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Optical force acting on a molecule near a metal sphere: effects of decay rate change and resonance frequency shift
Optics Communications 249 (2005) 329–337 www.elsevier.com/locate/optcom
Optical force acting on a molecule near a metal sphere: effects of decay rate change and resonance frequency shift Railing Chang
*
Institute of Optoelectronic Sciences, National Taiwan Ocean University Keelung 202, Taiwan, ROC Center of Nanostorage Research, National Taiwan University, Taipei 10617, Taiwan, ROC Received 2 October 2004; received in revised form 17 December 2004; accepted 21 December 2004
Abstract We present a theoretical analysis of optical force on a molecule near a metal sphere by taking into account the change of decay rate as well as the shift of resonance frequency in the response of the metal surface. As a result, the optical force is determined not only by the gradient of field but also the polarizability depending on the position and orientation of the molecule. In addition, we find the resulting reactive force cannot be expressed as a gradient of a scalar potential, in contrast to the usual situations. 2004 Elsevier B.V. All rights reserved. PACS: 32.80.Lg; 33.90.+h; 78.20.Bh Keywords: Optical force; Surface plasmon
1. Introduction Ever since the seminal works of Ashkin et al. [1,2], trapping and manipulating neutral particles have become techniques with a wide variety of applications [3,4]. The usual method to achieve trapping of particles is provided by a highly focused laser beam that has a strong gradient near the focal point. Due to the diffraction limit, the size of objects to be trapped is limited above sev*
Tel.: +886 2 24622192x6714; fax: +886 2 24634360. E-mail address: [email protected].
eral tens of nanometers. The extension of such technology to a region of several nanometers has been considered by making use of the techniques of near field optics. Novotny et al. [5] propose to use a laser-illuminated metal-tip to trap a dielectric particle in nanometer scale. Chaumet et al. [6] suggest a scheme to overcome the difficulty of Brownian motion and selectively capture a nanometer particle. Relevant work with an aperture is also proposed by Okamoto and Kawata [7]. In a recent work, Gu et al. [8] experimentally demonstrate another method of trapping that uses a focused evanescent field from an objective and the axial
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.12.046
330
R. Chang / Optics Communications 249 (2005) 329–337
trapping size can be as small as 60 nm. The main progress in the understanding of optical force in near field has been reviewed by Nieto-Vesperinas et al. [9]. A pertinent development is to use evanescent field generated by nanostructures to manipulate a luminescent molecule [10–12]. Calander and Willander [10] discuss the effect of trapping of single molecule at laser-illuminated metal particles. Michaels et al. [11] found in an AFM measurements that the single molecules exhibiting surface enhanced Raman scattering (SERS) are located at the junctions of two or more aggregated Ag nanoparticles. The strong field at the junction sites has an extremely sharp gradient and pulling the molecule toward it is a natural consequence. Though the field strength decays by orders within short range, Xu and Ka¨ll [12] perform a theoretical investigation and evaluate the trapping potential near the junction and conclude the optical trapping is definitely likely to occur in normal conditions of experiments. In Xu et al.Õs theory, the trapping force is completely determined by the field around the metal clusters and the polarizability of the molecule is regarded as an intrinsic property of the molecule and independent of the position of the molecule. However, it is well known that the optical properties, including decay rate as well as energy level of an excited state in a molecule located at the vicinity of metal nanoparticle, are sensitively dependent on position and orientation of the molecule. Vast amount of theoretical investigations of optical properties based on different viewpoints have been performed [13–22]. The reason for such change of properties is mainly due to the alternation of local density of state for photons near the surface of metallic nanostructures. Thus, the molecule is expected to exhibit a position- and orientation-dependent polarizability (PODP). Consequently, the optical force acting on the molecule is not determined merely by the electric field profile around the metal sphere. In this paper, we will demonstrate theoretically the optical force on a molecule by taking into account the effect of PODP due to the metal. We will consider only a single metal sphere for simplicity. We found the force is qualitatively different from
that obtained by field gradient. In addition, it is also found that the reactive force is not a gradient of a scalar potential in general.
2. Theory The system under consideration consists of a metal sphere with radius a and a molecule located at position (r,h,u = 0) interacting with an incident light polarized along z-axis as shown in Fig. 1. The surrounding environment is assumed to be vacuum. With the excitation of surface plasmon, evanescent field with great strength and sharp gradient appears around the metal sphere. Since the considered value of a is smaller than the wavelength of the incident light by order of magnitude, the simplest way to evaluate the field is to use electrostatics in which the incident field around the metal sphere is approximated by a uniform field EI^z. The evanscent field thus takes a form of *R
^ r þ ER E ¼ ER r^ h h;
ð1Þ
where 2P me P me cos h; and ER sin h ð2Þ h ¼ r3 r3 1 3 I Þa E and eme the dielectric conwith P me ¼ ðeeme me þ2 stant of the metal. Obviously, this first order solution has already exhibited the steep spatial ER r ¼
Fig. 1. The system considered for calculation.
