Optical forces on small particles: attractive and ... - OSA Publishing

J. R. Arias-Gonza´lez and M. Nieto-Vesperinas

Vol. 20, No. 7 / July 2003 / J. Opt. Soc. Am. A

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Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions J. R. Arias-Gonza´lez and M. Nieto-Vesperinas Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientı´ficas, Cantoblanco, Madrid 28049, Spain Received November 14, 2002; revised manuscript received February 28, 2003; accepted March 5, 2003 A detailed study of time-averaged electromagnetic forces on subwavelength-sized particles is presented. An analytical decomposition of the force into gradient and scattering-plus-absorption components is carried out, on the basis of which the attractive or repulsive behavior of the force is explained. Small metallic particles are shown to experience both kinds of forces; which kind also depends on the excitation of surface plasmons. Resonances give rise to enhancements of both the scattering and the absorption forces, but the gradient force can become negligible. Also, close to resonant wavelengths, the gradient force can be maximum, while both the scattering and the absorption forces remain large. Comparisons of analytic results with rigorous calculations allow the establishment of ranges of validity of the dipolar approximation for these forces. © 2003 Optical Society of America OCIS codes: 290.5850, 260.5740, 240.6680.

1. INTRODUCTION The mechanical action of light on particles permits researchers to hold and manipulate them by means of optical tweezers1 in a variety of techniques such as spectroscopy,2–4 studies of phase transitions in polymer molecules,5 and light-force microscopy of cells6,7 and biomolecules.8 It has been proven that dielectric particles are attracted both toward the axis of focused beams1 and to surfaces on which evanescent waves propagate,9–11 i.e., to regions of higher field intensity. Though it has initially been reported that small metallic particles are subjected to repulsive gradient forces,12–15 the action of attractive forces on them has subsequently been observed.16–20 These studies also showed repulsive gradient forces acting on larger metallic particles (i.e., with diameters greater than 50 nm). On the other hand, it has been proven that scattering and absorption forces push the particle along the wave propagation direction.21 Reference 19 explains, by means of ray-tracing arguments, that repulsive forces arise from surface reflection, whereas attractive forces come from creeping waves on the particle surface. Also, Ref. 22 calculates the forces exerted by evanescent fields on metallic particles, showing the action of both attractive and repulsive forces. The decomposition of electromagnetic forces into gradient, scattering, and absorption components was done in Refs. 1 and 23. A determination of these components on particles with size of the order of, or smaller than, the wavelength was presented in Ref. 24 up to the second order of the scattering events in the particle. A complete determination of the total force on a dipolar particle under a generic time-harmonic field is done in Ref. 25. This paper addresses dipolar metallic particles. The forces are split into gradient and scattering-plus1084-7529/2003/071201-09$15.00

absorption components, and we examine the conditions under which they become either attractive or repulsive. We thus show that the sign of the force is related to that of the particle polarizability and to the generation of a surface plasmon on it and that this resonance enhances both its scattering and its absorption components; however, the gradient force at this resonant wavelength can decrease with respect to its magnitude under nonresonant illumination. In addition, on comparing the analytic results with exact numerical calculations, (integral method, which gives the same result for isolated particles as the partial-wave series like that of Mie for spheres or Rayleigh’s series in terms of Bessel functions for cylinders; see Section 2), we establish ranges of validity of the dipole approximation for these forces. It should be pointed out, however, that the validity of the aforementioned series calculations in the presence of interfaces involves neglecting multiple scattering between particle and interface. This has been already studied in several papers,11,26–28 and it is further discussed in Section 6. We have recently found an analysis on dielectric dipolar particles near resonant absorption in Ref. 29. After introducing the configuration in Section 2, we study in Subsection 3.A the decomposition of the force on a dipolar particle into gradient and scattering-plusabsorption components. We analyze the plasmonresonant conditions in Subsection 3.B and present numerical results for both spherical and cylindrical particles in Subsection 3.C. The potential energy generated by the gradient forces is studied in Section 4. The validity of the dipole approximation is discussed in Section 5, and it is shown how it depends on the field intensity uniformity inside the particle. Finally, we consider in Section 6 larger particles, addressing the connection of these results with previous studies when the interaction with an interface cannot be neglected. © 2003 Optical Society of America

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J. Opt. Soc. Am. A / Vol. 20, No. 7 / July 2003

2. FORMULATION We establish here a basic configuration, useful for studying a particle under the influence of either a Gaussian beam or an evanescent wave created under total internal reflection (TIR) at an interface. Figure 1 illustrates this scattering geometry. We consider incidence of either a plane wave or a Gaussian beam of half-width at halfmaximum W, with polarization either S (electric field along the Y axis) or P (magnetic field along the Y axis), impinging from a half-space filled by glass ( 冑⑀ 0 ⫽ 1.51) on a flat interface, above which there is another dielectric of permittivity ⑀ 1 , where the particle (i.e., either a sphere or a cylinder with its axis normal to the XZ plane) is placed. We shall study the scattering effects in this system when the transmitted wave is either propagating (incidence angle ␪ 0 ⫽ 0°) or evanescent ( ␪ 0 ⬎ ␪ c , where ␪ c is the critical angle). Metallic30 particles of radius a are situated at a distance z ⫽ d ⫹ a from its center to the interface. For an incident beam, the electric and magnetic vectors are, respectively, E共 inc兲共 r, t 兲 ⫽ 关 0, ⌽ 共Sinc兲共 r兲 , 0兴 exp共 ⫺i ␻ t 兲 ,

