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Radiation forces on a Mie particle in the evanescent field of an optical waveguide
Optics Communications 246 (2005) 373–383 www.elsevier.com/locate/optcom
Radiation forces on a Mie particle in the evanescent field of an optical waveguide H.Y. Jaising *, O.G. Hellesø Department of Physics, University of Tromsø, Nordlysobservatoriet, 9037 Tromsø, Norway Received 11 December 2003; received in revised form 17 September 2004; accepted 4 November 2004
Abstract We use a generalization of Mie theory – the so-called Arbitrary Beam Theory – to calculate the scattered field and radiation forces on a dielectric spherical particle in the evanescent field of an optical waveguide with a step index profile. The guiding velocities calculated by us agree well with experimental results reported by other groups. Moreover, the occurrence of resonances in the scattered field has the consequence that the particle is repelled from, instead of being attracted to, the waveguide for certain values of particle refractive index and radius. 2004 Elsevier B.V. All rights reserved. PACS: 42.82.E; 42.25.F; 32.80.P; 33.80.P Keywords: Evanescent field; Optical waveguide; Mie scattering; Radiation forces; Morphology dependent resonances
1. Introduction The experiments by Ashkin and others in 1970s in optical trapping and manipulation laid the groundwork for the rapid development of the field of optical trapping and optical tweezers [1]. Since the experiments by AshkinÕs group, several different schemes have been used to trap and manipulate small particles. In the experiment by Kawata and Sugiura [2] the radiation force due to the *
Corresponding author. Tel.: +47 77 64 51 14; fax: + 47 77 64 55 80. E-mail address: [email protected] (H.Y. Jaising).
evanescent field produced by totally reflecting a laser beam at a prism surface was used to propel dielectric microparticles. Later, the evanescent field of an optical waveguide was used to guide dielectric microparticles [3] and gold Rayleigh particles [4] along a channel waveguide. This opens up the possibility of combining optical trapping with integrated optics to make new and compact devices for manipulation, detection and sorting of micro- and nanoparticles. In order to understand optical trapping it is necessary to calculate the radiation force acting on a particle. Geometrical optics is by far the simplest
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.11.016
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approach [5], but Mie theory is needed for a rigorous calculation of the radiation force due to a plane wave incident on a spherical scatterer [6]. Several groups have developed generalizations of Mie theory which have been used to calculate radiation forces due to arbitrarily shaped beams. The Generalized Lorentz–Mie Theory (GLMT) developed by Gouesbet et al. [7] is one such generalization of Mie theory. Another such generalization is the formalism of Barton et al. [8,9], referred to as Arbitrary Beam Theory (ABT). ABT was used by Almaas and Brevik [10] to calculate forces on a Mie particle in an evanescent field at a prism surface. In this paper we have used ABT to calculate the scattered field and forces on a spherical dielectric particle in the evanescent field of a single mode optical waveguide, as shown in Fig. 1. The calculations are for a polystyrene latex particle in water. Our calculations predict some unexpected behaviour for the particle in the evanescent field of a waveguide. Normally, a particle having a higher refractive index than that of the surrounding medium is attracted to the interface where total internal reflection takes place – in this case the waveguide–cover interface. We illustrate this point for the case of a Rayleigh particle. The radiation force acting on a Rayleigh particle has a component which acts in the direction of the gradient of the electric field, and another component called the scattering force, which is proportional to the scattered intensity. The gradient force on a Rayleigh particle with polarizability a1 is given by [1] a1 f ¼ rhE2 i: ð1Þ 2
The intensity of the incident evanescent field is strongest at the waveguide–cover interface, and dominates in the total intensity ÆE2æ, since the intensity of the scattered field is weak. The gradient force therefore, attracts the particle to the waveguide. Referring to Fig. 1, this means that the gradient force will act in the downward or negative x-direction. However, we shall show that for certain values of radii corresponding to Morphology Dependent Resonance (MDR) peaks appearing in the plots for the force in the x-direction, this force can repel the particle from the waveguide–cover interface. The effect of resonances on radiation pressure has been studied both in theory and experiments [11,12] for plane wave incidence. Our calculations for an incident evanescent field show that the radiation force on the particle in x-direction can, for certain values of radius, be directed in the direction opposite to that suggested by the above example. In the following section we derive expressions for the partial wave expansion coefficients for scattering of a spherical particle in the evanescent field of an optical waveguide, which will allow us to calculate the scattered field and the forces on the particle. To do this we use the expressions for the evanescent field from an optical waveguide with a slab geometry. In Section 3 we give an account of the assumptions and parameters used in our calculations, and discuss the results of our calculations.
2. Theoretical background 2.1. Forces on a Mie particle
x
z
n1
cover
a
n3 h
y x= 0
n2
waveguide (core)
The plane-wave partial wave expansion coefficients Al and Bl of Mie theory are replaced in ABT by the partial wave coefficients Alm and Blm. The radial parts of the E and H field vectors of the incident wave, denoted by the index (i), are therefore as follows:
x= -d
ns
substrate
Fig. 1. Particle in the evanescent field of a waveguide, n3 > n1 and n2 > ns > n1.
EðiÞ r ¼
1 X l E0 X lðl þ 1ÞAlm wl ðk 1 rÞY lm ðh; uÞ; r2 l¼1 m¼l
ð2Þ
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
use the orthogonality properties of the spherical harmonics Ylm: Z b2 Alm ¼ EðiÞ ðb; h; uÞY lm ðh; uÞ dX; E0 lðl þ 1Þwl ðk 1 bÞ X r ð5Þ
x
ϕ
b2 Blm ¼ H 0 lðl þ 1Þwl ðk 1 bÞ
r θ z
Fig. 2. Definition of polar angle h and azimuthal angle u.