R. Chang / Optics Communications 249 (2005) 329–337
variation in the vicinity of metal surface and exertion of an optical force on a molecule is anticipated. Although the luminescent molecule usually has a finite size, we assume that the description of interaction between field and molecule by dipole approximation still holds true for simplicity. As a result, the optical force exerted on the molecule is given as *
F ¼ pj rEj ; ð3Þ P * where p j pj^j is the dipole moment of the mol* *R P ecule and E j Ej^j ¼ EI^z þ E is the electric field of the optical wave. With electric field expressed as 1 * E ¼ ðE0 eixt þ c:c:Þ; 2
*
ð4Þ
and molecular linear polarizability a, the induced dipole moment is 1 * p ¼ ðp 0 eixt þ c:c:Þ 2 1 * ¼ ðaðE0 ^ pÞ^ peixt þ c:c:Þ; 2 i.e.,
*
c=Cnat ¼ 1 þ 3qImðGÞ=2k 3 ;
ð12Þ
where q is the quantum efficiency of radiation, k ” xnat/c, xnat the natural frequency and Cnat the natural decay rate. According to electrostatic theory, the r-dependent G-function is obtained as [13] X ðl þ 1Þ2 Dl a2lþ1 r2ðlþ1Þ l
ð14Þ
X lðl þ 1Þ Dl a2lþ1 ; 2r2ðlþ1Þ l
ð15Þ
lðeme 1Þ : lðeme þ 1Þ þ l
ð16Þ
G? ¼ and Gk ¼ where
ð8Þ
ð9Þ
and i pj ^ pk ðE0k ri E0j E0k ri E0j Þ hF idissip i ¼ a00 ^ 4
ð11Þ
ð13Þ
Dl ¼ a0 ^ p^ p ðE ri E0j þ E0k ri E0j Þ 4 j k 0k
e2 =m : x2 þ icx x20
The subsequent work is straightforward if a is a constant and the force acting on molecule will have a radial coordinate dependence as 1/r7. However as explained above, the decay rate c and frequency x0 of molecule are sensitive functions of position and dipole orientation when it is near a metal sphere. As a result, a(x) is not a constant anymore and the functional form of the forces will have position and orientation dependence qualitatively different from what is expected. In classical theory, the decay rate and resonance frequency varied by the boundary condition [24] are
ð7Þ
where hF ireact i ¼
aðxÞ ¼
ð5Þ
This can be separated into reactive and dissipative parts as hF i i ¼ hF ireact i þ hF idissip i;
Next, we model the molecule by a damping harmonic oscillator with resonance frequency x0 and damping rate constant c. The polarizability is simply
dx=Cnat ðx0 xnat Þ=Cnat ¼ 3qReðGÞ=4k 3 ;
1 aij E0j eixt þ c:c:Þ; pi ¼ ð~ ð6Þ 2 ~ij ¼ a^ pi ^ pi is the projection of unit pj and ^ where a vector of ^ p along ith axis. Thus, the time-averaged force can be written as 1 a E ri E0j þ c:c:Þ: hF i i ¼ hpj ri Ej i ¼ ð~ 4 jk 0k
331
ð10Þ
with a 0 (a00 ) being the real (imaginary) part of a.
In general case, Ggeneral ¼ G? cos2 hp þ Gk sin2 hp , where hp is the angle between dipole moment and radial direction. In the following, we will use ap(x,r) to denote PODP calculated from Eqs. (11)–(16). The position- and orientation-independent polarizability (POIP) from Eq. (11) with x0 = xnat and c = Cnat will be denoted by ao(x).
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R. Chang / Optics Communications 249 (2005) 329–337
Given this simple formulation, we next proceed to perform numerical calculation for the optical force in a few cases.
3. Results and discussions
δω /Γ
γ /Γ
In the numerical calculation, we assume that the ˚ and the natural metal sphere has radius a = 50 A lifetime of the molecule, 1/Cnat is fixed at 1.0 ns, a typical value for most of luminescent molecules. The dielectric constant of the metal is simply described by the Drude model, eme ðxÞ ¼ 1 x2p =xðx þ icme Þ, for which we assume xp = 1.36 · 1016 s1, corresponding to the case of Ag, and cme = 0.0025xp. Figs. 2(a) and (b) present the changed decay rate c and the shift of resonance frequency dx as functions of natural frequency xnat of the molecule. We fix the quantum efficiency at q = 1. The radial coordinate of the molecule is ˚ . Results for dipole along assumed to be r = 60 A radial as well as tangential directions are presented together and radial dipole exhibits stronger enhancement for c and dx. The positions of sharp peaks for c and zeros for dx correspond to the frequencies of surface plasmon modes given by the condition eme(x) =p(l pffiffiffi [25] with lower ffiffiffi + 1)/l (upper) limit at xp = 3ðxp = 2Þ. Around these resonance frequencies, the values of c and dx can be
10
5
10
4
10
3
10
2
1x10
5
extremely large in magnitude and can even become spurious if a smaller value of r is chosen. The unphysical result is due to the limited validity of Eqs. (12) and (13) for resonance case. More specific theory is required to reach realistic results. In the following, in order to demonstrate the effect of PODP on force, we will consider a case xnat = 0.5xp, which is substantially off resonance with any mode frequency and Eqs. (12) and (13) can be used. Furthermore, we assume the molecule has a finite size so that the minimum value of r ˚ , namely 5 A ˚ away from the meconsidered is 55 A tal surface. This choice is reasonable for dye molecule such as Rhodamine 6G. In Fig. 3, c and dx are displayed as functions of r. The maximum shift in resonance frequency for radial dipole is approximately 2.6 · 105Cnat, corresponding to an energy level shift 0.17 eV while the decay rate change is relatively small i.e. 3000Cnat corresponding to a lifetime 0.33 ps. Next, we may consider the force acting on the molecule by the electric field around the metal sphere. We will compare the results of calculation, by using: (A) POIP ao and (B) PODP ap. First we show in Figs. 4(a) and (b) the radial component of forces versus frequency of the field x for cases (A) and (B), respectively, where the position of mole˚ and h = 0, and the dipole cule is fixed at r = 60 A * moment of molecule p is along the radial direc-
(a)
(b)
0
-1x10
5
0.4
0.5
0.6
0.7
0.8
ωnat /ωp
Fig. 2. (a) The change of decay rates and (b) the shift of resonance frequency of the molecule as a function of natural frequency for dipole moment along radial (solid line) and tangential (dashed line) directions. The radial coordinate of ˚. position for molecule is r = 60 A
Fig. 3. (a) The change of decay rate and (b) the shift of resonance frequency as a function of radial coordinate of molecule for dipole moment along radial (solid line) and tangential (dashed line) directions. The natural frequency of the molecule is fixed at xnat = 0.5xp.
