR reorients at the wavelength λr0 that satisfiesα0L�λr0� � α0S�λr0�. However, this wavelength depends on the ra-tio of short lS and long lL-axis of the ellipsoidal NR. Figure 4reveals that in the case of metallic NRs, there can be morethan one λr0 for certain ratios lS∕lL. In the case of Au NR, λr0overlaps with the wavelength of the plasmon resonance alongthe long NR axis. For Ag NR, λr0 coincides with the plasmonresonances along long and short NR axes. Although there is noplasmon resonance in the case of dielectric silica NRs, thewavelength λr0 coincides well with the maximum of imaginarypart of the NR polarizability, too. Figure 4 then plainly revealsthat the long-axis plasmon resonance in metal NR determinesthe longest wavelength where the NR reorients.

B. Longitudinal TrappingWe have concluded in the previous part that the NR orienta-tion does not depend on the longitudinal position of the NR. Inthis subsection, we investigate conditions for stable longitudi-nal confinement of an NR placed on the optical axis. UsingEq. (32), which describes the longitudinal optical force in thescalar Gaussian beam approximation, one can find one stablez0 and one unstable z00 equilibrium positions of the NR, inde-pendent on the trapping field intensity:

z0 �a0 −

����D

p

2ka00; z00 �

a0 �����D

p

2ka00;

D � �a0�2 � 4�a00�2kzR�1 − kzR�: (39)

However, they exist only if the discriminant D is positive,D > 0, i.e., in a tightly focused beam. If the optical confine-ment is to be stable against thermal fluctuations of the sur-rounding medium, the work needed to get the NR from z0to z00 > z0 must be higher than the energy of the thermalmotion kBT , where kB is the Boltzmann constant and T isthe absolute temperature:

Table 1. Physical Conditions for the Stable Alignment of the Nanorod Corresponding to its Parallel ∥ and

Perpendicular ⊥ Orientation with Respect to the Electric Field E of the Incident Beam

E direction NR orientation symbol α ψ θ a0 a00

↑ ▮ ∥ α0L > α0S 0 0 α0L α00L↑ ▬ ⊥ α0L < α0S 0 π

2 α0S α00S

1228 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Trojek et al.

Uz � −

Zz00

z0

Fz�0; 0; z�dz > kBT: (40)

Within the considered scalar description, the work Uz can befound in the analytical form

Uz �14E20

� ����D

p− a00�2kzR − 1� arctan

����D

p

a00�2kzR − 1�

�: (41)

Because of the stochastic nature of the NR motion, the higherthe ratioUz∕�kBT�, the longer the so-called first mean passagetime [45] (under certain conditions also called Kramers time[46]) corresponding to the mean time needed by the particle toescape across z00.

Considering the vector description of the Gaussian beam,the analyses must be done numerically. Figure 5 comparesthe longitudinal position of the optical trap obtained fromEq. (39) and from numerical analysis of Eq. (36) for gold NR.The illumination wavelength and the angular aperture are

chosen as the free parameters; however, the wavelength rangeis chosen such that the NR reaches stable alignment with re-spect to the beam polarization (see Table 1 and Fig. 3). Thewhite regions denote a combination of parameters where nostable longitudinal equilibrium position (optical trap) exists.Comparing the results from the scalar and vector beam de-scriptions, one can immediately see that the vector descrip-tion provides optical traps even for low angular aperturesand wavelengths closer to the plasmon resonance. At thesame time, the optical traps are longitudinally more displacedfrom the beam focus in the vector description. This corre-sponds to the different longitudinal profiles of the opticalintensity of Gaussian beams described by SD and VD, asFig. 1 demonstrated. However, both descriptions predict nolongitudinal NR confinement for red detuned wavelengthsin the vicinity of the plasmon resonance and for angular aper-tures Θ lower than about 20 deg. Since the optical forces andtorques are directly proportional to the incident laser power,we choose for all the figures the same intensity of the electric