(1)

H共 inc兲共 r, t 兲 ⫽ 关 0, ⌽ 共Pinc兲共 r兲 , 0兴 exp共 ⫺i ␻ t 兲 ,

(2)

where ⌽ 共␥inc兲共 r兲 ⫽ exp关 i 冑⑀ 1 k 0 共 x sin ␪ 0 ⫹ z cos ␪ 0 兲 g 共 x, z 兲兴 ⫻ exp关 ⫺共 x cos ␪ 0 ⫺ z sin ␪ 0 兲 2 /W 2 兴 ,

(3)

where k 0 ⫽ ␻ /c ⫽ 2 ␲ /␭, ␻ being the angular frequency, c as the light velocity in vacuum and ␭ the wavelength in vacuum; ␥ stands for either S or P polarization, and g 共 x, z 兲 ⫽ 1 ⫹

1

⑀ 1 k 02 W 2

冋

2 W2

册

共 x cos ␪ 0 ⫺ z sin ␪ 0 兲 2 ⫺ 1 .

(4)

The field is then exactly calculated by means of the extinction theorem boundary condition (ET), as described in previous papers,31 and compared with that obtained from the dipole approximation. Our ET calculation will not consider the multiple scattering between the particle and the interface, which of course provides the same results as those from the partial-wave series (namely, Mie’s series for an isolated sphere or the Bessel-function series for an isolated cylinder). For this configuration, this will be justified in Section 6 on comparison with calculations that address multiple scattering. The electromagnetic forces are then obtained from Maxwell’s stress tensor.32 At optical frequencies involved in many experiments, however,

the time average of the electromagnetic force is observed. In an isotropic medium this reads as

具 F典 ⫽

1 8␲

Re

冠冕再 ⌺

d2 r 关 ⑀ E共 r, ␻ 兲 • n兴 E* 共 r, ␻ 兲

⫹ 关 ␮ H共 r, ␻ 兲 • n兴 H* 共 r, ␻ 兲 ⫺ ⫹ ␮ 兩 H共 r, ␻ 兲兩 2 兴 n

冎冡

1 2

关 ⑀ 兩 E共 r, ␻ 兲兩 2

,

(5)

where ⌺ is a surface enclosing the particle, n stands for the local outward unit normal, * denotes the complex conjugate, ␮ is the permeability, and Re represents the real part of a complex number. Equation (5) is written in Gaussian units.

3. FORCE ON A DIPOLE A. Gradient, Scattering, and Absorption Components of the Force The dipolar approximation is valid for particles of size much smaller than ␭/n (where n is their refractive index). However, the qualitative behavior of the forces obtained with this approach exhibits good agreement with partialwave-series calculations for particles well beyond Rayleigh’s condition: ka Ⰶ 1. Also, it possesses a considerable interpretative value for analyzing the contribution of the gradient, scattering, and absorption forces, as well as for explaining their evolution in the neighborhood of a resonance. In this section we study these force components on a small metallic particle, close to the excitation of a surface plasmon. Starting from the expression of the time-averaged total force exerted on a dipolar particle in a time-harmonic field25 [a time dependence exp(⫺i␻t) is assumed], we obtain F i 共 r兲 ⫽

1 2

冋

Re ␣ E j 共 r兲

⳵ E j* 共 r兲 ⳵xi

册

,

(6)

where ␣ is the particle polarizability, F i (r) (i ⫽ 1, 2, 3) are the time-averaged force Cartesian components, and E j are the complex-amplitude components of the electric field vector. A sum over repeated indices is assumed in Eq. (6). The particle polarizability accounts for the linear coupling that exists between the dipolar moment of the particle and the incident field. In the case of a small sphere, it can be expressed as follows33:

␣共 ␻ 兲 ⫽

␣ 0共 ␻ 兲 1⫺

2 3 3 ik ␣ 0 共 ␻ 兲

,

(7)

where ␣ 0 ⫽ a 3 ( ⑀ ⫺ 1)/( ⑀ ⫹ 2) is the well-known Clausius–Mossotti relation. ⑀ ⫽ ⑀ 2 / ⑀ 0 stands for the ratio of permittivities of the particle and the surrounding medium. The light field can be expressed by its paraxial form; e.g., it is either a beam or a plane wave (either propagating or evanescent), so that it has a main propagation direction along k. Then it can be described by Fig. 1.

Geometry of the system.

E共 r兲 ⫽ E0 共 r兲 exp共 ik – r兲 .

(8)



Substituting Eq. (8) into Eq. (6), one obtains for the averaged force

具 F典 ⫽

1 4

Re兵 ␣ 其 ⵜ 兩 E0 兩 2 ⫹

⫺

1 2

1 2 2 k Im兵 ␣ 其兩 E0 兩

Im兵 ␣ 其 Im兵 E0 • ⵜE0* 其 .