R. Chang / Optics Communications 249 (2005) 329–337 2
4000
0
2
-4
1000
-6 -8
0
-10
-1000
-12
-2000
-14
-4000
-18 0.499995
0.499998
0.500001
0.500004
-20
ω /ωp
(a)
-0.005
2
-0.010
0
-0.015
-2
-0.020
-4
-0.025
-6
-0.030
2
4
0.483
0.484
(b)
0.485
0.486
2
0.000
r
6
F dissip [fN/(mW/µm )]
r
2
-16 -3000
r
r
-2 2000
F dissip [fN/(mW/µm )]
Freact [fN/(mW/µm )]
3000
F react[fN/(mW/µm )]
333
0.487
ω/ωp
Fig. 4. The force on molecule with radial dipole moment calculated by using (a) POIP ao and (b) PODP ap. The molecule is located at ˚ and h = 0. r = 60 A
tion. Throughout the present paper the force is measured in unit of femto-Newton per unit incident power in mW/lm2. Beside the difference in resonance frequency, the result of case (B) has a largely reduced magnitude compared to that of case (A) due to the enhanced decay rate c. The reactive force is directed towards the metal sphere where field strength is high, as the field is tuned below the resonance frequency of the molecule. While with a blue-shifted field, the molecule experiences a reactive force with a direction pointing outward. Now we fix the frequency of field at x = 0.49xp and calculate the PODP ap(x,r) as well as the optical force on a molecule as functions of position. Due to the shift of resonance frequency dx shown in Fig. 3(b), the field detuning relative to molecule turns from red to blue as the molecule approaches the sphere. The resulting PODP ap(x,r) is shown in ˚ Fig. 5. The resonance occurs at r = rrad 61.43 A
˚ for dipoles oriented along and r = rtan 58.52 A radial and tangential directions, respectively. In the following, we will mainly consider a dipole
Fig. 5. (a) Real part and (b) imaginary part of PODP ap as functions of radial coordinate of molecule with dipole moment along radial (solid line) and tangential (dot line) directions.
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R. Chang / Optics Communications 249 (2005) 329–337
moment along the radial direction. Figs. 6(a) and (b) present, for cases (A) and (B), respectively, the radial component of reactive forces F rreact as functions of r for different values of h. Since c and dx do not depend on h, the magnitudes of F rreact for both cases have similar h-dependence and reach maxima (minima) at h = 0(p/2) with symmetry F rreact ðhÞ ¼ F rreact ðp hÞ. However, one can observe the distinctive characters in r-dependence of F rreact between the cases (A) as well as (B) in Fig. 6. F rreact of case (A) is purely attractive that it tends to pull the molecule all the way to the surface of the metal sphere. However in case (B), the calculated F rreact exhibits totally different feature as shown in Fig. 6(b). The attraction toward the metal surface occurs only if the molecule is located at a position with r > rrad. As the molecule is shifted to a position with r < rrad, F rreact becomes repulsive. Hence the molecule is likely to be ˚ positioned around r = rrad. Within a range of 5 A
around r = rrad, F rreact for case (B) is essentially greater than that for case (A) with maxima of magnitude 8 fN/(mW/lm2). Figs. 6(c) and (d) show the radial component of dissipative force F rdissip evaluated for cases (A) and (B), respectively. The magnitude of F rdissip is significantly weaker than that of F rreact for both cases of (A) and (B). However, F rdissip of case (B) is still much greater than (A) since at r = rrad the frequency of field is in resonance with the molecular oscillation and thus absorption rate of light by the molecule is higher than case (A). As to the h-component of the force, we plot the calculated results with respect to angle position h for different r. In the case (A), the calculated reactive force F hreact tends to pull the molecule towards the regions near the pole, i.e., h = 0 and h = p, no matter what the value of r is, as shown in Fig. 7a. In case (B), however, the behavior of calculated F hreact sensitively depends on r as shown in Fig.
Fig. 6. Radial component of calculated reactive force F rreact by (a) POIP ao and (b) PODP ap and dissipative force F rdissip by (c) POIP ao and (d) PODP ap for various angle h with increment p/10.
R. Chang / Optics Communications 249 (2005) 329–337
335
Fig. 7. h-Component of calculated reactive force F hreact by (a) POIP ao and (b) PODP ap and dissipative force F hdissip by (c) POIP ao and ˚ (solid line) r = 61 A ˚ (dashed line) r = 62 A ˚ (dot line) r = 62 A ˚ (dashed dot). (d) PODP ap for r = 60 A
7(b). When the molecule is located at a position with r < rrad, F hreact tends to push the molecule to the equilateral region, i.e., h = p/2. As r > rrad, F hreact turns to pull the molecule to the poles of the sphere. The dissipative forces of h-component, F hdissip , in both cases (A) and (B), shown in Figs. 7(c) and (d), respectively, tend to push the molecule towards the equilateral region. The latter is different from the former only in magnitude, but without qualitatively different feature. Since the dissipative force is rather small, we may pay more attention to the reactive force exclusively. Fig. 8*shows the distribution of reactive force vectors F react F rreact^r þ F hreact ^ h near the critical radial coordinate r = rrad for molecule with radial dipole moment, as PODP is taken into * account. The distribution of F react on the molecule with a tangential dipole moment has been calculated as well and the result near r = rtan is shown in Fig. 9. Similar to the radial dipole case, both
of ^r- and ^h-components of force are opposite on both sides of the critical radial coordinate r = rtan. Usually one prefers to express the reactive force in
*
Fig. 8. Force vectors F react on molecule with a dipole moment along radial direction located at position near r = rrad. Line integral along the thin dotted loop will give a nonzero value.
