corresponding electric field is given by (k0 = 2Ï/λ0 is the vacuum wavenumber). Eport(Ï, z) ... the interface between the glass slide and the sample chamber.
Mie Ressonance Effects in Optical Tweezers R. S. Dutra, N. B. Viana, P. A. Maia Neto and H. M. Nussenzveig Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brazil and LPO-COPEA, Instituto de Ciˆencias Biom´edicas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21941-590, Brazil
Abstract We extend the MDSA (Mie-Debye-Spherical Aberration) theory of trapping forces in optical tweezers, previously developed for circularly polarized trapping beams, to linear polarization. Although it does not significantly affect the trap stiffness, linear polarization may introduce a strong axial asymmetry of the optical forces near the edge of a trapped microsphere, arising from Mie resonance effects.
1
Introduction
Optical tweezers are single-beam optical traps of dielectric microparticles [1]. Applications range from cell biology to the development of micromachines [2]. In most quantitative applications, dielectric microspheres are employed as handles, and forces in the piconewton range are measured against the calibrated optical force. The feasibility of an absolute calibration of optical tweezers was recently demonstrated [3] [4], by comparing experimental results for the transverse stiffness with the Mie-Debye (MD) theory [5] taking into account the spherical aberration (SA) introduced by refraction at the interface between the slide and the sample solution. Good agreement with the Mie-Debye-Spherical Aberration (MDSA) theory was found, with all relevant parameters independently measured (no fitting), for a large range of sphere radii a. The MDSA theory presented in Ref. [4] assumes the trapping laser beam to be circularly polarized, since this is the simplest situation from a theoretical viewpoint. However, in most applications the trapping beam is linearly polarized. Rohrbach [6] treated linear polarization employing the RayleighGans approximation to Mie scattering. The results are not applicable to the geometrical optics size range [4], which is of particular interest in connection with polarization and Mie resonance effects. Numerical results are presented in Sec. 3. Polarization effects are relatively small as far as the transverse stiffness is concerned, but they become more important for a lateral beam displacement of the order of a. Recently, Fontes et al. [7] showed that Mie resonances can be selectively excited by manipulating the angle φ. As an application of our theory, we show that the resonant enhancement of the transverse optical force takes place when the sphere is displaced off-axis by a distance slightly larger than a. This agrees with the interpretation of Mie resonances as arising from tunneling of light rays propagating outside the sphere near its edge [8]. We demonstrate the connection between above-edge rays and the resonant force enhancement by analyzing the effect of individual multipole contributions.
2
Optical force
We assume that the trapping laser beam at the entrance port of the infinity-corrected objective is Gaussian with a plane wavefront and waist w. We take the x axis along the direction of linear polarization. The corresponding electric field is given by (k0 = 2π/λ0 is the vacuum wavenumber) 2
Eport (ρ, z) = Ep eik0 z e−ρ
/w2
x ˆ.
(1)
After the objective, the paraxial approximation is no longer applicable to the strongly focused beam. We assume that the objective is corrected for spherical aberration when the beam is focused at the interface between the glass slide and the sample chamber. The electric field in the glass medium, of index n, may be written as a Debye-type superposition of plane waves [9] with wavevectors k(θ, ϕ)
covering a conical angular sector with |k(θ, ϕ)| = nk0 . The angular aperture θ0 of the angular sector is determined by the objective numerical aperture (NA): sin θ0 = NA/n. We assume that the objective satisfies the Abbe sine condition: sin θ = ρ/f, where f is the focal length. The parameter γ = f /w determines the fraction of available power that fills the objective aperture. Each plane-wave component is linearly polarized along the direction of the unit vector xˆ0 (θ, ϕ), which is obtained from x ˆ by rotation with Euler angles (ϕ, θ, −ϕ): Z
Z
2π
Eglass (r) = E0
θ0
dϕ 0
√ 2 2 dθ sin θ cos θe−γ sin θ e−ik·ra eik·r xˆ0 (θ, ϕ).