1 X l H0 X lðl þ 1ÞBlm wl ðk 1 rÞY lm ðh; uÞ; r2 l¼1 m¼l
ð3Þ where k1 = 2pn1/k, in which k is the wavelength of the incident wave in the medium with refractive index n1. E0 and H0 are the field amplitudes, Ylm(h,u) are the spherical harmonics, the polar angle h and the azimuthal angle u being defined in Fig. 2. The wls are the real Riccati–Bessel functions given by wl ðxÞ ¼ xjl ðxÞ ¼
px1=2 J lþ1=2 ðxÞ; 2
Z
H ðiÞ r ðb;h;uÞY lm ðh;uÞ dX; X
ð6Þ
y
H ðiÞ r ¼
375
ð4Þ
where jl is the spherical Bessel function and Jl + 1/2 is the ordinary Bessel function of the first kind. The electric and magnetic pffiffiffiffiffiffiffi field amplitudes are related through H 0 ¼ =lE0 . The corresponding expressions for the h and u components of E and H are given in [8]. For the scattered fields the partial wave coefficients are denoted as alm and blm in place of Alm and Blm, respectively. The Ricatti–Bessel functions for the scattered field are written as nl, which are expressed in terms of the spherical and ordinary Hankel functions hl and Hl + 1/2, respectively, in place of jl and Jl + 1/2 in Eq. (4). The expansion coefficients Alm and Blm are obtained from the following expressions, which
where dX = sinh dh du. The integration is carried out over the surface of a sphere of radius b; the index (i) refers to the field incident on the particle. The expansions for the field components Er ; Eh ; Eu ; H r ; H h and H u are then substituted in the expression for the radiation force F, which is given in terms of MaxwellÕs stress tensor Tij [8]: I T ij nj ds; ð7Þ F¼ S I h 1 2 n3 Er Er Eh Eh Eu Eu ¼ 2 S i þ H r H r H h H h H u H u ^r þ n23 Er Eh þ H r H h ^h ^ ds; þ n23 Er Eu þ H r H u u ð8Þ I ^ ds: ¼ fr^r þ fh ^h þ fu u ð9Þ S
According to definition Tij is the momentum flux tensor, and therefore fi = Tijnj ds is the ith component of force on the surface element ds, when n is the normal directed outwards from the closed surface S [13]. The x, y and z components of the radiation force can then be written in terms of a nested sum involving binary products of Alm, Blm, alm and blm. The final expressions for the forces Fx and Fz are given in S.I. units as Eqs. (21) and (22) in [10], and these are our basic expressions for calculating radiation forces. In order to calculate Fx and Fz on a particle we need the explicit expression for the evanescent field of a waveguide. In particular, we require the amplitude E0 be used in the expressions for Fx and Fz.
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2.2. Evanescent field of a waveguide
Ez ðx; zÞ ¼
jr H y: x
ð18Þ
We consider an asymmetric slab waveguide with a step index profile as illustrated in Fig. 1. We start with a wave polarized along the ydirection (TE), having amplitude A and angular frequency x. Then the TE evanescent field components in the cover region (x > 0) are [14]:
Note that the prefactor in the TM case is 0 A0 cosðk 2x d=2 þ /0 Þ, in which / is given by the dual of the TE expression in Eq. (13). In spherical polar coordinates the radial components of the E and H fields for TM polarization are:
Ey ðx; zÞ ¼ A cosðk 2x d=2 þ /Þ erx ejkz z ;
0 0 rx jk z z H ðiÞ e sin h sin u; r ¼ A cosðk 2x d=2 þ / Þ e
H x ðx; zÞ ¼
j oEy j k z ðjk z ÞEy ¼ ¼ Ey ; xl oz xl xl
ð10Þ
ð19Þ ð11Þ EðiÞ r ¼
j oEy j jr ðrÞ Ey ¼ Ey ; ð12Þ ¼ H z ðx; zÞ ¼ xl ox xl xl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k 2x ¼ k 22 k 2z is the x component of the wave vector k2 for the guiding region having refractive index n2, and kz is the propagation constant. The decay constantqofffiffiffiffiffiffiffiffiffiffiffiffiffiffi the evanescent field in ffi 2 2 the cover region is r ¼ k z k 1 , while / is obtained by applying the boundary conditions for the fields at the waveguide–cover interface [14] l2 r k 2x d np; n ¼ 0; 1; 2 . . . / ¼ tan1 l1 k 2x 2 ð13Þ On transforming to spherical polar coordinates (r, h, u), we get the radial components of the evanescent TE field which is incident on the particle, and therefore denoted by index (i): rx jk z z EðiÞ e sin h sin u r ¼A cosðk 2x d=2 þ /Þ e k z jr Ey cos h H ðiÞ Ey sin h cos u þ r ¼ xl xl A ¼ cosðk 2x d=2 þ /Þerx ejkz z xl
kz Hy; x
ð20Þ
In order to calculate the partial wave coefficients Alm and Blm, the radial components of the E and H fields are inserted in the expressions for the expansion coefficients (5) and (6). For the TE case this gives: ATE lm ¼
b2 A cosðk 2x d=2 þ /Þ rh e E0 lðl þ 1Þwl ðk 1 bÞ Z sin h sin u erx ejkz z Y lm ðh; uÞ dX;
ð21Þ
X
BTE lm ¼
b2 A cosðk 2x d=2 þ /Þ rh e H 0 lðl þ 1Þwl ðk 1 bÞ xl Z ðk z sin h cos u þ jr cos hÞ erx X
ejkz z Y lm ðh; uÞ dX;
ð22Þ
where we have shifted the origin to the centre of the sphere with the translation
ð15Þ
x ! x þ h;
In a similar manner we start with the TM incident fields, which are:
Ex ðx; zÞ ¼
kz jr H y cos h: H y sin h cos u þ x x
ð14Þ
ðk z sin h cos u þ jr cos hÞ:
H y ðx; zÞ ¼ A0 cosðk 2x d=2 þ /0 Þ erx ejkz z ;
ð16Þ ð17Þ
h being the distance from the waveguide to the centre of the particle, as shown in Fig. 1. The amplitude E0 of the electric vector of the evanescent field at x = 0 is simply Acos(k2xd/ 2 + /), as can be seen from the expression (14). We make this substitution both in the expression TE for ATE lm and in the expression pffiffiffiffiffiffiffi for Blm above, keeping in mind that H 0 ¼ =lE0 . In the cover region ¼ n21 0 . After making the substitutions x ¼ b sin h cos u;
z ¼ b cos h;
ð23Þ
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
the integrals can then be put in a form that lends itself to numerical evaluation [10]. This leads us to the TE partial wave coefficients: rh ATE Q3 ðl; mÞ; ð24Þ lm ¼ a1 ðl; mÞ e BTE lm
a1 ðl; mÞ ¼ pffiffiffiffiffi erh ½k z Q1 ðl; mÞ þ jrQ2 ðl; mÞ; x l ð25Þ
where
1=2 2l þ 1 ðl 1Þ! b2 a1 ðl; mÞ ¼ ; 4p ðl þ mÞ! lðl þ 1Þwl ðk 2 bÞ ð26Þ and Q1,
Q2 and Q3 are the integrals, for the cases even lþm¼ , respectively: odd
Z p=2 cos jm1j 2 Q1 ðl; mÞ ¼ 2pð1Þ dh sin h j sin 0 P ml ðcos hÞ½I jm1j ðrb sin hÞ
ðk z b cos hÞ þ I jmþ1j ðrb sin hÞ; Q2 ðl; mÞ ¼ 2pð1Þ
jmj
Z
ð27Þ
p=2
dh sin h cos h
j sin
cos m ðk z b cos hÞ P l ðcos hÞI jmj ðrb sin hÞ; 0
ð28Þ Q3 ðl; mÞ ¼ 2pjð1Þ
jmj
m rb
Z
ðk z b cos hÞ
p=2
dh sin h 0
cos
j sin
P ml ðcos hÞI jmj ðrb sin hÞ;
ð29Þ are the associated Legendre funcwhere tions, and Ijmj(rb sin h) the modified Bessel functions. The partial wave coefficients for TM polarization can similarly be expressed in terms of the integrals Q1, Q2 and Q3: P ml ðcos hÞ
ATM lm ¼
a1 ðl; mÞ rh pffiffiffiffiffi e ½k z Q1 ðl; mÞ jrQ2 ðl; mÞ; x l ð30Þ
rh BTM Q3 ðl; mÞ; lm ¼ a1 ðl; mÞ e
ð31Þ