336
R. Chang / Optics Communications 249 (2005) 329–337
*
Fig. 9. Force vectors F react on molecule with a dipole moment along the tangential direction located at position near r = rtan. Line integral along the thin dotted loop will give a nonzero value.
terms of a gradient of a scalar potential. This is no longer true in case of PODP as implied by the feature shown in Figs. 8 and 9. If one performs a line integral over a loop enclosing a portion of the critical radial coordinate (e.g. dotted lines in Figs. 8 and 9), a nonzero value will be obtained due to the contribution of F hreact ^ h. One can observe this more explicitly by taking the curl of the reactive force in Eq. (9) * 1 ðr hF react iÞi ¼ nlm eijk rj rk ðEm El þ Em El Þ 2 1 þ eijk rj nlm rk ðEm El þ Em El Þ; 2 ð17Þ
where nlm ¼ 14a0 p^l p^m . Note that the first term in Eq. (17) is zero while the second term is not, in contrast to the usual situation. In the absence of the position-dependent polarizability, there are two situations that make the second term vanish. The first situation is to have a constant dipole direction ^p that leads to $jnlm = 0. The second one is to choose a position-dependent di* pole direction but with ^ pk E then rnlm kr ðEm El þ Em El Þ and the cross product in the second term is zero. In the case we have discussed in the present paper, a 0 = ap(x,r) is a complicated function of position of the molecule, the vanishing of the second term can hardly occur in general.
In conclusion, we have studied the optical force on a molecule near a metal sphere by taking into account the change of lifetime and the shift of resonance frequency. Several possible directions can be considered in order to extend our work. First the classical model for molecule is not a realistic description as far as the optical force is concerned. The saturation effect that would weaken the optical force as discussed by Calander and Willander [10] can be treated only by two-level model. In addition, it is essential to consider the vibronic coupling that results in the broadening of energy levels in a molecular system. On the other hand, the Drude model for the dielectric constant of metal is not sufficient in particular for the metal sphere in nanometer size. Modification of this by taking into account nonlocalilty using hydrodynamic model [13,14] or other more advanced approach [22,23] should be considered. Furthermore, as discussed in [11,12], the trapping centers for molecules that exhibit SERS are located at the junctions of cluster of metallic nanoparticles in realistic situations of experiment. We will extend our consideration to systems of cluster in the next step and a direct comparison with experiment can be performed.
Acknowledgments This research is supported by National Science Council, Taiwan, under Grant number NSC-932112-M-019-004. The author also thanks professor P.T. Leung for encouragement.
References [1] A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. [2] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288. [3] S. Chu, Science 253 (1991) 861. [4] A. Ashkin, J.M. Dziedzic, Science 235 (1987) 1517. [5] L. Novotny, R.X. Bian, X.S. Xie, Phys. Rev. Lett. 79 (1997) 645. [6] P.C. Chaumet, A. Rahmani, M. Nieto-Vesperinas, Phys. Rev. Lett. 88 (2002) 123601. [7] K. Okamoto, S. Kawata, Phys. Rev. Lett. 83 (1999) 4534.
R. Chang / Optics Communications 249 (2005) 329–337 [8] M. Gu, J.-B. Haumonte, Y. Micheau, J.W.M. Chon, X. Gan, Appl. Phys. Lett. 84 (2004) 4236. [9] M. Nieto-Vesperinas, P.C. Chaumet, A. Rahmani, Philos. Trans. R. Soc. Lond. A 362 (2004) 719. [10] N. Calander, M. Willander, Phys. Rev. Lett. 89 (2002) 143603. [11] A. Michaels, J. Jiang, L. Brus, J. Phys. Chem. B 104 (2000) 11965. [12] H. Xu, M. Ka¨ll, Phys. Rev. Lett. 89 (2002) 246802. [13] J.I. Gersten, A. Nitzan, Surf. Sci. 158 (1985) 165. [14] P.T. Leung, Phys. Rev. B 42 (1990) 7622. [15] P.T. Leung, M.H. Hider, J. Chem. Phys. 98 (1993) 5019. [16] H. Chew, J. Chem. Phys. 87 (1987) 1355.
337
[17] N. Liver, A. Nitzan, K.F. Freed, J. Chem. Phys. 82 (1985) 3831. [18] P.T. Leung, T.F. George, J. Chem. Phys. 87 (1987) 6722. [19] P.T. Leung, Y.S. Kim, T.F. George, J. Phys. Chem. 92 (1988) 6206. [20] A. Nitzan, L.E. Brus, J. Chem. Phys. 75 (1981) 2205. [21] J. Gersten, A. Nitzan, J. Chem. Phys. 75 (1981) 1139. [22] W. Ekardt, Z. Penzar, Phys. Rev. B 34 (1986) 8444. [23] S. Corni, J. Tomasi, J. Chem. Phys. 118 (2003) 6481. [24] R.R. Chance, A. Prock, R. Silbey, Adv. Chem. Phys. 37 (1978) 1. [25] H. Raether, Surface Plasmons on Smooth and Rough Surface and on Gratings, Springer, Berlin, 1988.