(2)
0
The field amplitude E0 is written in terms of the objective transmission coefficient Tobj : E0 = −i
nf Tobj Ep . λ0
(3)
For simplicity, we have assumed that the objective transmission loss is uniform. Radial dependence of the objective transmittance in the infrared can be taken into account by the introduction of an effective beam waist [10]. The trapped microsphere (refractive index n2 ) is usually within a sample chamber filled with water (index n1 ) over the glass slide (index n). The focus would be at position ra if there were no refractive index mismatch between the sample fluid and the glass slide. In this case, ra and its conjugate at infinity form an aplanatic pair. However, the sample (index n1 ) is usually a water solution, and refraction at the glass-water interface introduces spherical aberration [11]. Each plane wave component in Eq. (2) is refracted at the interface at z = −h. Wavevectors in the sample region are k1 (θ1 , ϕ1 ), with |k1 | = n1 k0 , ϕ1 = ϕ and sin θ1 = sin θ/N (N = n1 /n is the relative refractive index). Each plane wave amplitude is multiplied by the Fresnel coefficient 1 T (θ) =
2 cos θ . cos θ + N cos θ1
(4)
We must take into account propagation inside the glass slide up to z = −h when writing the field in the sample region: Z Ein (r) = E0
Z
2π
θ0
dϕ 0
√ 2 2 dθ sin θ cos θe−γ sin θ T (θ)e−i(kz −k1 z )h e−ik·ra eik1 ·r xˆ0 (θ1 , ϕ1 ).
(5)
0
When the numerical aperture is larger than n1 , part of the angular spectrum is totally reflected at the interface. We neglect contributions from evanescent waves assuming that the microsphere is a few wavelengths away from the glass slide. Hence, we need to replace θ0 by arcsin(N ) in (5) if NA > n1 . Since the relative refractive index N is close to one, we may neglect multiple reflections between the microsphere and the glass slide (reverberation), provided that they are not too close. Thus, we solve the Mie boundary conditions for the scattered field Es with Ein as the incident field. We take a coordinate system with origin at the sphere center. Hence h represents the distance between the center and the glasswater plane interface. The optical force is computed from the Maxwell stress tensor. Momentum conservation allows us to take a spherical surface S(r) at infinity: " # Z r 2 2 F = lim − dΩ r (εE + µ0 H ) , (6) r→∞ 2 S(r) where E = Ein + Es is the total field. We normalize the force in the usual way, defining the dimensionless efficiency factor Q=
F , n1 P/c
(7)
1 We neglect the difference between the Fresnel transmission coefficients for transverse electric and transverse magnetic polarizations, thereby assuming that the angle between the direction of linear polarization and the plane of incidence is conserved. The same assumption is inherent in the construction of the electromagnetic Debye-type representation in Ref. [9].
where P is the power at the sample region and c is the speed of light in vacuum. P is a fraction of 2 2 the total laser power Pport available at the entrance port of the objective: P = A Tobj Pport , with Tobj representing the intrinsic transmission loss of the objective. The loss at the objective entrance aperture (filling factor) is contained in the coefficient Z A = 16 0
q S
2
dξ ξ exp (−2ξ ) ³q
(1 − 1−
ξ2 γ2
ξ2 2 γ 2 )(N
+
−
ξ2 γ2 )
q N2 −
ξ2 γ2
´2 ,
(8)
which also takes into account the Fresnel coefficients at the glass-water interface. Since the relative refractive index N is close to one, the interface effect is small. In most situations, A ≈ 1−exp(−2γ 2 sin2 θ0 ) (value for N = 1), which represents the fraction of the total laser power that fills the entrance aperture. The efficiency factor Q is not modified by an uniform objective transmission loss, since both 2 the optical force and the power P are reduced by the same factor Tobj in Eq. (7). On the other hand, non-uniform losses might modify the trapping efficiency, because the contributions of different plane-wave components depend strongly on the corresponding angles with respect to z axis. For the setup analyzed in Ref. [10], the objective transmittance radial variation was successfully modeled by a Gaussian function. This effect is easily incorporated into MDSA theory by replacing the trapping beam waist by an effective (reduced) waist, resulting in better agreement with the experimental results [4] Q is obtained as a sum of two separate contributions: Q = Qe + Qs .