where we have inserted H0=A 0 cos(k2xd/2 + / 0 ). Comparing Eq. (24) with (31), and (25) with (30)
377
gives us the following equalities between the partial wave coefficients: TM ATE lm ¼ Blm ;
ð32Þ
TM BTE lm ¼ Alm :
ð33Þ
These equalities are valid for the waveguide geometry shown in Fig. 1, and differ from the corresponding equalities obtained in [10] for a particle in the evanescent field of a totally reflected beam at an interface. The above equalities greatly facilitate computations, as we now only need to perform numerical integrations to find the coefficients Alm and Blm for either TE or TM polarization. Finally, the prefactors A and A 0 appearing in the expressions for the amplitudes E0 and H0 are obtained from the normalizations [14]: P TE ¼
Wk z j Aj2 d TE eff ; 4xl
ð34Þ
P TM ¼
Wk z 2 j A0 j d TM eff ; 4x2
ð35Þ
where PTE and PTM are the guided mode Poynting powers for each polarization, W is the width of the TM waveguide, and d TE are the effective eff and d eff thicknesses of the waveguide which take into account the Goos–Ha¨nchen shift. Once we have found A and A 0 , and thereby E0 and H0, we can calculate the forces Fx and Fz, which may then be plotted against the radius a of the particle, or against the size parameter a = k1a. We then use Stokes Law to find the terminal velocity vz for a particle of radius a F z ¼ 6pgavz ;
ð36Þ
where we make use of the following modified expression for the viscosity g which is valid for a microparticle near a surface [15]: 9 1 3 g g0 1 ða=hÞ þ ða=hÞ 16 8 1 45 1 4 5 ða=hÞ ða=hÞ : ð37Þ 256 16 In this expression g0 is the unmodified viscosity of the solvent in which the particles are suspended, which in our case is water (g0 = 103 Pa s).
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3. Numerical calculations and discussion ABT has been used by Almaas and Brevik [10] to calculate forces on a particle in the evanescent field due to total internal reflection of a plane wave arriving at a glass–fluid interface. In that case, the angle of incidence entirely determines the values of the components of the wave vector k1,2 for a given set of refractive indices. In a waveguide, however, it is the refractive indices of cover, substrate and core, as well as the thickness of the core, that determine the wave vector. Therefore, we have to take as our starting point the value of the effective index neff obtained from the guidance condition, given here for the TE case [14]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! n2eff n2s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k 2x d tan l1 n22 n2eff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! l n2eff n21 ffi ¼ pp; tan1 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ls n22 n2eff 1
l2
ð38Þ
where p = 0,1,. . .. is the mode index, and ls is the permeability of the substrate with index ns. The effective index neff is the index ‘‘seen’’ by the propagating wave and is related to the propagation constant kz for the waveguide through: k z ¼ k 0 neff :
ð39Þ
The decay constant r of the evanescent field ffi is then qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 obtained from the relation r ¼ k z k 1 . In our calculations we have assumed a waveguide with a step index profile, as shown in Fig. 1, with the values of refractive indices ns = 1.51, n2 = 1.52 and n1 = 1.33. These values make our model similar to a waveguide formed by K+ ion-exchange on glass, and for which the cover medium is water [16]. The transverse field amplitude is taken to be constant along the width of the waveguide: this is a reasonable approximation for particle diameters that are comparable to the channel width W. The value of waveguide thickness d used in our calculations is the optimized thickness dopt, i.e. that thickness for which neff optimizes the gradient of the evanescent field [17] when the propagating wave is TE polarized. We used this value of dopt to calculate forces on a particle for TE as well as TM polarization. For the values of refractive indi-
ces specified above, the optimized thickness was found using the expression in [17] to be dopt = 1.9 lm at k = 1.064 lm for a single mode waveguide and TE polarization. The power P in the waveguide was taken to be 100 mW, and the channel width W = 5 lm, while the distance h from the surface of the waveguide to the centre of the sphere was set at h = a. This means that there is no gap between waveguide and particle. The value of the effective index we arrived at by solving Eq. (38) with n1, n2, ns and dopt as above was neff = 1.5116. For all our calculations the value of the sphere of integration b was set at 1.1a. A Matlab program for calculating forces was written and used to generate the plots shown here, which are for a polystyrene microsphere with refractive index n3 = 1.59. As a check, we ensured that our numerical procedure was able to reproduce the plots obtained by Almaas and Brevik [10]. We also compared our calculations based on ABT with the expressions for Rayleigh theory [18]. The plots in Fig. 3 show that there is good agreement, especially for TE polarization, in the predicted order of magnitude of velocities for the range of radii up to 100 nm, i.e. for the range in which Rayleigh theory can be applied [19]. According to Eqs. (7) and (8) the force on the particle is determined by the field distribution on the surface of the particle. We shall therefore examine the contribution of the scattered field to the force at a single point on the particle. It is convenient to choose the point (r = 1.01a, h = p/2, u = p), because at this point the radial component of the integrand, fr, directly determines the vertical force fx on ds, but with opposite sign, since fx ¼ fr sin h cos u þ fh cos h cos u fu sin u ¼ fr
ðh ¼ p=2; u ¼ pÞ:
ð40Þ
Fig. 4 shows a plot of the scattered TE field component jEuj, at the point with coordinates (r = 1.01a, h = p/2,u = p) on the spherical particle, when the axes are defined as in Fig. 2. With increasing radius, the loss for the internal field decreases, the internal field builds up, resonances occur, and the internal field becomes larger than the incident field. As these resonances originate in the electromagnetic modes of a sphere [20], they are referred to
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
379
0.07 TM pol: Mie TE pol: Mie TM pol: Rayleigh TE pol: Rayleigh
Forward velocity (µm/s)
0.06
0.05
0.04
0.03
0.02
0.01
0 0
0.02
0.04 0.06 Particle radius (µm)
0.08
0.1
Fig. 3. Comparison of forward particle velocities using Mie and Rayleigh theory, n1 = 1.33, n2 = 1.52, ns = 1.51, n3 = 1.59.