−400

−200

0

200

400

α [1

0−5 p

F n

m2 ]

Au

400 600 800 1000 12004

6

8

10

12

14

λ0 [nm]

−400

−200

0

200

400Ag

400 600 800 1000 1200−20

−10

0

10

20

30

λ0 [nm]

0

2

4

6

8

10SiO

2

400 600 800 1000 12003.4

4

4.6

λ0 [nm]

αS’

αL’

⊥⊥

⊥ ⊥

Fig. 3. (Color online) Spectral dependence of the polarizabilities α0L and α0S described by Eq. (17). The bottom row presents the same data as thetop row with a different scaling of the vertical axis. Spectral regions where (α0L − α0S) does not change its sign are denoted by the horizontal greenarrows. In these regions, the NR is aligned parallel to the beam polarization by its long axis (marked as ∥) or short axis (marked as⊥). We considerellipsoidal Au, Ag, and SiO2 NRs with long axis lL � 40 nm and short axes lS � 10 nm immersed in water. The values of the refractive indices aretaken from [44).

0 0.5 1

400

600

800

1000

1200

lS / l

L

λ 0r [nm

]

Au

0 0.5 1

400

600

800

1000

1200

lS / l

L

Ag

0 0.5 1

400

600

800

1000

1200

lS / l

L

SiO2

Fig. 4. (Color online) Critical parameters under which the NR reorients. The ratio of lengths of the short lS and long lL-axis of the ellipsoidal NRdetermines the wavelength λr0 at which (α0L − α0S � 0) and the NR reorients (black dots). For comparison, the combinations of lS∕lL and λr0 for whichthe imaginary part of the NR polarizability α00L and α00S reaches maximum are denoted by a red × and a green�, respectively. In the case of metal NRs,they correspond to the plasmon resonances along the long or short NR axis, respectively. We again consider Au, Ag, and SiO2 NRs immersed inwater and having long axis lengths equal to lL � 40 nm.

Trojek et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1229

field at the beam focus E0 � 19 MV∕m. It corresponds to theoptical intensity I0 � 637 mW∕μm2, which is obtained if thelaser power of 1 W is focused into the beam waist w0 � 1 μm.

Figure 6 compares the work Uz needed to free the NR long-itudinally for the scalar and vector descriptions of the incidentGaussian beams. One can find that the vector description pro-vides slightly higher values of Uz and consequently morestable longitudinal NR confinement. One also immediately re-cognizes that the NR confinement is much weaker below theplasmon resonance (at shorter wavelengths), i.e., if the NRlong axis is oriented perpendicularly to the beam polarization.This corresponds to our previous results in Eqs. (29) and (30)and Table 1 because for this NR orientation, much weaker po-larizabilities α0S and α00S are responsible for the magnitude ofthe optical force.

C. Lateral TrappingIn the previous parts, we have assumed without a proof thatthe NR is placed on the optical axis. In this part, we focus onthe lateral stability of the trapped NR. Let us assume first thescalar description of the incident Gaussian beam usingEqs. (23)–(25), and let us express the lateral work Ur neededto move the NR from the beam axis to infinity (so-called lat-eral depth of the optical trap):

Ur �−

Z∞

0Fr�r;z�dr�E2

0zR�a0zR −a00z�4�z2R�z2� �Kr

z2R�z2

2kzR; (42)

where Fr and Kr denote the lateral optical force and trap stiff-ness, respectively. Since the force Fr does not change its signfor all radial distances r at a fixed z, positive values of Ur

indicate the NR is confined at the optical axis. However,for certain wavelengths and longitudinal positions, the brack-et (a0zR − a00z) can become negative and, subsequently, the NRwill be repelled from the optical axis. This happens ifz > zRa0∕a00, assuming positive a00 (note that a00 varies in therange between α00S and α00L depending on the NR orientation;therefore a00 > 0). A detailed analysis for the scalar Gaussianbeam reveals that the longitudinal stable equilibrium positionof the optical trap z0, done by Eq. (39), always satisfiesz0 < zRa0∕a00. If a0 ≤ 0, there is no stable longitudinal positionz0, and the NR is repelled from the optical axis. Therefore, wecan conclude that the NR is stably trapped laterally on the op-tical axis if the NR is stably aligned with respect to the beampolarization and stably trapped longitudinally.