(9)

Im denotes the imaginary part. The first term is the gradient force acting on the particle, whereas the second term represents the radiation pressure contribution to the scattering force that, on substituting the above approximation for ␣, ␣ ⫽ ␣ 0 (1 ⫹ 32 ik 3 兩 ␣ 0 兩 2 ), can also be expressed for a Rayleigh particle (ka Ⰶ 1) as21 ( 兩 E0 兩 2 /8␲ )Ck/k, where C is the particle-scattering cross section: C ⫽ (8/3) ␲ k 4 兩 ␣ 0 兩 2 . Notice that the last term of Eq. (9) is zero only when either ␣ or E0 is real (this is the case for a propagating or evanescent plane wave but not for a beam, in general). As observed in Eq. (9), the real part of the polarizability accounts for the gradient force, whereas the imaginary part is responsible for the scattering-plus-absorption force. To illustrate this, let us consider a sphere with dielectric permittivity ⑀ 2 and radius a Ⰶ ␭/ 冑⑀ 2 . For an evanescent field created under TIR of an incident plane wave at the flat interface (see Fig. 1), the field components above that interface are expressed as E ⫽ 共 0, 1, 0 兲 T⬜ exp共 iKx 兲 exp共 ⫺qz 兲

(10)

for S polarization and E ⫽ 共 ⫺iq, 0, K 兲

T储 k

exp共 iKx 兲 exp共 ⫺qz 兲

(11)

for P polarization. K denotes the component of the wave vector k, parallel to the interface, and iq is its perpendicular component, such that k ⫽ (K, iq), q 2 2 冑冑冑 ⫽ K ⫺ k 0 , K ⬎ k 0 , K ⬍ ⑀ k 0 , and k ⫽ ⑀ 0 k 0 . T⬜ and T 储 are the transmission coefficients for the electric and the magnetic vectors in S and P polarizations, respectively. In the absence of multiple scattering between the particle and the interface, the force from Eqs. (9)–(11) splits into a scattering-plus-absorption force (along OX) and a purely gradient force (along OZ), as follows: Fx ⫽

兩T兩2

2

K Im兵 ␣ 其 exp共 ⫺2qz 兲 ,

兩T兩 Fz ⫽ ⫺ q Re兵 ␣ 其 exp共 ⫺2qz 兲 . 2

(12)

兩T兩2 K

8␲ k

exp共 ⫺2qz 兲 C ext ,

(14)

where the particle-extinction cross section C ext has been introduced as

⑀⫹2

Fz ⫽

⫹

8␲ 3

冏冏

k 4a 6

⑀⫺1

⑀⫹2

2

. (15)

兩T兩2

2

k Im兵 ␣ 其 ,

(17)

where k ⫽ 冑⑀ 0 k 0 . In Eqs. (16) and (17) the Gaussian factors of F x and F z is exp(⫺2x2/W2) ⬇ 1; also, g(x) ⬇ 1 [cf. Eqs. (3) and (4)] in the range of W and ␭ considered here. For smaller widths W, both components of F exhibit a Gaussian dependence; thus both F x and F z increase on approaching the beam axis. It is shown, again, that the scattering-plus-absorption component of the force becomes dependent on Im兵␣其, while the gradient component of F depends on Re兵␣其. B. Attractive and Repulsive Forces: PlasmonResonance Conditions Examining the pairs Eqs. (12) and (13) and Eqs. (16) and (17), we note that the attractive and repulsive natures of the force components depend on the sign of Re兵␣其 and Im兵␣其. If ka Ⰶ 1, we approximate for a spherical particle ␣ by ␣ 0 for which the real and imaginary parts read as 3

(13)

T stands for either T⬜ or T 储 , depending on whether the polarization is S or P, respectively. The exponential decay of the force from the surface plane accounts for the evanescent field intensity behavior. Equation (12) can be expanded into ka powers by using Eq. (7). Then the force parallel to OX transforms into

再冎 ⑀⫺1

The first term of Eq. (15), substituted into Eq. (14), leads to the radiation pressure of the evanescent wave on the particle that is due to absorption, whereas the second term corresponds to the scattering contribution. Notice that Eq. (15) coincides with the value obtained from Mie’s theory for small particles in the low-order expansion of the size parameter ka of the extinction cross section,21 as it should. A second example of this decomposition is provided by the Gaussian beam given by Eqs. (1) and (2). Let us now assume the small sphere illuminated by this beam, isolated in a medium of dielectric permittivity ⑀ 0 (i.e., ⑀ 0 ⫽ ⑀ 1 in Fig. 1). Without loss of generality, we choose ␪ 0 ⫽ 0°. Under the influence of this field, the particle must feel a scattering-plus-absorption force along the axis of the beam and a gradient force perpendicular to this axis. Thus the particle is confined in the OX direction. For W large, the force exerted on the small particle becomes x F x ⫽ ⫺兩 T 兩 2 Re兵 ␣ 其 , (16) W2

Re兵 ␣ 0 其 ⫽ a

2

Fx ⫽

C ext ⫽ 4 ␲ ka 3 Im

1203

Im兵 ␣ 0 其 ⫽ a 3

共 ⑀ ⬘ ⫺ 1 兲共 ⑀ ⬘ ⫹ 2 兲 ⫹ ⑀ ⬙ 2 共 ⑀ ⬘ ⫹ 2 兲2 ⫹ ⑀ ⬙2

3⑀⬙ 共 ⑀ ⬘ ⫹ 2 兲2 ⫹ ⑀ ⬙2

,

,

(18)