2
TE Electric Field Amplitude (a.u.)
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
4.5
5 5.5 Particle radius (µm)
6
6.5
Fig. 4. Amplitude of scattered TE field component jEuj at the point (r = 1.01a, h = p/2, u = p) vs. particle radius.
as Morphology Dependent Resonances (MDRs). The MDRs show up as ripples in Fig. 4. Fig. 5 is obtained by inserting the TE field components Eu, Hr and Hh into Eq. (8). As noted earlier, the integrand in Eq. (8) represents the the force fr on an elemental surface ds. Our numerical data show that the maxima and minima in Fig. 4 coincide exactly with those in Fig.
5, indicating that the MDRs are the origin of the oscillatory behaviour of the vertical force fx at the point (1.01a, p/2, p). However, the force fr changes sign around 5 lm, indicating that the sum jEuj2 + jHhj2 increases at the expense of jHrj2 in Eq. (8). This means that fx = fr changes sign from negative to positive, with the consequence that the particle is repelled from the waveguide.
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H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383 0.7 0.6 0.5
r
f (a.u.)
0.4 0.3 0.2 0.1 0 0.1 0.2 4.5
5 5.5 Particle radius (µm)
6
6.5
Fig. 5. Force fr = fx in Eq. (40) for TE polarization vs. particle radius.
This is unexpected according to the arguments presented in Section 1. Furthermore, the maxima and minima in Fig. 7, which shows the dependence of the vertical force Fx on particle radius, also coincide exactly with the maxima and minima in Figs. 4 and 5. Therefore, the MDRs give rise to the oscillatory behaviour in the radiation force through Eq. (8).
Figs. 6 and 7 show that the MDR maxima lead to positive values for Fx for some radii above 5.7 lm for TE polarization, and for some radii above 6.5 lm for TM polarization, thereby predicting a repulsive force between particle and waveguide for those values of radius. However, the sign change in Fig. 5 does not occur at the same place as in Fig. 7. This is because Fig. 5 shows the
4 TM pol TE pol
Vertical force Fx (pN)
2
0
2
4
6
8
10
0
1
2
3 4 Particle radius (µm)
5
6
Fig. 6. Vertical force Fx for k = 1.064 lm vs. particle radius.
7
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381
4 TM po l TE po l
0
x
Vertical force F (pN)
2
2
4
6
8 10 5.5
6
6.5
7
Particle radius (µm)
Fig. 7. Vertical force Fx for k = 1.064 lm vs. particle radius for the range 5.5–7 lm.
30 TM pol TE pol
20
z
Horizontal force F (pN)
25
15
10
5
0 0
1
2
3 4 Particle radius (µm)
5
6
7
Fig. 8. Horizontal force Fz for k = 1.064 lm vs. particle radius.
contribution of the force fx at one single point (r = 1.01a, h = p/2, u = p) to the vertical force Fx (see Fig. 8). Turning now to the plots for velocities we find that the forward velocities of the guided particles shown in Figs. 9 and 10 are of the same order of magnitude reported in the letter by Kawata and Tani [3], which involved polystyrene latex parti-
cles of diameter 1–5 lm in the evanescent field of a waveguide. Figs. 6, 9 and 10 show that optical waveguides can efficiently trap and propel microparticles for radii between 1 and 5.5 lm. Efficient trapping and propulsion of small particles with Cs+ ion-exchanged waveguides has recently been shown in experiments done by our group [21].
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H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383 70
Forward velocity (µm/s)
60
50
40
30
20
10
0 0
1
2
3
4
5
6
7
Particle radius (µm)
Fig. 9. Forward particle velocity for TE polarization for k = 1.064 lm vs. particle radius.
70
Forward velocity (µm/s)
60
50
40
30
20
10
0 0
1
2
3 4 Particle radius (µm)
5
6
7
Fig. 10. Forward particle velocity for TM polarization for k = 1.064 lm vs. particle radius.
For radii larger than around 3.5 lm, closely spaced resonances begin to dominate the velocity curves in Figs. 9 and 10. The velocity maxima increase strongly with radius. However, our numerical data show that the velocity maxima do not coincide exactly with the maxima in the plot of Fx in Fig. 6. For the velocity maxima, the force Fx is negative for the range considered, and the
particle is thus attracted to the waveguide, while being propelled at high speed.
4. Conclusion We have shown that Arbitrary Beam Theory allows us to calculate guiding velocities for spherical
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
particles in the evanescent field of an optical waveguide. The order of magnitude of these velocities calculated by us agrees well with the observed guiding velocities reported so far. Our calculations also show that for values of particle radius corresponding to MDR maxima beyond a certain value of radius the particle is repelled from the waveguide, instead of being attracted to it. References [1] K. Svoboda, S.M. Block, Ann. Rev. Biophys. Biomol. Struct. 23 (1994) 247. [2] S. Kawata, T. Sugiura, Opt. Lett. 17 (1992) 772. [3] S. Kawata, T. Tani, Opt. Lett. 21 (1996) 1768. [4] L.N. Ng, B.J. Luff, M.N. Zervas, J.S. Wilkinson, Opt. Commun. 208 (2002) 117. [5] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156. [6] H.C. van Hulst, Light Scattering by Small Particles, Dover, New York, 1981. [7] G. Gouesbet, B. Maheu, G. Gre´han, J. Opt. Soc. Am. A 5 (1988) 1427.