Figure 7 compares the lateral works for the scalar and vec-tor descriptions of the incident Gaussian beam. The compar-ison of the top left panel of Fig. 7 with the left panel of Fig. 6calculated for the scalar beam description proves that oncethe NR is trapped longitudinally, it is also trapped laterally be-cause the lateral work is always positive. However, the vectordescription diverts from this rule for lower angular aperturesnear the plasmon resonance, as illustrated by the comparisonof the remaining three panels of Fig. 7 with the right panel ofFig. 6. In this region, the NR can be trapped longitudinally;however, it will be repelled from the optical axis. A similareffect was observed by Simpson and Hanna [47] in the caseof a dielectric microrod optically trapped in a tightly focusedGaussian beam. They calculated the coupling between lateraltranslations and rotations of the rod, which rose from theasymmetry of the stiffness matrix, and they showed thatthe rod was laterally repelled from the optical axis for certainrod orientations.

400 600 800 1000 120010

20

30

40

50

60

70

λ [nm]

Θ [

deg]

SD

2

2

3

3

400 600 800 1000 120010

20

30

40

50

60

70

λ [nm]

VD

2

3

3

1e1

1e2

1e3

1e4 Au

lL = 40nm

lS = 10nm

⊥ || ⊥ ||

z0 [nm]

Fig. 5. (Color online) Longitudinal equilibrium positions z0 of Au NR as a function of the vacuum trapping wavelength λ0 and the angularaperture Θ. The scalar (SD, left) and vector (VD, right) descriptions of the incident Gaussian beam are compared; contours of identical z0are in logarithmic scale. The vertical green line divides the range of wavelengths into two regions where the NR is oriented by its long axis parallel(∥) or perpendicular (⊥) to the beam polarization (along x-axis). No equilibrium positions z0 exist for parameters falling into the white regions. Theintensity of the electric field at the beam focus E0 was equal to E0 � 19 MV∕m; the Au NR of the dimensions lL � 40 nm and lS � 10 nm wasimmersed in water.

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

Θ [

deg]

SD

1 1

1

1 4

8

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

VD

1

1

4

8

10

0

5

10Au

lL = 40nm

lS = 10nm

⊥ || ⊥ ||

Uz [kBT]

Fig. 6. (Color online) The work Uz needed to free the Au NR in longitudinal direction as a function of the vacuum trapping wavelength λ0 and theangular aperture Θ. The same conditions and symbols are used as in Fig. 5.

1230 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Trojek et al.

D. Stability of the Nanorod OrientationUp to now, we have mainly assumed that the NR is orientedparallel or perpendicularly to the beam polarization, but wehave not addressed the stability of such NR orientation. Letus suppose the NR is trapped on the optical axis at the equili-brium position z0 and that it is aligned parallel or perpendicu-larly to the beam polarization. The work needed to deflect theNR from its equilibrium alignment ψ0, θ0 (see Table 1) by anangle δ is expressed as

UψR � −

Z ψ0�δ

ψ0

Mz�0; 0; z0;ψ ; 0�dψ ;

UθR � −

Z θ0�δ

θ0My�0; 0; z0; 0; θ�dθ: (43)

It results in a general form

UR � 14jEx�0; 0; z0�j2jα0L − α0Sjsin2 δ: (44)

Figure 8 compares this quantity calculated for δ � π∕2 andboth descriptions of the incident Gaussian beams and for suchcombination of parameters that ensures the longitudinal NRconfinement. Therefore, the white areas correspond againto such sets of parameters that do not provide longitudinalNR confinement. The numerical values reveal that the workUR needed to reorient the NR by π∕2 from its equilibrium or-ientation is typically equal to tens of kBT . This value is highenough to keep the NR aligned stably against the thermal ac-tivation of the Brownian motion at room temperature T andconsidered incident field intensity E0.