(19)

where we have written ⑀ ⫽ ⑀ ⬘ ⫹ i ⑀ ⬙ . Equation (18) shows that the gradient force F z is positive, i.e., repulsive from the interface [cf. Eq. (13)] or from the beam axis [cf. Eq. (17)], for ⑀⬘ between (⫺2, 1). Otherwise, this force component is negative. On the other hand, the absorption component F x is shown by Eq. (19) to be always repulsive. Hence the repulsive nature of the gradient force on a metallic particle (which affects the trapping potential) is not only a question of size.16,19 For ⑀ ⬙ ⬍ 1, but ⑀ ⬙ ⫽ 0), the plasmon condition for a

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small sphere is ⑀ ⫽ ⫺2. 34,35 This is nearly satisfied by silver, for which ⑀ ⫽ ⫺2 ⫹ i0.54 at ␭ ⫽ 353.39 nm. If ⑀⬙ were zero, on taking the limit ⑀ ⬘ → ⫺2 and ⑀ ⬘ → 0, both Im兵␣0其 and Re兵␣0其 become infinite. The latter is discontinuous in this limit, thus exhibiting a saltus from ⫺⬁ to ⬁ as the resonant wavelength is approached. However, since ⑀ ⬙ ⫽ 0, the shape of Re兵␣0其 broadens about the saltus, so that this variable becomes continuous and passes through a zero close to the plasmon-excitation wavelength. Therefore the gradient force may become minimum at this wavelength. Thus resonant conditions cause both the absorption and the scattering forces [see also Eqs. (14) and (17), together with Eq. (15)] to become maximum but cause the magnitude of the gradient force to be minimum. This has previously been found in larger particles in Ref. 20. In the case of a differently shaped particle, the dipolar plasmon-resonance condition changes. Then all equations examined above hold, except those of the polarizability. For instance, for a cylinder, this last quantity (which is influenced by the plasmon excitation in P polarization20) with the radiative-reaction term from Ref. 11 is

␣

共 p兲

共␻兲 ⫽

␣ 共0 p 兲共 ␻ 兲 1 ⫺ ik 2 ␲ ␣ 共0 p 兲共 ␻ 兲 /2

,

methods of calculation justify the use of the dipole approximation for this particle size and material. The same applies to other similar metal particles like silver. The dipole approximation considers the field uniform inside the particle, which is not consistent when the gradient of the field in this region is large within length scales smaller than the particle diameter. This occurs, for instance, for evanescent wave illumination under large angles of incidence or when the width of an impinging Gaussian beam is too small. We shall come back to this point in Section 4. Figure 3 gives details of the relationship between the gradient force and the real part of the dielectric permittivity, on the one hand, and between the scattering-plusabsorption force and the imaginary part of this optical constant, addressed above in Subsection 3.B, on the other

(20)

where

␣ 共0 p 兲共 ␻ 兲 ⫽

a2 ⑀共 ␻ 兲 ⫺ 1 2 ⑀共 ␻ 兲 ⫹ 1

.

(21)

The superindex indicates P polarization. Now the resonant condition is found at a wavelength that makes ⑀ ⫽ ⫺1, and the interval of ⑀⬘ for which the gradient force becomes repulsive is (⫺1, 1). Otherwise, the gradient force becomes attractive under P-polarized illumination. This characterization of the force components under resonant conditions, though basic, has never been previously reported nor discussed from experiments, to our knowledge. C. Numerical Examples Let us consider a particle of radius a ⫽ 10 nm, made of either silver or gold. Figure 2 shows the real and imaginary parts of the dielectric permittivity of both materials versus ␭ when they are immersed in either vacuum (thick curves) or water (thin curves). As shown, the value of ␭ at which ⑀ ⫽ ⫺ 2 for a sphere (and that at which ⑀ ⫽ ⫺1 for a cylinder) corresponds to a relatively small ⑀⬙ for the case of silver, whereas for gold ⑀⬙ remains high around this ␭. Immersing the particle in water produces the resonant condition at smaller ⑀⬙. The eigenmode excitation is therefore more efficient. The top figure shows the force versus wavelength on a gold cylinder either in vacuum (thick curves) or in water (thin curves), illuminated by a plane-propagating wave. Throughout this paper, the force on a cylinder is expressed in N/nm, namely, as the magnitude per unit length (in nanometers) along its axis. The calculation was performed with the partialwave series for cylinders (plain curves), which is analogous to that of Mie for spheres, and by the dipole approximation (curves with symbols). The matching of both

Fig. 2. (a) Real and (b) imaginary parts of the dielectric permittivity for silver in vacuum (black thick curves) and in water (black thin curves) and gold in vacuum (gray thick curves) and in water (gray thin curves). The top figure shows the force on a gold cylinder (a ⫽ 10 nm) in vacuum (thick curve) and in water (thin curve) exactly calculated by means of the partial-wave series. The same calculation is performed via the dipole approximation: a gold cylinder in vacuum (thin curve with inverted triangles) and in water (thin curve with up-pointing triangles). The illumination is done with a propagating plane wave.

Fig. 3. Dipole approximation. (From left to right) top: real part, imaginary part, and modulus of the polarizability for a silver sphere (a ⫽ 10 nm) in vacuum. Middle: vertical component, horizontal component, and modulus of the force on the same sphere (d ⫽ 20 nm) under evanescent plane-wave incidence ( ␪ 0 ⫽ 50° ⬎ ␪ c ⫽ 41.47°). Bottom: horizontal component, vertical component, and modulus of the force on the same sphere under Gaussian beam illumination (W ⫽ 6000 nm, ␪ 0 ⫽ 0) at x 0 ⫽ ⫺200 nm. Plain curves, Clausius–Mossotti polarizability with the radiative-reaction term; curves with symbol, polarizability of Dungey and Bohren.