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[8] J.P. Barton, D.R. Alexander, S.A. Schaub, J. Appl. Phys. 64 (1988) 1632. [9] J.P. Barton, D.R. Alexander, S.A. Schaub, J. Appl. Phys. 66 (1989) 4594. [10] E. Almaas, I. Brevik, J. Opt. Soc. Am. B 12 (1995) 2429. [11] A. Ashkin, J.M. Dziedzic, Phys. Rev. Lett. 38 (1977) 1351. [12] P. Chy´lek, J.T. Kiehl, M.K.W. Ko, Phys. Rev. A 18 (1978) 2229. [13] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960. [14] Donald L. Lee, Electromagnetic Principles of Integrated Optics, Wiley, New York, 1986. [15] L.P. Faucheux, A.J. Libchaber, Phys. Rev. E 49 (1994) 5158. [16] Iraj Najafi, Introduction to Glass Integrated Optics, Arctech House, Boston, 1992. [17] L.N. Ng, B.J. Luff, M.N. Zervas, J.S. Wilkinson, J. Lightwave Technol. 18 (2000) 388. [18] Y. Harada, T. Asakura, Opt. Commun. 124 (1996) 529. [19] A.J. Cox, A.J. DeWeerd, J. Linden, Am. J. Phys. 70 (2002) 620. [20] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983. [21] K. Grujic, O.G. Hellesø, J.S. Wilkinson, J.P. Hole, Opt. Commun. 239 (2004) 227.
Radiation forces on a Mie particle in the evanescent field of an optical waveguide H.Y. Jaising *, O.G. Hellesø Department of Physics, University of Tromsø, Nordlysobservatoriet, 9037 Tromsø, Norway Received 11 December 2003; received in revised form 17 September 2004; accepted 4 November 2004
Abstract We use a generalization of Mie theory – the so-called Arbitrary Beam Theory – to calculate the scattered field and radiation forces on a dielectric spherical particle in the evanescent field of an optical waveguide with a step index profile. The guiding velocities calculated by us agree well with experimental results reported by other groups. Moreover, the occurrence of resonances in the scattered field has the consequence that the particle is repelled from, instead of being attracted to, the waveguide for certain values of particle refractive index and radius. 2004 Elsevier B.V. All rights reserved. PACS: 42.82.E; 42.25.F; 32.80.P; 33.80.P Keywords: Evanescent field; Optical waveguide; Mie scattering; Radiation forces; Morphology dependent resonances
1. Introduction The experiments by Ashkin and others in 1970s in optical trapping and manipulation laid the groundwork for the rapid development of the field of optical trapping and optical tweezers [1]. Since the experiments by AshkinÕs group, several different schemes have been used to trap and manipulate small particles. In the experiment by Kawata and Sugiura [2] the radiation force due to the *
Corresponding author. Tel.: +47 77 64 51 14; fax: + 47 77 64 55 80. E-mail address: [email protected] (H.Y. Jaising).
evanescent field produced by totally reflecting a laser beam at a prism surface was used to propel dielectric microparticles. Later, the evanescent field of an optical waveguide was used to guide dielectric microparticles [3] and gold Rayleigh particles [4] along a channel waveguide. This opens up the possibility of combining optical trapping with integrated optics to make new and compact devices for manipulation, detection and sorting of micro- and nanoparticles. In order to understand optical trapping it is necessary to calculate the radiation force acting on a particle. Geometrical optics is by far the simplest
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.11.016
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approach [5], but Mie theory is needed for a rigorous calculation of the radiation force due to a plane wave incident on a spherical scatterer [6]. Several groups have developed generalizations of Mie theory which have been used to calculate radiation forces due to arbitrarily shaped beams. The Generalized Lorentz–Mie Theory (GLMT) developed by Gouesbet et al. [7] is one such generalization of Mie theory. Another such generalization is the formalism of Barton et al. [8,9], referred to as Arbitrary Beam Theory (ABT). ABT was used by Almaas and Brevik [10] to calculate forces on a Mie particle in an evanescent field at a prism surface. In this paper we have used ABT to calculate the scattered field and forces on a spherical dielectric particle in the evanescent field of a single mode optical waveguide, as shown in Fig. 1. The calculations are for a polystyrene latex particle in water. Our calculations predict some unexpected behaviour for the particle in the evanescent field of a waveguide. Normally, a particle having a higher refractive index than that of the surrounding medium is attracted to the interface where total internal reflection takes place – in this case the waveguide–cover interface. We illustrate this point for the case of a Rayleigh particle. The radiation force acting on a Rayleigh particle has a component which acts in the direction of the gradient of the electric field, and another component called the scattering force, which is proportional to the scattered intensity. The gradient force on a Rayleigh particle with polarizability a1 is given by [1] a1 f ¼ rhE2 i: ð1Þ 2
The intensity of the incident evanescent field is strongest at the waveguide–cover interface, and dominates in the total intensity ÆE2æ, since the intensity of the scattered field is weak. The gradient force therefore, attracts the particle to the waveguide. Referring to Fig. 1, this means that the gradient force will act in the downward or negative x-direction. However, we shall show that for certain values of radii corresponding to Morphology Dependent Resonance (MDR) peaks appearing in the plots for the force in the x-direction, this force can repel the particle from the waveguide–cover interface. The effect of resonances on radiation pressure has been studied both in theory and experiments [11,12] for plane wave incidence. Our calculations for an incident evanescent field show that the radiation force on the particle in x-direction can, for certain values of radius, be directed in the direction opposite to that suggested by the above example. In the following section we derive expressions for the partial wave expansion coefficients for scattering of a spherical particle in the evanescent field of an optical waveguide, which will allow us to calculate the scattered field and the forces on the particle. To do this we use the expressions for the evanescent field from an optical waveguide with a slab geometry. In Section 3 we give an account of the assumptions and parameters used in our calculations, and discuss the results of our calculations.
2. Theoretical background 2.1. Forces on a Mie particle
x
z
n1
cover
a
n3 h
y x= 0
n2
waveguide (core)
The plane-wave partial wave expansion coefficients Al and Bl of Mie theory are replaced in ABT by the partial wave coefficients Alm and Blm. The radial parts of the E and H field vectors of the incident wave, denoted by the index (i), are therefore as follows:
x= -d
ns
substrate
Fig. 1. Particle in the evanescent field of a waveguide, n3 > n1 and n2 > ns > n1.
EðiÞ r ¼
1 X l E0 X lðl þ 1ÞAlm wl ðk 1 rÞY lm ðh; uÞ; r2 l¼1 m¼l
ð2Þ
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
use the orthogonality properties of the spherical harmonics Ylm: Z b2 Alm ¼ EðiÞ ðb; h; uÞY lm ðh; uÞ dX; E0 lðl þ 1Þwl ðk 1 bÞ X r ð5Þ
x
ϕ
b2 Blm ¼ H 0 lðl þ 1Þwl ðk 1 bÞ
r θ z
Fig. 2. Definition of polar angle h and azimuthal angle u.