However, the NR can still suffer from small deflectionsfrom its stable alignment due to the thermal activation. Sincethe polarizability a0 from Eq. (29) depends on the NR orienta-tion, the optical forces [expressed by Eqs. (23)–(25) and (36)]acting upon the NR depend on the NR orientation, too, and theanalysis becomes practically intractable. Therefore, let usfocus here only on the lateral and longitudinal trap stiffnessesat the equilibrium position r0 � �0; 0; z0� and investigate howthey are influenced by the NR alignment.

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

Θ [

deg]

SD

2

12

15

18

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

VD

0 022

8

12

15

0

10

20

0

10

20Au

lL = 40nm

lS = 10nm

⊥ || ⊥ ||

Uy [k

BT]U

r [k

BT]

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

Θ [

deg]

VD

8

12

15

18

2

2

2

2

00

0

00

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

VD

8

812

15

182

2

0

0

0

10

20

0

10

20Au

lL = 40nm

lS = 10nm

Ux [k

BT]U

x [k

BT]

||||⊥y ⊥z

Fig. 7. (Color online) The work needed to transfer the NR from the optical axis to infinity in the lateral direction as a function of the vacuumtrapping wavelength λ0 and the angular apertureΘ. The optical trap is laterally symmetrical in the scalar description (SD) and, therefore, the workis denoted as Ur and expressed by Eq. (42). However, in the vector description, the work generally differs for the NR transport along x and y-axis,denoted as Ux and Uy, respectively. In the case of Ux, two orientations of the NR must be considered, along y-axis (⊥y) and z-axis (⊥z). The sameconditions and symbols are used as in Fig. 5.

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

Θ [

deg]

SD

10 20

400 600 800 1000 120010

20

30

40

50

60

70

λ0 [nm]

VD

10

10

20

0

20

40

60 Au

lL = 40nm

lS = 10nm

⊥ || ⊥ ||

UR

[kBT]

Fig. 8. (Color online) The work UR needed to reorient the NR from the equilibrium position by π∕2 as a function of the vacuum trappingwavelength λ0 and the angular aperture Θ. The same conditions and symbols are used as in Fig. 5.

Trojek et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1231

In the case of the scalar description, the lateral trapstiffnesses along x and y-axis are identical and can be ex-pressed as

Kr � E20kz

2Ra0zR − a00z2�z2R � z2�2 : (45)

In connection with Eq. (42), the NR behavior has been dis-cussed for a stable alignment of the NR. However, the lateralstiffness Kr changes its sign for certain NR orientations and,consequently, the NR is repelled from the optical axis.Figure 9, top left, demonstrates such modification of the lat-eral stiffness Kr during deflection of the Au NR from its equi-librium alignment for both stable NR orientations (∥ and ⊥).In the parallel case (∥) the stiffness Kr decreases with increas-ing NR deflection but stays positive and, therefore, the NRstays confined on the optical axis. In the perpendicular case(⊥), the lateral stiffness changes its sign to negative even forsmall NR deflections (about 10 deg) and causes the NR repul-sion from the optical axis.