Optical force acting on a molecule near a metal sphere: effects of decay rate change and resonance frequency shift Railing Chang
*
Institute of Optoelectronic Sciences, National Taiwan Ocean University Keelung 202, Taiwan, ROC Center of Nanostorage Research, National Taiwan University, Taipei 10617, Taiwan, ROC Received 2 October 2004; received in revised form 17 December 2004; accepted 21 December 2004
Abstract We present a theoretical analysis of optical force on a molecule near a metal sphere by taking into account the change of decay rate as well as the shift of resonance frequency in the response of the metal surface. As a result, the optical force is determined not only by the gradient of field but also the polarizability depending on the position and orientation of the molecule. In addition, we find the resulting reactive force cannot be expressed as a gradient of a scalar potential, in contrast to the usual situations. 2004 Elsevier B.V. All rights reserved. PACS: 32.80.Lg; 33.90.+h; 78.20.Bh Keywords: Optical force; Surface plasmon
1. Introduction Ever since the seminal works of Ashkin et al. [1,2], trapping and manipulating neutral particles have become techniques with a wide variety of applications [3,4]. The usual method to achieve trapping of particles is provided by a highly focused laser beam that has a strong gradient near the focal point. Due to the diffraction limit, the size of objects to be trapped is limited above sev*
Tel.: +886 2 24622192x6714; fax: +886 2 24634360. E-mail address: [email protected].
eral tens of nanometers. The extension of such technology to a region of several nanometers has been considered by making use of the techniques of near field optics. Novotny et al. [5] propose to use a laser-illuminated metal-tip to trap a dielectric particle in nanometer scale. Chaumet et al. [6] suggest a scheme to overcome the difficulty of Brownian motion and selectively capture a nanometer particle. Relevant work with an aperture is also proposed by Okamoto and Kawata [7]. In a recent work, Gu et al. [8] experimentally demonstrate another method of trapping that uses a focused evanescent field from an objective and the axial
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.12.046
330
R. Chang / Optics Communications 249 (2005) 329–337
trapping size can be as small as 60 nm. The main progress in the understanding of optical force in near field has been reviewed by Nieto-Vesperinas et al. [9]. A pertinent development is to use evanescent field generated by nanostructures to manipulate a luminescent molecule [10–12]. Calander and Willander [10] discuss the effect of trapping of single molecule at laser-illuminated metal particles. Michaels et al. [11] found in an AFM measurements that the single molecules exhibiting surface enhanced Raman scattering (SERS) are located at the junctions of two or more aggregated Ag nanoparticles. The strong field at the junction sites has an extremely sharp gradient and pulling the molecule toward it is a natural consequence. Though the field strength decays by orders within short range, Xu and Ka¨ll [12] perform a theoretical investigation and evaluate the trapping potential near the junction and conclude the optical trapping is definitely likely to occur in normal conditions of experiments. In Xu et al.Õs theory, the trapping force is completely determined by the field around the metal clusters and the polarizability of the molecule is regarded as an intrinsic property of the molecule and independent of the position of the molecule. However, it is well known that the optical properties, including decay rate as well as energy level of an excited state in a molecule located at the vicinity of metal nanoparticle, are sensitively dependent on position and orientation of the molecule. Vast amount of theoretical investigations of optical properties based on different viewpoints have been performed [13–22]. The reason for such change of properties is mainly due to the alternation of local density of state for photons near the surface of metallic nanostructures. Thus, the molecule is expected to exhibit a position- and orientation-dependent polarizability (PODP). Consequently, the optical force acting on the molecule is not determined merely by the electric field profile around the metal sphere. In this paper, we will demonstrate theoretically the optical force on a molecule by taking into account the effect of PODP due to the metal. We will consider only a single metal sphere for simplicity. We found the force is qualitatively different from
that obtained by field gradient. In addition, it is also found that the reactive force is not a gradient of a scalar potential in general.
2. Theory The system under consideration consists of a metal sphere with radius a and a molecule located at position (r,h,u = 0) interacting with an incident light polarized along z-axis as shown in Fig. 1. The surrounding environment is assumed to be vacuum. With the excitation of surface plasmon, evanescent field with great strength and sharp gradient appears around the metal sphere. Since the considered value of a is smaller than the wavelength of the incident light by order of magnitude, the simplest way to evaluate the field is to use electrostatics in which the incident field around the metal sphere is approximated by a uniform field EI^z. The evanscent field thus takes a form of *R
^ r þ ER E ¼ ER r^ h h;
ð1Þ
where 2P me P me cos h; and ER sin h ð2Þ h ¼ r3 r3 1 3 I Þa E and eme the dielectric conwith P me ¼ ðeeme me þ2 stant of the metal. Obviously, this first order solution has already exhibited the steep spatial ER r ¼
Fig. 1. The system considered for calculation.