Fig. 4. Dipole approximation. (From left to right) top: real part, imaginary part, and modulus of the P-wave polarizability (Clausius–Mossotti equation with the radiative-reaction term) for a gold cylinder (a ⫽ 10 nm) in water. Middle: vertical component, horizontal component, and modulus of the force on this cylinder in water under P-polarized evanescent plane-wave incidence at d ⫽ 20 nm. Solid curves, ␪ 0 ⫽ 62.5°; dashed curves, ␪ 0 ⫽ 70°. ␪ c ⫽ 61.98°. Bottom: horizontal component, vertical component, and modulus of the force on this cylinder in water under P-polarized Gaussian beam incidence ( ␪ 0 ⫽ 0) at x 0 ⫽ 20 nm. Solid curves, W ⫽ 6000 nm; dashed curves, W ⫽ 10,000 nm.

hand. A silver sphere in vacuum has been considered. Within the dipole approximation, two models of polarizability are plotted: the Clausius–Mossotti equation with the radiative-reaction term of Draine [see Eq. (7)] and that of Dungey and Bohren.36 First, evanescent wave illumination [cf. Eqs. (10) and (11)] is created under TIR at a flat interface separating glass ( 冑⑀ 0 ⫽ 1.51) from vacuum ( 冑⑀ 0 ⫽ 1). The angle of incidence is ␪ 0 ⫽ 50° (critical angle: ␪ c ⫽ 41.47°). The particle is placed at a height d ⫽ 20 nm. Of course, S and P polarizations are indistinguishable for a sphere in this configuration, and the interaction with the flat interface is considered negligible for this small particle at the distance d used. Second, these magnitudes are plotted in Fig. 3 when the illumination is done with a Gaussian beam of width W ⫽ 6000 nm at ␪ 0 ⫽ 0° [cf. Eqs. (16) and (17)]. The calculation has been performed at a displacement x 0 ⫽ ⫺200 nm of the particle from the axis of the beam (x ⫽ 0). As expressed by Eqs. (12) and (13) and Eqs. (16) and (17), the Cartesian components of the gradient and of the scattering-plus-absorption forces follow the behavior of the real and imaginary parts of the polarizability, respectively, and are interchanged when one switches from evanescent to Gaussian beam illumination. In particular, Fig. 3 illustrates how the gradient component of the force abruptly changes its sign at the resonant wavelength, thus becoming very small within a narrow interval of width governed by ⑀⬙, about this particular wavelength. Notice that, taking into account the relative position of the particle with respect to either an incident evanescent wave or an incident beam, considered here, an attractive force toward the higher-intensity region is negative in the former case and positive in the latter. Equivalent results are shown in Fig. 4 for a gold cylinder immersed in water under P polarization. Now the


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variation of the polarizability near resonance is less abrupt than in Fig. 3, owing to the larger losses introduced by the higher value of ⑀⬙ for gold, and this has consequences on the sign of the gradient force. The critical angle in the configuration glass and water ( 冑⑀ 0 ⫽ 1.51 and 冑⑀ 1 ⫽ 1.333, respectively) is ␪ c ⫽ 61.98°. The second row in Fig. 4, corresponding to evanescent wave illumination, displays two angles of incidence: ␪ 0 ⫽ 62.5° (solid curves) and ␪ 0 ⫽ 70° (dashed curves). The lower forces obtained at the larger angle of incidence are due to the corresponding smaller amplitude of the transmitted evanescent field at the position of the particle and incident on it. However, since the gradient of this magnitude increases with ␪ 0 , the gradient force increases with ␪ 0 when the amplitude of the evanescent field, incident on the particle, is normalized to unity. Figure 4 also shows the forces on the gold cylinder in water, created by an incident Gaussian beam: W ⫽ 6000 nm (solid curves) and W ⫽ 10,000 nm (dashed curves). The narrower the width of the beam, the stronger the force is. The scattering-plus-absorption force is less sensitive than the gradient force to this small variation in the beam width W [see Eqs. (16) and (17)]; thus the variation of this force component with ␭ is similar to that of incidence with a plane wave (cf. top inset in Fig. 2). Analyzing the curves of the gradient and of the scattering-plus-absorption components in both examples of incident wave, we can distinguish three situations. Fixing the plasmon-resonance wavelength at which the scattering cross section has the peak, there are (a) the case out of resonance and large wavelength at which both the gradient and the scattering-plus-absorption forces are small, (b) the case in which the wavelength is that of the plasmon resonance and both force components are large, and (c) the situation in which the wavelength is in the neighborhood of the plasmon resonance, so that the scattering-plus-absorption component of the force is larger than the corresponding component of case (a), whereas the gradient component is smaller than that in case (a). In the case of the silver sphere, only cases (a) and (c) are practically possible, as we see in Fig. 4 (when the scattering-plus-absorption force is maximum, the gradient force is almost zero). Thus resonance conditions always mean a large scattering-plus-absorption force, but this does not always apply to the gradient force. Next we study the spatial variation of the forces and potentials for the gold cylinder in water in the three situations, (a), (b), and (c), described above.