A similar NR behavior is also observed in the vector descrip-tion of the incident Gaussian beam, exploiting Eq. (47). Thevector description of the incident beam also enables us to ex-press the stiffnesses as a function of z-coordinate analytically:

Kx�z� � −∂xFx�x; y; z�jx�0y�0 �

132

k4RfaxH1H3� − 4azH2H2

�g;(46)

Ky�z� � −∂yFy�x; y; z�jx�0y�0 �

132

k4RfaxH1H4�g; (47)

Kz�z� � −∂zFz�x; y; z�jx�0y�0 �

18k4Rfax�H1H6

� −H5H5��g;(48)

where aξ denotes the polarizability of the NR along ξ-axis andthe integrals Hn have the form

H1�z� �Z Θ

0A�α� sin α�1� cos α� exp�ikz cos α�dα; (49)

H2�z� �Z Θ

0A�α�sin3 α exp�ikz cos α�dα; (50)

H3�z� �Z Θ

0A�α�sin3 α�1� 3 cos α� exp�ikz cos α�dα; (51)

H4�z� �Z Θ

0A�α�sin3 α�3� cos α� exp�ikz cos α�dα; (52)

H5�z� �Z Θ

0A�α� sin α cos α�1� cos α� exp�ikz cos α�dα;

(53)

H6�z� �Z Θ

0A�α� sin α cos 2 α�1� cos α� exp�ikz cos α�dα:

(54)

No significant differences are observed betweenKx andKy va-lues for the selected parameters, and therefore only Ky isplotted in Fig. 9, top right.

Figure 9, bottom row, also presents longitudinal trap stiff-ness Kz, which is obtained numerically at the NR equilibriumposition r0 � �0; 0; z0�. Noticeable differences between thescalar and vector descriptions are caused by different NRequilibrium positions z0 (see Fig. 5). Similarly to the lateral

0 30 60 90−1

−0.5

0

0.3

ψ [deg]

Kz [

pN/µ

m/W

]

0 30 60 90−1

−0.5

0

0.3

ψ [deg]

Kz [

pN/µ

m/W

]

0 30 60 90−6

−4

−2

0

2

Kr [

pN/µ

m/W

]

SD

λ0 = 1064nm (ll)

λ0 = 700nm (⊥)

0 30 60 90−6

−4

−2

0

2

Ky [

pN/µ

m/W

]

VD

deflectiondeflection

Fig. 9. Dependence of the lateral (Kr ,Ky) and longitudinal (Kz) stiffnesses on the deflection ψ of the NR from its equilibrium alignment. The NR isdeflected around the optical axis z by the azimuthal angle ψ , while the polar angle is kept fixed at θ � 0. Results of the scalar (left column, SD) andvector (right column, VD) descriptions of the incident Gaussian beam are compared for two vacuum wavelengths λ0 � 1064 nm (solid curve) andλ0 � 700 nm (dashed curve) corresponding to the NR alignment parallel (∥) and perpendicular (⊥) to the beam polarization (along x-axis), re-spectively. The stiffnesses are normalized to the unit incident power and calculated for Au NR (lL � 40 nm, lS � 10 nm) placed on the optical axisat its original stable longitudinal position z0 (if no deflection is considered). The angular aperture is set to Θ � 50°.

1232 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Trojek et al.

stiffness, Kz changes sign for small NR deflections from thestable perpendicular orientation (⊥), and consequently theNR is repelled from z0 longitudinally.

5. HEATING OF THE NANORODSince the absorption of the incident light causes heating of theNR that could destroy the NR or cause boiling of the liquidenvironment, we focus here on the temperature increase ofthe optically trapped NR. Seol et al. [48] assumed the follow-ing radial dependence for the temperature decrease outsidethe nanosphere (of radius R):

T�r� � T0 �ΔTRr; �55�

where r is the radial distance from the nanoparticle center,and ΔT is the temperature difference between the nanopar-ticle and the initial temperature T0. The light power absorbedby the object is equal to Pabs � I�r0�σabs, where I�r0� �1∕2ϵ0cnmjE�r0�j2 is the irradiance of the object at the objectposition r0 and σabs � 3Vkα1 00 corresponds to the Rayleigh ab-sorption cross-section of the object with volume V and polar-izability α1 (the index1 signifies here only the dimensionlesspart of the polarizability; see below). In the steady state,the light power Pabs absorbed by the object is equal to the heatflow throughout the surface S of the object into the surround-ing medium with thermal conductivity κ and refractive indexnm. Then the temperature rise of the NR can be expressedas [49]