R. Chang / Optics Communications 249 (2005) 329–337
variation in the vicinity of metal surface and exertion of an optical force on a molecule is anticipated. Although the luminescent molecule usually has a finite size, we assume that the description of interaction between field and molecule by dipole approximation still holds true for simplicity. As a result, the optical force exerted on the molecule is given as *
F ¼ pj rEj ; ð3Þ P * where p j pj^j is the dipole moment of the mol* *R P ecule and E j Ej^j ¼ EI^z þ E is the electric field of the optical wave. With electric field expressed as 1 * E ¼ ðE0 eixt þ c:c:Þ; 2
*
ð4Þ
and molecular linear polarizability a, the induced dipole moment is 1 * p ¼ ðp 0 eixt þ c:c:Þ 2 1 * ¼ ðaðE0 ^ pÞ^ peixt þ c:c:Þ; 2 i.e.,
*
c=Cnat ¼ 1 þ 3qImðGÞ=2k 3 ;
ð12Þ
where q is the quantum efficiency of radiation, k ” xnat/c, xnat the natural frequency and Cnat the natural decay rate. According to electrostatic theory, the r-dependent G-function is obtained as [13] X ðl þ 1Þ2 Dl a2lþ1 r2ðlþ1Þ l
ð14Þ
X lðl þ 1Þ Dl a2lþ1 ; 2r2ðlþ1Þ l
ð15Þ
lðeme 1Þ : lðeme þ 1Þ þ l
ð16Þ
G? ¼ and Gk ¼ where
ð8Þ
ð9Þ
and i pj ^ pk ðE0k ri E0j E0k ri E0j Þ hF idissip i ¼ a00 ^ 4
ð11Þ
ð13Þ
Dl ¼ a0 ^ p^ p ðE ri E0j þ E0k ri E0j Þ 4 j k 0k
e2 =m : x2 þ icx x20
The subsequent work is straightforward if a is a constant and the force acting on molecule will have a radial coordinate dependence as 1/r7. However as explained above, the decay rate c and frequency x0 of molecule are sensitive functions of position and dipole orientation when it is near a metal sphere. As a result, a(x) is not a constant anymore and the functional form of the forces will have position and orientation dependence qualitatively different from what is expected. In classical theory, the decay rate and resonance frequency varied by the boundary condition [24] are
ð7Þ
where hF ireact i ¼
aðxÞ ¼
ð5Þ
This can be separated into reactive and dissipative parts as hF i i ¼ hF ireact i þ hF idissip i;
Next, we model the molecule by a damping harmonic oscillator with resonance frequency x0 and damping rate constant c. The polarizability is simply
dx=Cnat ðx0 xnat Þ=Cnat ¼ 3qReðGÞ=4k 3 ;
1 aij E0j eixt þ c:c:Þ; pi ¼ ð~ ð6Þ 2 ~ij ¼ a^ pi ^ pi is the projection of unit pj and ^ where a vector of ^ p along ith axis. Thus, the time-averaged force can be written as 1 a E ri E0j þ c:c:Þ: hF i i ¼ hpj ri Ej i ¼ ð~ 4 jk 0k
331
ð10Þ
with a 0 (a00 ) being the real (imaginary) part of a.
In general case, Ggeneral ¼ G? cos2 hp þ Gk sin2 hp , where hp is the angle between dipole moment and radial direction. In the following, we will use ap(x,r) to denote PODP calculated from Eqs. (11)–(16). The position- and orientation-independent polarizability (POIP) from Eq. (11) with x0 = xnat and c = Cnat will be denoted by ao(x).
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Given this simple formulation, we next proceed to perform numerical calculation for the optical force in a few cases.
3. Results and discussions
δω /Γ
γ /Γ
In the numerical calculation, we assume that the ˚ and the natural metal sphere has radius a = 50 A lifetime of the molecule, 1/Cnat is fixed at 1.0 ns, a typical value for most of luminescent molecules. The dielectric constant of the metal is simply described by the Drude model, eme ðxÞ ¼ 1 x2p =xðx þ icme Þ, for which we assume xp = 1.36 · 1016 s1, corresponding to the case of Ag, and cme = 0.0025xp. Figs. 2(a) and (b) present the changed decay rate c and the shift of resonance frequency dx as functions of natural frequency xnat of the molecule. We fix the quantum efficiency at q = 1. The radial coordinate of the molecule is ˚ . Results for dipole along assumed to be r = 60 A radial as well as tangential directions are presented together and radial dipole exhibits stronger enhancement for c and dx. The positions of sharp peaks for c and zeros for dx correspond to the frequencies of surface plasmon modes given by the condition eme(x) =p(l pffiffiffi [25] with lower ffiffiffi + 1)/l (upper) limit at xp = 3ðxp = 2Þ. Around these resonance frequencies, the values of c and dx can be
10
5
10
4
10
3
10
2
1x10
5
extremely large in magnitude and can even become spurious if a smaller value of r is chosen. The unphysical result is due to the limited validity of Eqs. (12) and (13) for resonance case. More specific theory is required to reach realistic results. In the following, in order to demonstrate the effect of PODP on force, we will consider a case xnat = 0.5xp, which is substantially off resonance with any mode frequency and Eqs. (12) and (13) can be used. Furthermore, we assume the molecule has a finite size so that the minimum value of r ˚ , namely 5 A ˚ away from the meconsidered is 55 A tal surface. This choice is reasonable for dye molecule such as Rhodamine 6G. In Fig. 3, c and dx are displayed as functions of r. The maximum shift in resonance frequency for radial dipole is approximately 2.6 · 105Cnat, corresponding to an energy level shift 0.17 eV while the decay rate change is relatively small i.e. 3000Cnat corresponding to a lifetime 0.33 ps. Next, we may consider the force acting on the molecule by the electric field around the metal sphere. We will compare the results of calculation, by using: (A) POIP ao and (B) PODP ap. First we show in Figs. 4(a) and (b) the radial component of forces versus frequency of the field x for cases (A) and (B), respectively, where the position of mole˚ and h = 0, and the dipole cule is fixed at r = 60 A * moment of molecule p is along the radial direc-
(a)
(b)
0
-1x10
5
0.4
0.5
0.6
0.7
0.8
ωnat /ωp
Fig. 2. (a) The change of decay rates and (b) the shift of resonance frequency of the molecule as a function of natural frequency for dipole moment along radial (solid line) and tangential (dashed line) directions. The radial coordinate of ˚. position for molecule is r = 60 A
Fig. 3. (a) The change of decay rate and (b) the shift of resonance frequency as a function of radial coordinate of molecule for dipole moment along radial (solid line) and tangential (dashed line) directions. The natural frequency of the molecule is fixed at xnat = 0.5xp.