4. POTENTIAL OF A DIPOLAR PARTICLE IN THE LIGHT FIELD Next, we study the mechanical action of the electromagnetic field in terms of the potential energy. We perform calculations with the dipolar approximation and compare them with those from the ET and Eq. (5). Gaussian beam illumination creates a potential well (which can be considered a square shape for small beam widths W), giving rise to a critical position of the particle along the beam axis, which is either stable or unstable depending on its polarizability. Figure 5 shows this potential, obtained by the dipolar approximation and compared with the more

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rigorous result yielded by the ET. A small gold cylinder in water is illuminated by a P-polarized Gaussian beam, like the one described in Eqs. (1)–(4), with ␪ 0 ⫽ 0°. This complements the study of Ref. 20, which was done for larger cylinders for which there exists multiple scattering with the plane interface. For comparison with the influence of the Brownian motion, we have normalized the potential to the thermal energy k B T ⫽ 4.04 ⫻ 10⫺21 J, where k B is the Boltzmann constant and T ⫽ 293 K is the room temperature. The particle is displaced parallel to the OX axis. We have set the zero of potential energy at the beam axis (i.e., along OZ). Within the optical wavelength range, and with W considered in the range of several micrometers (so that W Ⰷ x max , x max being the maximum of x), the potential energy varies as the square of the x coordinate, as can be observed from Eq. (17). Namely, it can be written as V共 x 兲 ⫽ 兩T兩

2

x2 2W 2

Re兵 ␣ 其 .

(22)

Three wavelengths have been addressed, corresponding to those cases described in Subsection 3.C. All of them yield attraction of the particle toward the beam axis in P polarization (see bottom of Fig. 4). Regarding Fig. 5, for ␭ ⫽ 500 nm, ⑀ 2 ⫽ ⫺2.81 ⫹ i3.18, both the gradient and scattering-plus-absorption forces are maxima. At ␭ ⫽ 1064 nm, ⑀ 2 ⫽ ⫺53.65 ⫹ i4.18, both the gradient and the scattering-plus-absorption forces are minima. And for ␭ ⫽ 479 nm, ⑀ 2 ⫽ ⫺1.75 ⫹ i4.34, the gradient force is minimum, whereas the scattering-plus-absorption force is maximum. The strength of the gradient force makes the potential well deeper. The force along the OZ axis (scattering-plus-absorption force) is 5.72 ⫻ 10⫺20 N/nm for ␭ ⫽ 500 nm, 2.89 ⫻ 10⫺22 N/nm for ␭ ⫽ 1064 nm, and 4.81 ⫻ 10⫺20 N/nm for ␭ ⫽ 479 nm, as can be seen from Fig. 4 (bottom row). Figure 5 also shows how the depth of the potential well increases as the width W of the Gaussian beam diminishes. As can be seen, the accuracy of the dipole approximation for this particle size and composition is excellent. Figure 6 studies the potential, as well as the validity of its computation from the dipole approximation, versus separation d between the particle and the interface, at

Fig. 6. ET calculation of the potential energy of an isolated cylinder (a ⫽ 10 nm) immersed in water under the influence of a P-polarized evanescent wave (plane of incidence, OXZ). The cylinder moves along the OZ axis. (a) Gold, ␭ ⫽ 479 nm; (b) gold, ␭ ⫽ 500 nm (on resonance); (c) gold, ␭ ⫽ 1064 nm; (d) glass, ␭ ⫽ 632.8 nm. Plain lines, ET calculation; symbols, dipole approximation. Black curves and crosses, ␪ 0 ⫽ 62.5°; gray curves and inverse triangles, ␪ 0 ⫽ 65°; dashed curves and triangles, ␪ 0 ⫽ 70°. ␪ c ⫽ 61.98°. k B is the Boltzmann constant, and T ⫽ 293 K is the temperature.

different angles of incidence ␪ 0 at TIR. A small gold cylinder (a ⫽ 10 nm) in water is exposed to a P-polarized evanescent field created under TIR at the flat surface [Figs. 6(a)–6(c)]. To give a reference with a nonabsorbing particle, the same calculations are also performed, with a glass cylinder [Fig. 6(d)]. When the zero of potential energy is set at the flat surface, for a dipolar particle that is at distance z, V共 z 兲 ⫽

兩T兩2

4

Re兵 ␣ 其关 exp共 ⫺2qa 兲 ⫺ exp共 ⫺2qz 兲兴 . (23)

Figures 6(a)–6(c) show the potential at the wavelengths used in Fig. 5. As already seen in Fig. 4, the behavior of the polarizability for these wavelengths and polarization explains the vertical force attracting the particle to the surface. At ␭ ⫽ 500 nm (which corresponds to plasmon excitation) the potential, and hence the gradient force in the vertical direction and the scattering-plus-absorption force in the horizontal direction, is maximum (see the peaks of Fig. 4). This occurs for all angles of incidence and is similar to the behavior exhibited by the Gaussian beam, except that now both the potential energy and the gradient force at ␭ ⫽ 479 nm are higher than at ␭ ⫽ 1064 nm. This is explained as follows: Both the force and the potential energy depend on the wavelength, in the case of an incident evanescent wave, through both the polarizability 兵i.e., ␣ ⫽ ␣ 关 ⑀ (␭), 2␲/␭]其 and the factor q(␭) ⫽ (2 ␲ /␭) ⫻ ( ⑀ 1 sin2 ␪0 ⫺ ⑀0)1/2, and, in the case of beam illumination, through the polarizability only [see Eqs. (16) and (22)]. When q is small enough (i.e., ␪ 0 → ␪ c is larger but near ␪ c ) Eq. (23) linearizes as V共 z 兲 ⫽