ΔT � I�r0�σabs4πRβκ ; (56)

where R corresponds to the radius of equivalent sphere havingthe same volume as the NR. Assuming an ellipsoidal NR withaxes lL, lS , lS , the following relation is valid: �2R�3 � lLlS2. Thefactor β strongly depends on the aspect ratio lL∕lS , but it isvery close to 1 for NRs with aspect ratios from 1 to 5. Fora nanosphere, the dimensionless polarizability is expressed as

α1 �m2 − 1

m2 � 2; (57)

while for an NR, we use the following expression:

α1 �m2 − 1

3L�m2 − 1� � 3; (58)

where L is done by Eq. (18) along one of the NR axes.For the sake of consistency, we consider here the same in-

tensity of the electric field at the beam focus E0 � 19 MV∕mas in the previous sections dealing with the optical trapping.This value is on average 1 order of magnitude higher com-pared to those used in the experimental optical manipulationwith gold NRs [22,23]. Since optical force and also heating arelinearly scalable with the incident power, the presented re-sults can be easily rescaled to desired incident laser power.The effects of NR heating due to the absorption of the incidentlight are summarized in Fig. 10. The right part of Fig. 10 showsthe temperature increaseΔT of a gold nanosphere and an NRof the same volume is placed at the beam focus. In reality, thenano-object is confined in an optical trap that is longitudinallydisplaced from the beam waist. At the trap location, the cor-responding optical intensity is lower; however, the order ofΔT is not changed. Comparison of the spectral profile of NRheating with the spectral dependence of the imaginary part ofthe polarizability α1 (Fig. 10, left) clearly shows the correlationof the two quantities. Figure 10 also confirms that the NR tem-perature increase is an order of magnitude higher comparedto the nanosphere of the same volume. This is caused by astronger polarizability α001 of the NR (see the left part ofFig. 10).

Based on the analytical expression (56), the temperatureincrease of the gold NR is enormous and water should boilthere; however, it is not reported experimentally. Utilizationof the finite element method gives the same results for thesame NR. Recent theoretical results [50,51] based on the mo-lecular dynamics provide an explanation based on the as-sumption that around a metallic nanoparticle there exists athin fluid layer of high pressure and very low molecular den-sity that could ensure nanoparticle melting without boiling ofthe surrounding liquid.

If the NR orientation θ is changed with respect to the po-larization of the incident beam, the NR temperature decreasesby several orders of magnitude (see Fig. 11). However, the NRis not optically confined in these orientations.

6. DISCUSSIONEven though we use the simplest theoretical description of theoptical forces and torques acting upon metal NR, we find new

400 600 800 1000 120010

−4

10−2

100

102

λ0 [nm]

α 1’’

Au

400 600 800 1000 120010

0

101

102

103

104

105

λ0 [nm]

∆T [

° C]

Au

NSNR

Fig. 10. Left: spectral dependence of the imaginary part of the polarizability α1 for a gold nanosphere [R � 16 nm, dashed curve, Eq. (57)] and agold ellipsoidal nanorod [lL � 40 nm, lS � 10 nm, solid curve, Eq. (58)] of the same volumes. Right: spectral dependence of the temperature in-crease for the same objects as in the left figure. The nanoparticles are immersed in water, and the electric field intensity at the beam focus isE0 � 19 MV∕m.

Trojek et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1233

and interesting conclusions that are coherent with experimen-tally observed NR behavior.