R. Chang / Optics Communications 249 (2005) 329–337 2
4000
0
2
-4
1000
-6 -8
0
-10
-1000
-12
-2000
-14
-4000
-18 0.499995
0.499998
0.500001
0.500004
-20
ω /ωp
(a)
-0.005
2
-0.010
0
-0.015
-2
-0.020
-4
-0.025
-6
-0.030
2
4
0.483
0.484
(b)
0.485
0.486
2
0.000
r
6
F dissip [fN/(mW/µm )]
r
2
-16 -3000
r
r
-2 2000
F dissip [fN/(mW/µm )]
Freact [fN/(mW/µm )]
3000
F react[fN/(mW/µm )]
333
0.487
ω/ωp
Fig. 4. The force on molecule with radial dipole moment calculated by using (a) POIP ao and (b) PODP ap. The molecule is located at ˚ and h = 0. r = 60 A
tion. Throughout the present paper the force is measured in unit of femto-Newton per unit incident power in mW/lm2. Beside the difference in resonance frequency, the result of case (B) has a largely reduced magnitude compared to that of case (A) due to the enhanced decay rate c. The reactive force is directed towards the metal sphere where field strength is high, as the field is tuned below the resonance frequency of the molecule. While with a blue-shifted field, the molecule experiences a reactive force with a direction pointing outward. Now we fix the frequency of field at x = 0.49xp and calculate the PODP ap(x,r) as well as the optical force on a molecule as functions of position. Due to the shift of resonance frequency dx shown in Fig. 3(b), the field detuning relative to molecule turns from red to blue as the molecule approaches the sphere. The resulting PODP ap(x,r) is shown in ˚ Fig. 5. The resonance occurs at r = rrad 61.43 A
˚ for dipoles oriented along and r = rtan 58.52 A radial and tangential directions, respectively. In the following, we will mainly consider a dipole
Fig. 5. (a) Real part and (b) imaginary part of PODP ap as functions of radial coordinate of molecule with dipole moment along radial (solid line) and tangential (dot line) directions.
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moment along the radial direction. Figs. 6(a) and (b) present, for cases (A) and (B), respectively, the radial component of reactive forces F rreact as functions of r for different values of h. Since c and dx do not depend on h, the magnitudes of F rreact for both cases have similar h-dependence and reach maxima (minima) at h = 0(p/2) with symmetry F rreact ðhÞ ¼ F rreact ðp hÞ. However, one can observe the distinctive characters in r-dependence of F rreact between the cases (A) as well as (B) in Fig. 6. F rreact of case (A) is purely attractive that it tends to pull the molecule all the way to the surface of the metal sphere. However in case (B), the calculated F rreact exhibits totally different feature as shown in Fig. 6(b). The attraction toward the metal surface occurs only if the molecule is located at a position with r > rrad. As the molecule is shifted to a position with r < rrad, F rreact becomes repulsive. Hence the molecule is likely to be ˚ positioned around r = rrad. Within a range of 5 A
around r = rrad, F rreact for case (B) is essentially greater than that for case (A) with maxima of magnitude 8 fN/(mW/lm2). Figs. 6(c) and (d) show the radial component of dissipative force F rdissip evaluated for cases (A) and (B), respectively. The magnitude of F rdissip is significantly weaker than that of F rreact for both cases of (A) and (B). However, F rdissip of case (B) is still much greater than (A) since at r = rrad the frequency of field is in resonance with the molecular oscillation and thus absorption rate of light by the molecule is higher than case (A). As to the h-component of the force, we plot the calculated results with respect to angle position h for different r. In the case (A), the calculated reactive force F hreact tends to pull the molecule towards the regions near the pole, i.e., h = 0 and h = p, no matter what the value of r is, as shown in Fig. 7a. In case (B), however, the behavior of calculated F hreact sensitively depends on r as shown in Fig.
Fig. 6. Radial component of calculated reactive force F rreact by (a) POIP ao and (b) PODP ap and dissipative force F rdissip by (c) POIP ao and (d) PODP ap for various angle h with increment p/10.
R. Chang / Optics Communications 249 (2005) 329–337
335
Fig. 7. h-Component of calculated reactive force F hreact by (a) POIP ao and (b) PODP ap and dissipative force F hdissip by (c) POIP ao and ˚ (solid line) r = 61 A ˚ (dashed line) r = 62 A ˚ (dot line) r = 62 A ˚ (dashed dot). (d) PODP ap for r = 60 A
7(b). When the molecule is located at a position with r < rrad, F hreact tends to push the molecule to the equilateral region, i.e., h = p/2. As r > rrad, F hreact turns to pull the molecule to the poles of the sphere. The dissipative forces of h-component, F hdissip , in both cases (A) and (B), shown in Figs. 7(c) and (d), respectively, tend to push the molecule towards the equilateral region. The latter is different from the former only in magnitude, but without qualitatively different feature. Since the dissipative force is rather small, we may pay more attention to the reactive force exclusively. Fig. 8*shows the distribution of reactive force vectors F react F rreact^r þ F hreact ^ h near the critical radial coordinate r = rrad for molecule with radial dipole moment, as PODP is taken into * account. The distribution of F react on the molecule with a tangential dipole moment has been calculated as well and the result near r = rtan is shown in Fig. 9. Similar to the radial dipole case, both
of ^r- and ^h-components of force are opposite on both sides of the critical radial coordinate r = rtan. Usually one prefers to express the reactive force in
*
Fig. 8. Force vectors F react on molecule with a dipole moment along radial direction located at position near r = rrad. Line integral along the thin dotted loop will give a nonzero value.