Fig. 5. ET calculation of the potential energy of a gold cylinder (a ⫽ 10 nm) immersed in water under the influence of a P-polarized Gaussian beam with axis parallel to the cylinder axis along the line x ⫽ 0 ( ␪ 0 ⫽ 0). The cylinder moves along OX. Black thick curve, ␭ ⫽ 500 nm and W ⫽ 6000 nm; black thin curve, ␭ ⫽ 479 nm and W ⫽ 6000 nm; dashed curve, ␭ ⫽ 1064 nm and W ⫽ 6000 nm; gray thick curve, ␭ ⫽ 479 nm and W ⫽ 10,000 nm. Curves with crosses denote the same calculations performed with the dipole approximation. k B is the Boltzmann constant, and T ⫽ 293 K is the temperature.

兩T兩2

2

Re兵 ␣ 其 q 共 z ⫺ a 兲 ,

(24)

which explicitly shows that the slope of V(z) depends on both ␣(␭) and q(␭). For ␭ ⫽ 1064 nm, q is smaller than half of its value at ␭ ⫽ 479 nm, whereas ␣ does not undergo such a change. This result can be noticed in the graph of F z in Fig. 4. As stated in Subsection 3.C, the dipole approximation assumes that the field is uniform within the particle. When ␪ 0 increases beyond ␪ c , the field intensity gradient may become large within the par-


ticle, thus breaking the validity of this approximation at a given size and refractive index. (The same happens with a Gaussian beam by diminishing W.) This is clearly shown in Fig. 6 as ␪ 0 grows. We see, however, that indeed the behavior of the force is well understood within this approximation, even though it does not provide its exact value. The force is shown to dramatically depend on the decay length, especially when this length becomes very small. This explains the large changes that the force manifests when the particle is very close to an interface22—the multiple scattering effects can make the field highly inhomogeneous in the gap region. The simulations for both the sphere and the cylinder presented in this paper have been done assuming a unit incident field amplitude (Gaussian units) on the surface. This means an incident intensity on this surface of the order of 10⫺3 mW/␮m2 . Since the calculations for the cylinder are per unit length, if this is L ⫽ 10 ␮ m, the illuminated area is 102 ␮ m2 , and the incident power P 0 ⫽ 0.1 mW, then the order of magnitude of the potential in Fig. 5 is V/k B T ⬃ 10⫺5 , whereas it is V/k B T ⬃ 10⫺3 in Fig. 6.

5. ON THE VALIDITY OF THE DIPOLE APPROXIMATION Whereas Fig. 6 shows how the dipole approximation is broken in terms of the loss of uniformity of the field intensity inside the particle, owing to increase of the angle of incidence at TIR, we now plot in Fig. 7 the departure of this approximation in terms of the particle radius, keeping fixed the angle of incidence ␪ 0 ⫽ 62.5°. The results for gold at ␭ ⫽ 500 nm and ␭ ⫽ 479 nm, both of which are in the vicinity of the resonance for a ⫽ 10 nm, are similar to each other [we show the result only for ␭ ⫽ 500 nm, Fig. 7(a)]. However, owing to the lower ratio a/␭, the approximation for gold at ␭ ⫽ 1064 nm, Fig. 7(b), remains accurate for larger sizes. Figure 7(c) shows that, within the same range of radii, the dipole approximation also works correctly for a glass cylinder. Next, we analyze the error of the dipole approximation versus the ET results, defined as 100(F ref ⫺ F pol)/F ref ,

Fig. 7. ET calculation of the potential energy of an isolated cylinder immersed in water under the influence of a P-polarized evanescent wave with ␪ 0 ⫽ 62.5° ( ⬎ ␪ c ⫽ 61.98°) (plane of incidence, OXZ). The cylinder moves along the OZ axis. (a) Gold, ␭ ⫽ 500 nm (on resonance); (b) gold, ␭ ⫽ 1064 nm; (c) glass, ␭ ⫽ 632.8 nm. Plain lines, ET calculation; symbols, dipole approximation. Black curves and crosses, a ⫽ 20 nm; gray curves and inverse triangles, a ⫽ 30 nm; dashed curves and triangles, a ⫽ 50 nm; dotted curves and circles, a ⫽ 70 nm; dotted–dashed curves and squares, a ⫽ 100 nm. k B is the Boltzmann constant, and T ⫽ 293 K is the temperature.


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Fig. 8. Relative difference between the normalized force 关 F/exp(⫺2qz), see the text for details] calculated from the ET and from the dipole approximation. A cylinder (a ⫽ 10 nm) is immersed in water and illuminated by a P-polarized evanescent wave (plane of incidence, OXZ). (a), (b) Error in F x ; (c), (d) error in F z . (a), (c) Error as a function of the decay length of the evanescent wave; (b), (d) error as a function of the angle of incidence. Black curves: gold, ␭ ⫽ 479 nm; gray curves: gold, ␭ ⫽ 500 nm (on resonance); dashed curves: gold, ␭ ⫽ 1064 nm; dotted curves: glass, ␭ ⫽ 632.8 nm.