Pelton et al. [22] reported stable trapping of a single goldNR (lL � 40 nm, lS � 10 nm) immersed in water. They usedtrapping wavelength 850 nm and objective lens with angularaperture 60 deg. Our results show that using the classical sca-lar description of the incident Gaussian beam one would not

be able to trap such NR. However, utilization of the more ap-propriate and precise vector description of the incident Gaus-sian beam predicts a potential well deep enough to providestable confinement of the NR under the reported experimentalconditions.

C. Selhuber-Unkel et al. measured lateral and longitudinaltrap stiffnesses for an ensemble of gold NRs trapped on theoptical axis [23]. Since the NR equilibrium position is indepen-dent of the incident trapping power and the trap stiffness islinearly proportional to the incident trapping power, they pre-sented the trap stiffnesses normalized to the incident trappingpower equal to 1 W. Taking their values and the NR para-meters from Table 1 in [23], we calculated theoretically thenormalized optical trap stiffnesses Kx, Ky, and Kz using thevector description of the incident Gaussian beam. The com-parison between the experimental measurements [23] andour theoretical results is presented in Fig. 12. The theoreticalresults corresponding to the mean experimental values of theNR sizes (red ⋄) predict in general higher lateral stiffnesses;some results fit within the experimental error, but otherresults differ by an order of magnitude. In contrast to the lat-eral stiffness, the measured longitudinal trap stiffness Kz is

0 30 60 9010

0

101

102

103

θ [deg]

∆T [

°C]

Fig. 11. Temperature increase of Au NR (lL � 40 nm, lS � 10 nm)illuminated at the wavelength λ0 � 850 nm under different anglesθ. The NR is immersed in water; the electric field at the beam focusis E0 � 19 MV∕m. These results were obtained by finite elementsmethods using Comsol Multiphysics software.

4 5 6 7 8 9 10 15 20 2510

−3

10−2

10−1

Kx [

pN/n

m/W

]

experimenttheory: middletheory: uppertheory: lower

4 5 6 7 8 9 10 15 20 2510

−3

10−2

10−1

experimenttheory: 50°, 0 kTtheory: 60°, 0 kTtheory: 50°, 2 kT

4 5 6 7 8 9 10 15 20 2510

−3

10−2

10−1

Ky [

pN/n

m/W

]

4 5 6 7 8 9 10 15 20 2510

−3

10−2

10−1

4 5 6 7 8 9 10 15 20 2510

−4

10−3

10−2

R [nm]

Kz [

pN/n

m/W

]

4 5 6 7 8 9 10 15 20 2510

−4

10−3

10−2

R [nm]

Fig. 12. (Color online) Comparison between the experimental trap stiffness measurements [23] and the theoretical predictions based on the vectordescription of the incident beam if the NRs are trapped on the optical axis at z0. The horizontal axis gives the mean value of the NRs radii R � lS∕2;both lS and lL of compared NRs are taken from Table 1 in [23]. The left column demonstrates the influence of the spread of the NR sizes on thestiffnesses. Here, ⋄ denotes the theoretical results corresponding to the average measured values of the NR size andΘ � 50°;▿ and▵ correspondto the smallest and biggest NRs within the experimental error reported in [23], respectively. The right column shows the dependence of the stiffnesson the angular apertureΘ (⋄ versus⊳) and deflection of the NR from its equilibrium orientation corresponding to the reorientation workUR � 2kT(⋄ versus ⊲).

1234 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Trojek et al.