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R. Chang / Optics Communications 249 (2005) 329–337
*
Fig. 9. Force vectors F react on molecule with a dipole moment along the tangential direction located at position near r = rtan. Line integral along the thin dotted loop will give a nonzero value.
terms of a gradient of a scalar potential. This is no longer true in case of PODP as implied by the feature shown in Figs. 8 and 9. If one performs a line integral over a loop enclosing a portion of the critical radial coordinate (e.g. dotted lines in Figs. 8 and 9), a nonzero value will be obtained due to the contribution of F hreact ^ h. One can observe this more explicitly by taking the curl of the reactive force in Eq. (9) * 1 ðr hF react iÞi ¼ nlm eijk rj rk ðEm El þ Em El Þ 2 1 þ eijk rj nlm rk ðEm El þ Em El Þ; 2 ð17Þ
where nlm ¼ 14a0 p^l p^m . Note that the first term in Eq. (17) is zero while the second term is not, in contrast to the usual situation. In the absence of the position-dependent polarizability, there are two situations that make the second term vanish. The first situation is to have a constant dipole direction ^p that leads to $jnlm = 0. The second one is to choose a position-dependent di* pole direction but with ^ pk E then rnlm kr ðEm El þ Em El Þ and the cross product in the second term is zero. In the case we have discussed in the present paper, a 0 = ap(x,r) is a complicated function of position of the molecule, the vanishing of the second term can hardly occur in general.
In conclusion, we have studied the optical force on a molecule near a metal sphere by taking into account the change of lifetime and the shift of resonance frequency. Several possible directions can be considered in order to extend our work. First the classical model for molecule is not a realistic description as far as the optical force is concerned. The saturation effect that would weaken the optical force as discussed by Calander and Willander [10] can be treated only by two-level model. In addition, it is essential to consider the vibronic coupling that results in the broadening of energy levels in a molecular system. On the other hand, the Drude model for the dielectric constant of metal is not sufficient in particular for the metal sphere in nanometer size. Modification of this by taking into account nonlocalilty using hydrodynamic model [13,14] or other more advanced approach [22,23] should be considered. Furthermore, as discussed in [11,12], the trapping centers for molecules that exhibit SERS are located at the junctions of cluster of metallic nanoparticles in realistic situations of experiment. We will extend our consideration to systems of cluster in the next step and a direct comparison with experiment can be performed.
Acknowledgments This research is supported by National Science Council, Taiwan, under Grant number NSC-932112-M-019-004. The author also thanks professor P.T. Leung for encouragement.
References [1] A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. [2] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288. [3] S. Chu, Science 253 (1991) 861. [4] A. Ashkin, J.M. Dziedzic, Science 235 (1987) 1517. [5] L. Novotny, R.X. Bian, X.S. Xie, Phys. Rev. Lett. 79 (1997) 645. [6] P.C. Chaumet, A. Rahmani, M. Nieto-Vesperinas, Phys. Rev. Lett. 88 (2002) 123601. [7] K. Okamoto, S. Kawata, Phys. Rev. Lett. 83 (1999) 4534.
R. Chang / Optics Communications 249 (2005) 329–337 [8] M. Gu, J.-B. Haumonte, Y. Micheau, J.W.M. Chon, X. Gan, Appl. Phys. Lett. 84 (2004) 4236. [9] M. Nieto-Vesperinas, P.C. Chaumet, A. Rahmani, Philos. Trans. R. Soc. Lond. A 362 (2004) 719. [10] N. Calander, M. Willander, Phys. Rev. Lett. 89 (2002) 143603. [11] A. Michaels, J. Jiang, L. Brus, J. Phys. Chem. B 104 (2000) 11965. [12] H. Xu, M. Ka¨ll, Phys. Rev. Lett. 89 (2002) 246802. [13] J.I. Gersten, A. Nitzan, Surf. Sci. 158 (1985) 165. [14] P.T. Leung, Phys. Rev. B 42 (1990) 7622. [15] P.T. Leung, M.H. Hider, J. Chem. Phys. 98 (1993) 5019. [16] H. Chew, J. Chem. Phys. 87 (1987) 1355.
337
[17] N. Liver, A. Nitzan, K.F. Freed, J. Chem. Phys. 82 (1985) 3831. [18] P.T. Leung, T.F. George, J. Chem. Phys. 87 (1987) 6722. [19] P.T. Leung, Y.S. Kim, T.F. George, J. Phys. Chem. 92 (1988) 6206. [20] A. Nitzan, L.E. Brus, J. Chem. Phys. 75 (1981) 2205. [21] J. Gersten, A. Nitzan, J. Chem. Phys. 75 (1981) 1139. [22] W. Ekardt, Z. Penzar, Phys. Rev. B 34 (1986) 8444. [23] S. Corni, J. Tomasi, J. Chem. Phys. 118 (2003) 6481. [24] R.R. Chance, A. Prock, R. Silbey, Adv. Chem. Phys. 37 (1978) 1. [25] H. Raether, Surface Plasmons on Smooth and Rough Surface and on Gratings, Springer, Berlin, 1988.