where the subindex ref denotes the force derived from the ET calculation and pol stands for that obtained with the dipole approximation. This quantity is studied as a function of the angle of incidence, ␪ 0 ⬎ ␪ c , and the decay length of the evanescent wave, defined as ⌳ ⫽ 1/q. Both quantities characterize the uniformity of the field and, as mentioned before, measure the validity of the dipole approximation. Figure 8 displays this error for a gold cylinder immersed in water. Since the force due to an evanescent wave decays exponentially with distance to the interface, its magnitude in this calculation has been normalized by dividing it by exp(⫺2qz). As the size of the particle and the wavelength are kept constant for each case, one can see how the decrease of the field uniformity inside the particle causes the dipole approximation to fail. This failure increases as the decay length of the evanescent field decreases, as expected. The curves of Fig. 8 also show that the dipolar model works better for small refractive-index dielectric particles, compared with metallic particles. Concerning those of gold, it is shown that for the nonresonant wavelength ␭ ⫽ 1064 nm the dipole approximation is worse than for those cases with ␭ in the vicinity of the resonance, although for small field variations, namely, large decay length ⌳ or ␪ 0 ⲏ ␪ c , the error in F z is similar in and off resonance.

6. EFFECTS OF MULTIPLE SCATTERING WITH THE PLANE INTERFACE Rigorous numerical calculations of potentials, including the wave multiple scattering between the particle and a dielectric plane, have been performed before at the above

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wavelengths by means of the ET,28 for both dielectric and metallic particles of radii up to 200 nm. The previous calculations show no appreciable differences with those based on the assumption that the particle isolated when the distance d to the interface is not smaller than its radius. (At smaller distances and for particles, say, with a ⫽ 10 nm, the optical force may compete with atomic forces, however.) In fact, we have previously investigated20 cylinders with radius a ⫽ 125 nm, either in the presence of or without a flat interface, that showed only small modifications in the potential curves. With respect to the forces for d ⭓ a, calculations considering the multiple interactions between the object and the surface found small amplitude oscillations as the particle approaches a flat interface. They have recently been experimentally detected in Ref. 37. This multiplescattering effect is, however, less appreciable in the potential, as a consequence of the smearing due to the integration of the force. Since the dipolar approximation fails when the field variations are large within the particle, it will be valid in the presence of an interface only if the multiple-scattering effects are not large. However, a qualitative description that includes the sign of the force and its behavior at resonance (which, in the presence of a plane, is found at shifted wavelengths with respect to those for an isolated particle28) is fruitful as a first approach to understanding the behavior of the force components and the influence of the particle polarizability. Another effect of multiple scattering is the mixing up in all directions of both the gradient and the scattering-plus-absorption forces, on which rests validity for such component decomposition that holds so well for small particles. The competition and transformation of these components may change the resulting force from attractive into repulsive, and vice versa. This was observed in Ref. 28 for both silver and silicon cylinders with radius a ⫽ 200 nm on glass surfaces with defects, although no experimental or theoretical research has yet systematically studied this effect.

We have explained in terms of polarizability how metallic particles may be subjected to forces that attract them or push them away from the higher-intensity regions. For the sake of brevity we have not explicitly shown the latter cases but state that they are present at the same wavelengths as those used for the cylinder in Figs. 4–6, although now in S polarization.20 These cases can also be found in cylinders in P polarization for ⫺1 ⬍ ⑀ ⬘ ⬍ 1 (with ⑀ ⬙ ⬍ 1) and in spheres for ⫺2 ⬍ ⑀ ⬘ ⬍ 1 (also with ⑀ ⬙ ⬍ 1), as has been explained in Eqs. (18) and (20). Thus we see that small metallic particles can suffer attractive or repulsive behaviors. We have proven that the excitation of a plasmon in a small metallic particle is associated with an abrupt sign change of the gradient force from extreme values of its magnitude in its wavelength-dependence curve and with a large scattering-plus-absorption force. Hence, in the close vicinity of this resonant wavelength, the magnitude of the gradient trapping force can abruptly vary from large to small values. This fact must be taken into account in connection with experiments that, like those recently reported,38,39 aim to create strong trapping potentials around metallic spheres by means of plasmon excitation. Similar results can be found in dielectric dipolar particles near resonant absorption (cf., Ref. 29).

ACKNOWLEDGMENTS We acknowledge a grant to J. R. Arias-Gonza´lez from Comunidad de Madrid, as well as partial support from Direccioń General de Investigacioń Cientı´fica y Tećnica and the European Union. Corresponding author M. Nieto-Vesperinas can be reached by phone, 34 91 3349044, or e-mail, mnieto @icmm.csic.es.

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7. CONCLUSIONS We have studied the range of validity of the dipole approximation on comparison with exact calculations and have shown its usefulness for interpreting the attractive or repulsive nature of the electromagnetic force, which can then be decomposed into a gradient component and a scattering-plus-absorption component. In establishing this validity, we have also shown the importance of the field amplitude uniformity inside the particle. In this respect, the evanescent wave decay length is a parameter to be taken into account. It is shown that the magnitude of the force is very sensitive to the lack of this uniformity. Dielectric particles (nonabsorbing) have been shown to be attracted toward the higher-intensity regions. This is so because the real part of the polarizability is always positive (assuming these particles to be optically denser than the surrounding medium, i.e., Re兵⑀2 /⑀0其 ⬎ 1). However, resonant conditions, e.g., an analysis of whisperinggallery modes,21 cannot be studied under the dipole approximation, since they involve multiple scattering in the particle.

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