usually higher than the corresponding theoretical predictions.The above summary indicates that in this case, the expectedvector description of the beam is not perfectly suitable for theexperimental configuration. The reason for the discrepancycould stem from the fact that the authors used a special im-mersion oil with a carefully selected refractive index to sup-press the spherical aberrations at the trapping plane placed5 μm above the lower cell surface and thus to maximize thelongitudinal trap stiffness [23,52]. Such a procedure could leadto a field distribution with optimized and steeper intensity gra-dients in longitudinal direction but shallower gradients in thelateral direction, which corresponds to the results presentedin Fig. 12. Obviously, to get a better coincidence with the the-ory, we would need to use a lower angular aperture Θ for thelateral stiffnesses and a higher aperture for the longitudinalstiffness, as the right column of the figure illustrates. The rightcolumn also shows that the deflection of the NR from its equi-librium orientation (corresponding to the work UR � 2kBT)does not influence the stiffnesses significantly. As illustratedin the left column, a stronger influence comes from the errorin the experimental determination of the NR sizes. Here weplot the stiffness for the mean nanorod size and also for thesmallest and biggest estimated nanorod sizes reported inTable 1 in [23]. It can be seen that the spread of the obtainedstiffnesses is larger than the standard deviation of the mean ofthe experimental data.

Since the Rayleigh approximation is valid only for verysmall particles, its utilization for larger particles overestimatesthe optical forces [53]. Furthermore, we have not reduced theeffective NR volumes due to the skin effect [4], which alsoresults in overestimation of the optical forces. However, thesetrends are not clearly visible in Fig. 12.

The differences between the scalar and vector descriptionsof the incident beam presented in the previous sections illu-strated how strongly the spatial distribution of the field at thebeam focus influences the stable NR confinement even if onlyone parameter Θ is used to characterize the beam. In thisstudy, we have ignored all optical aberrations, even thoughthey are likely to be responsible for the discrepancies betweenthe measurement and simulation results presented in Fig. 12.The reason is that there is no exact procedure for how to ob-tain experimentally the spatial distribution of the electric fieldcomponents at the beam focus to justify the proper theoreticaldescription.

7. CONCLUSIONSWithin the Rayleigh approximation, we have presented a the-oretical description of optical forces and torques acting upona metal nanorod. To this end, we have adopted the scalar andvector descriptions of the focused Gaussian beam illuminatingthe NR and studied the behavior of metal NRs in respectivefields. The analysis of the results has revealed that an NR canbe aligned with its long axis parallel or perpendicular to thebeam polarization, depending on the real parts of the NRpolarizabilities along its long and short axis. We havedemonstrated that a gold NR tends to orient its long axisparallel (perpendicular) to the beam polarization for laserwavelengths longer (shorter) than the longitudinal plasmonresonance wavelength. In the case of silver NRs, the behavioris more complex and the NR reorients at another two shorter

wavelengths close to the transversal plasmon resonantwavelength.

In addition, conditions for longitudinal and lateral opticaltrapping of a gold NR have been investigated. Vector descrip-tion of the incident Gaussian beam predicts stable NR trap-ping even at low angular apertures and also at broaderrange of wavelengths. Only the vector description confirmsstable optical trapping under the conditions observed experi-mentally [22]. Both models predict that it is not possible totrap gold NRs at wavelengths slightly longer than the plasmonresonant wavelength. Both models also predict a larger opti-cal torque if the gold NR is illuminated by a wavelengthslightly shorter than the plasmon resonant wavelength,eventhough the optical confinement is weaker here. The vectormodel also predicts unstable on-axial optical trapping of goldNRs at low angular apertures and wavelengths close to theplasmon resonance. Agreement in the order of magnitudehas been reached with the experimental measurements of theNRs’ lateral stiffnesses [23]. Even though the theoretical re-sults differ from the experimental ones, they follow correctlythe observed trend for NRs of different sizes, and the pre-sented method enables rough and fast prediction of the NRbehavior. The observed discrepancy is mainly caused by thelack of knowledge about the exact spatial distribution of thetrapping electric field near the beam focus. A more versatilevector theoretical model would be more appropriate to simu-late a tightly focused beam in real experiments [54].

ACKNOWLEDGMENTSThe authors acknowledge valuable comments of Dr. A. Jonášand support from the Czech Scientific Foundation (202/09/0348), the Ministry of Education, Youth and Sports ofthe Czech Republic, and the European Commission(ALISI No. CZ.1.05/2.1.00/01.0017).

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