1 X l H0 X lðl þ 1ÞBlm wl ðk 1 rÞY lm ðh; uÞ; r2 l¼1 m¼l
ð3Þ where k1 = 2pn1/k, in which k is the wavelength of the incident wave in the medium with refractive index n1. E0 and H0 are the field amplitudes, Ylm(h,u) are the spherical harmonics, the polar angle h and the azimuthal angle u being defined in Fig. 2. The wls are the real Riccati–Bessel functions given by wl ðxÞ ¼ xjl ðxÞ ¼
px1=2 J lþ1=2 ðxÞ; 2
Z
H ðiÞ r ðb;h;uÞY lm ðh;uÞ dX; X
ð6Þ
y
H ðiÞ r ¼
375
ð4Þ
where jl is the spherical Bessel function and Jl + 1/2 is the ordinary Bessel function of the first kind. The electric and magnetic pffiffiffiffiffiffiffi field amplitudes are related through H 0 ¼ =lE0 . The corresponding expressions for the h and u components of E and H are given in [8]. For the scattered fields the partial wave coefficients are denoted as alm and blm in place of Alm and Blm, respectively. The Ricatti–Bessel functions for the scattered field are written as nl, which are expressed in terms of the spherical and ordinary Hankel functions hl and Hl + 1/2, respectively, in place of jl and Jl + 1/2 in Eq. (4). The expansion coefficients Alm and Blm are obtained from the following expressions, which
where dX = sinh dh du. The integration is carried out over the surface of a sphere of radius b; the index (i) refers to the field incident on the particle. The expansions for the field components Er ; Eh ; Eu ; H r ; H h and H u are then substituted in the expression for the radiation force F, which is given in terms of MaxwellÕs stress tensor Tij [8]: I T ij nj ds; ð7Þ F¼ S I h 1 2 n3 Er Er Eh Eh Eu Eu ¼ 2 S i þ H r H r H h H h H u H u ^r þ n23 Er Eh þ H r H h ^h ^ ds; þ n23 Er Eu þ H r H u u ð8Þ I ^ ds: ¼ fr^r þ fh ^h þ fu u ð9Þ S
According to definition Tij is the momentum flux tensor, and therefore fi = Tijnj ds is the ith component of force on the surface element ds, when n is the normal directed outwards from the closed surface S [13]. The x, y and z components of the radiation force can then be written in terms of a nested sum involving binary products of Alm, Blm, alm and blm. The final expressions for the forces Fx and Fz are given in S.I. units as Eqs. (21) and (22) in [10], and these are our basic expressions for calculating radiation forces. In order to calculate Fx and Fz on a particle we need the explicit expression for the evanescent field of a waveguide. In particular, we require the amplitude E0 be used in the expressions for Fx and Fz.
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2.2. Evanescent field of a waveguide
Ez ðx; zÞ ¼
jr H y: x
ð18Þ
We consider an asymmetric slab waveguide with a step index profile as illustrated in Fig. 1. We start with a wave polarized along the ydirection (TE), having amplitude A and angular frequency x. Then the TE evanescent field components in the cover region (x > 0) are [14]:
Note that the prefactor in the TM case is 0 A0 cosðk 2x d=2 þ /0 Þ, in which / is given by the dual of the TE expression in Eq. (13). In spherical polar coordinates the radial components of the E and H fields for TM polarization are:
Ey ðx; zÞ ¼ A cosðk 2x d=2 þ /Þ erx ejkz z ;
0 0 rx jk z z H ðiÞ e sin h sin u; r ¼ A cosðk 2x d=2 þ / Þ e
H x ðx; zÞ ¼
j oEy j k z ðjk z ÞEy ¼ ¼ Ey ; xl oz xl xl
ð10Þ
ð19Þ ð11Þ EðiÞ r ¼
j oEy j jr ðrÞ Ey ¼ Ey ; ð12Þ ¼ H z ðx; zÞ ¼ xl ox xl xl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k 2x ¼ k 22 k 2z is the x component of the wave vector k2 for the guiding region having refractive index n2, and kz is the propagation constant. The decay constantqofffiffiffiffiffiffiffiffiffiffiffiffiffiffi the evanescent field in ffi 2 2 the cover region is r ¼ k z k 1 , while / is obtained by applying the boundary conditions for the fields at the waveguide–cover interface [14] l2 r k 2x d np; n ¼ 0; 1; 2 . . . / ¼ tan1 l1 k 2x 2 ð13Þ On transforming to spherical polar coordinates (r, h, u), we get the radial components of the evanescent TE field which is incident on the particle, and therefore denoted by index (i): rx jk z z EðiÞ e sin h sin u r ¼A cosðk 2x d=2 þ /Þ e k z jr Ey cos h H ðiÞ Ey sin h cos u þ r ¼ xl xl A ¼ cosðk 2x d=2 þ /Þerx ejkz z xl
kz Hy; x
ð20Þ
In order to calculate the partial wave coefficients Alm and Blm, the radial components of the E and H fields are inserted in the expressions for the expansion coefficients (5) and (6). For the TE case this gives: ATE lm ¼
b2 A cosðk 2x d=2 þ /Þ rh e E0 lðl þ 1Þwl ðk 1 bÞ Z sin h sin u erx ejkz z Y lm ðh; uÞ dX;
ð21Þ
X
BTE lm ¼
b2 A cosðk 2x d=2 þ /Þ rh e H 0 lðl þ 1Þwl ðk 1 bÞ xl Z ðk z sin h cos u þ jr cos hÞ erx X
ejkz z Y lm ðh; uÞ dX;
ð22Þ
where we have shifted the origin to the centre of the sphere with the translation
ð15Þ
x ! x þ h;
In a similar manner we start with the TM incident fields, which are:
Ex ðx; zÞ ¼
kz jr H y cos h: H y sin h cos u þ x x
ð14Þ
ðk z sin h cos u þ jr cos hÞ:
H y ðx; zÞ ¼ A0 cosðk 2x d=2 þ /0 Þ erx ejkz z ;
ð16Þ ð17Þ
h being the distance from the waveguide to the centre of the particle, as shown in Fig. 1. The amplitude E0 of the electric vector of the evanescent field at x = 0 is simply Acos(k2xd/ 2 + /), as can be seen from the expression (14). We make this substitution both in the expression TE for ATE lm and in the expression pffiffiffiffiffiffiffi for Blm above, keeping in mind that H 0 ¼ =lE0 . In the cover region ¼ n21 0 . After making the substitutions x ¼ b sin h cos u;
z ¼ b cos h;
ð23Þ
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
the integrals can then be put in a form that lends itself to numerical evaluation [10]. This leads us to the TE partial wave coefficients: rh ATE Q3 ðl; mÞ; ð24Þ lm ¼ a1 ðl; mÞ e BTE lm
a1 ðl; mÞ ¼ pffiffiffiffiffi erh ½k z Q1 ðl; mÞ þ jrQ2 ðl; mÞ; x l ð25Þ
where
1=2 2l þ 1 ðl 1Þ! b2 a1 ðl; mÞ ¼ ; 4p ðl þ mÞ! lðl þ 1Þwl ðk 2 bÞ ð26Þ and Q1,
Q2 and Q3 are the integrals, for the cases even lþm¼ , respectively: odd
Z p=2 cos jm1j 2 Q1 ðl; mÞ ¼ 2pð1Þ dh sin h j sin 0 P ml ðcos hÞ½I jm1j ðrb sin hÞ
ðk z b cos hÞ þ I jmþ1j ðrb sin hÞ; Q2 ðl; mÞ ¼ 2pð1Þ
jmj
Z
ð27Þ
p=2
dh sin h cos h
j sin
cos m ðk z b cos hÞ P l ðcos hÞI jmj ðrb sin hÞ; 0
ð28Þ Q3 ðl; mÞ ¼ 2pjð1Þ
jmj
m rb
Z
ðk z b cos hÞ
p=2
dh sin h 0
cos
j sin
P ml ðcos hÞI jmj ðrb sin hÞ;
ð29Þ are the associated Legendre funcwhere tions, and Ijmj(rb sin h) the modified Bessel functions. The partial wave coefficients for TM polarization can similarly be expressed in terms of the integrals Q1, Q2 and Q3: P ml ðcos hÞ
ATM lm ¼
a1 ðl; mÞ rh pffiffiffiffiffi e ½k z Q1 ðl; mÞ jrQ2 ðl; mÞ; x l ð30Þ
rh BTM Q3 ðl; mÞ; lm ¼ a1 ðl; mÞ e
ð31Þ
where we have inserted H0=A 0 cos(k2xd/2 + / 0 ). Comparing Eq. (24) with (31), and (25) with (30)
377
gives us the following equalities between the partial wave coefficients: TM ATE lm ¼ Blm ;
ð32Þ
TM BTE lm ¼ Alm :
ð33Þ
These equalities are valid for the waveguide geometry shown in Fig. 1, and differ from the corresponding equalities obtained in [10] for a particle in the evanescent field of a totally reflected beam at an interface. The above equalities greatly facilitate computations, as we now only need to perform numerical integrations to find the coefficients Alm and Blm for either TE or TM polarization. Finally, the prefactors A and A 0 appearing in the expressions for the amplitudes E0 and H0 are obtained from the normalizations [14]: P TE ¼
Wk z j Aj2 d TE eff ; 4xl
ð34Þ
P TM ¼
Wk z 2 j A0 j d TM eff ; 4x2
ð35Þ
where PTE and PTM are the guided mode Poynting powers for each polarization, W is the width of the TM waveguide, and d TE are the effective eff and d eff thicknesses of the waveguide which take into account the Goos–Ha¨nchen shift. Once we have found A and A 0 , and thereby E0 and H0, we can calculate the forces Fx and Fz, which may then be plotted against the radius a of the particle, or against the size parameter a = k1a. We then use Stokes Law to find the terminal velocity vz for a particle of radius a F z ¼ 6pgavz ;
ð36Þ
where we make use of the following modified expression for the viscosity g which is valid for a microparticle near a surface [15]: 9 1 3 g g0 1 ða=hÞ þ ða=hÞ 16 8 1 45 1 4 5 ða=hÞ ða=hÞ : ð37Þ 256 16 In this expression g0 is the unmodified viscosity of the solvent in which the particles are suspended, which in our case is water (g0 = 103 Pa s).
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3. Numerical calculations and discussion ABT has been used by Almaas and Brevik [10] to calculate forces on a particle in the evanescent field due to total internal reflection of a plane wave arriving at a glass–fluid interface. In that case, the angle of incidence entirely determines the values of the components of the wave vector k1,2 for a given set of refractive indices. In a waveguide, however, it is the refractive indices of cover, substrate and core, as well as the thickness of the core, that determine the wave vector. Therefore, we have to take as our starting point the value of the effective index neff obtained from the guidance condition, given here for the TE case [14]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! n2eff n2s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k 2x d tan l1 n22 n2eff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! l n2eff n21 ffi ¼ pp; tan1 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ls n22 n2eff 1
l2
ð38Þ
where p = 0,1,. . .. is the mode index, and ls is the permeability of the substrate with index ns. The effective index neff is the index ‘‘seen’’ by the propagating wave and is related to the propagation constant kz for the waveguide through: k z ¼ k 0 neff :
ð39Þ
The decay constant r of the evanescent field ffi is then qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 obtained from the relation r ¼ k z k 1 . In our calculations we have assumed a waveguide with a step index profile, as shown in Fig. 1, with the values of refractive indices ns = 1.51, n2 = 1.52 and n1 = 1.33. These values make our model similar to a waveguide formed by K+ ion-exchange on glass, and for which the cover medium is water [16]. The transverse field amplitude is taken to be constant along the width of the waveguide: this is a reasonable approximation for particle diameters that are comparable to the channel width W. The value of waveguide thickness d used in our calculations is the optimized thickness dopt, i.e. that thickness for which neff optimizes the gradient of the evanescent field [17] when the propagating wave is TE polarized. We used this value of dopt to calculate forces on a particle for TE as well as TM polarization. For the values of refractive indi-
ces specified above, the optimized thickness was found using the expression in [17] to be dopt = 1.9 lm at k = 1.064 lm for a single mode waveguide and TE polarization. The power P in the waveguide was taken to be 100 mW, and the channel width W = 5 lm, while the distance h from the surface of the waveguide to the centre of the sphere was set at h = a. This means that there is no gap between waveguide and particle. The value of the effective index we arrived at by solving Eq. (38) with n1, n2, ns and dopt as above was neff = 1.5116. For all our calculations the value of the sphere of integration b was set at 1.1a. A Matlab program for calculating forces was written and used to generate the plots shown here, which are for a polystyrene microsphere with refractive index n3 = 1.59. As a check, we ensured that our numerical procedure was able to reproduce the plots obtained by Almaas and Brevik [10]. We also compared our calculations based on ABT with the expressions for Rayleigh theory [18]. The plots in Fig. 3 show that there is good agreement, especially for TE polarization, in the predicted order of magnitude of velocities for the range of radii up to 100 nm, i.e. for the range in which Rayleigh theory can be applied [19]. According to Eqs. (7) and (8) the force on the particle is determined by the field distribution on the surface of the particle. We shall therefore examine the contribution of the scattered field to the force at a single point on the particle. It is convenient to choose the point (r = 1.01a, h = p/2, u = p), because at this point the radial component of the integrand, fr, directly determines the vertical force fx on ds, but with opposite sign, since fx ¼ fr sin h cos u þ fh cos h cos u fu sin u ¼ fr
ðh ¼ p=2; u ¼ pÞ:
ð40Þ
Fig. 4 shows a plot of the scattered TE field component jEuj, at the point with coordinates (r = 1.01a, h = p/2,u = p) on the spherical particle, when the axes are defined as in Fig. 2. With increasing radius, the loss for the internal field decreases, the internal field builds up, resonances occur, and the internal field becomes larger than the incident field. As these resonances originate in the electromagnetic modes of a sphere [20], they are referred to
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
379
0.07 TM pol: Mie TE pol: Mie TM pol: Rayleigh TE pol: Rayleigh
Forward velocity (µm/s)
0.06
0.05
0.04
0.03
0.02
0.01
0 0
0.02
0.04 0.06 Particle radius (µm)
0.08
0.1
Fig. 3. Comparison of forward particle velocities using Mie and Rayleigh theory, n1 = 1.33, n2 = 1.52, ns = 1.51, n3 = 1.59.
2
TE Electric Field Amplitude (a.u.)
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
4.5
5 5.5 Particle radius (µm)
6
6.5
Fig. 4. Amplitude of scattered TE field component jEuj at the point (r = 1.01a, h = p/2, u = p) vs. particle radius.
as Morphology Dependent Resonances (MDRs). The MDRs show up as ripples in Fig. 4. Fig. 5 is obtained by inserting the TE field components Eu, Hr and Hh into Eq. (8). As noted earlier, the integrand in Eq. (8) represents the the force fr on an elemental surface ds. Our numerical data show that the maxima and minima in Fig. 4 coincide exactly with those in Fig.
5, indicating that the MDRs are the origin of the oscillatory behaviour of the vertical force fx at the point (1.01a, p/2, p). However, the force fr changes sign around 5 lm, indicating that the sum jEuj2 + jHhj2 increases at the expense of jHrj2 in Eq. (8). This means that fx = fr changes sign from negative to positive, with the consequence that the particle is repelled from the waveguide.
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H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383 0.7 0.6 0.5
r
f (a.u.)
0.4 0.3 0.2 0.1 0 0.1 0.2 4.5
5 5.5 Particle radius (µm)
6
6.5
Fig. 5. Force fr = fx in Eq. (40) for TE polarization vs. particle radius.
This is unexpected according to the arguments presented in Section 1. Furthermore, the maxima and minima in Fig. 7, which shows the dependence of the vertical force Fx on particle radius, also coincide exactly with the maxima and minima in Figs. 4 and 5. Therefore, the MDRs give rise to the oscillatory behaviour in the radiation force through Eq. (8).
Figs. 6 and 7 show that the MDR maxima lead to positive values for Fx for some radii above 5.7 lm for TE polarization, and for some radii above 6.5 lm for TM polarization, thereby predicting a repulsive force between particle and waveguide for those values of radius. However, the sign change in Fig. 5 does not occur at the same place as in Fig. 7. This is because Fig. 5 shows the
4 TM pol TE pol
Vertical force Fx (pN)
2
0
2
4
6
8
10
0
1
2
3 4 Particle radius (µm)
5
6
Fig. 6. Vertical force Fx for k = 1.064 lm vs. particle radius.
7
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
381
4 TM po l TE po l
0
x
Vertical force F (pN)
2
2
4
6
8 10 5.5
6
6.5
7
Particle radius (µm)
Fig. 7. Vertical force Fx for k = 1.064 lm vs. particle radius for the range 5.5–7 lm.
30 TM pol TE pol
20
z
Horizontal force F (pN)
25
15
10
5
0 0
1
2
3 4 Particle radius (µm)
5
6
7
Fig. 8. Horizontal force Fz for k = 1.064 lm vs. particle radius.
contribution of the force fx at one single point (r = 1.01a, h = p/2, u = p) to the vertical force Fx (see Fig. 8). Turning now to the plots for velocities we find that the forward velocities of the guided particles shown in Figs. 9 and 10 are of the same order of magnitude reported in the letter by Kawata and Tani [3], which involved polystyrene latex parti-
cles of diameter 1–5 lm in the evanescent field of a waveguide. Figs. 6, 9 and 10 show that optical waveguides can efficiently trap and propel microparticles for radii between 1 and 5.5 lm. Efficient trapping and propulsion of small particles with Cs+ ion-exchanged waveguides has recently been shown in experiments done by our group [21].
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H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383 70
Forward velocity (µm/s)
60
50
40
30
20
10
0 0
1
2
3
4
5
6
7
Particle radius (µm)
Fig. 9. Forward particle velocity for TE polarization for k = 1.064 lm vs. particle radius.
70
Forward velocity (µm/s)
60
50
40
30
20
10
0 0
1
2
3 4 Particle radius (µm)
5
6
7
Fig. 10. Forward particle velocity for TM polarization for k = 1.064 lm vs. particle radius.
For radii larger than around 3.5 lm, closely spaced resonances begin to dominate the velocity curves in Figs. 9 and 10. The velocity maxima increase strongly with radius. However, our numerical data show that the velocity maxima do not coincide exactly with the maxima in the plot of Fx in Fig. 6. For the velocity maxima, the force Fx is negative for the range considered, and the
particle is thus attracted to the waveguide, while being propelled at high speed.
4. Conclusion We have shown that Arbitrary Beam Theory allows us to calculate guiding velocities for spherical
H.Y. Jaising, O.G. Hellesø / Optics Communications 246 (2005) 373–383
particles in the evanescent field of an optical waveguide. The order of magnitude of these velocities calculated by us agrees well with the observed guiding velocities reported so far. Our calculations also show that for values of particle radius corresponding to MDR maxima beyond a certain value of radius the particle is repelled from the waveguide, instead of being attracted to it. References [1] K. Svoboda, S.M. Block, Ann. Rev. Biophys. Biomol. Struct. 23 (1994) 247. [2] S. Kawata, T. Sugiura, Opt. Lett. 17 (1992) 772. [3] S. Kawata, T. Tani, Opt. Lett. 21 (1996) 1768. [4] L.N. Ng, B.J. Luff, M.N. Zervas, J.S. Wilkinson, Opt. Commun. 208 (2002) 117. [5] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156. [6] H.C. van Hulst, Light Scattering by Small Particles, Dover, New York, 1981. [7] G. Gouesbet, B. Maheu, G. Gre´han, J. Opt. Soc. Am. A 5 (1988) 1427.
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