(9)
Qe represents the rate of momentum removal from the incident beam, which is closely related to energy removal (extinction). It contains the crossed terms of the form Ein Es corresponding to interference between the scattered and incident fields, whereas Qs results from the quadratic terms of the form Es2 . Part of the removed momentum is carried away by the scattered field, at the rate −Qs , and the rest is transferred to the microsphere resulting in optical force as given by (9). For presenting of the results, it is more intuitive to take the origin at the some fixed point along the symmetry axis of the optical system. We choose the paraxial focal point as the new origin. We express the position of the sphere center with respect to the paraxial focus in cylindrical coordinates: r = (ρ, φ, z). Hence, the length ρ represents the distance between the beam symmetry axis and the sphere center, as shown in Figure 1.
Figure 1: Schematic representation of the sphere (radius a) in the water of the sample chamber. Paraxial rays intersect at the paraxial focus F. Non-paraxial rays (not shown) intersect the symmetry z-axis in the interval between F and the glass-water interface. The position of the sphere is measured with respect to F (origin of the coordinate axis). The x direction corresponds to the polarization direction of the trapping beam at the entrance port of the objective. Hence φ represents the angle between the sphere radial displacement from the axis and the polarization direction.
In Fig. 2, we show how to implement this change of coordinate system. We denote by L the distance between the paraxial focal plane and the glass-water interface. Hence the sphere height h in (5) is written as h = L + z. From the law of refraction we find that the point ra is at a height L/N with respect to the interface. The figure shows that za = L/N − L − z. We insert this equation into (5) when deriving the multipole expansion of the incident field.
Figure 2: Length parameters characterizing the optical trap. The directions of propagation in the glass medium intersect at ra .
3
Numerical Results
In many quantitative applications of optical tweezers, the transverse trap stiffness κ⊥ is the variable of interest. It has recently been shown [3][4] that an absolute calibration of κ⊥ is possible when all relevant parameters (beam waist at the objective entrance port, power P at the sample region, etc) are independently measured. To determine the effect of polarization on the transverse stiffness, we first derive transverse component force κ⊥ = −
n1 P c
µ
∂Qρ ∂ρ
¶ (10) ρ=0,z=zeq
where zeq is the most stable [4] equilibrium position of the microsphere. We take parameters corresponding to one of the experiments reported in Ref. [4]: laser wavelength λ0 = 1.064µm, beam waist radius w = 4.2 mm at the objective entrance port (radius 3.5 mm) and angular aperture in the glass medium Θ0 = 61.9o (corresponding to an effective numerical aperture NAeff = n1 = 1.332). For the refractive indices, we take n1 = 1.332 (water), n2 = 1.576 (polystyrene microsphere) and n = 1.51 (glass). In order to simulate the experimental procedure [4], we determine the distance L between paraxial focus and interface as follows. We first calculate the distance Lc such that the equilibrium point corresponds to h = a (microsphere attached to the interface, see Fig. 2). Then, we take L = Lc + N d, where d is the objective upward displacement (d = 3 µm in the numerical example presented below). In Figure 3 we plot κ⊥ /P as a function of sphere radius a for linear polarizations along (φ = 0, solid line) and perpendicular (φ = π/2, dashed line) to the direction of (infinitesimal) transverse displacement near the equilibrium point on the z-axis. Experimental values (with error bars) from Refs. [3] and [4], corresponding to φ = 0, are also shown in the figure. In the geometrical optics (GO) limit (a À λ1 = λ0 /n1 ), the transverse stiffness is virtually independent of polarization. From scaling, the stiffness decays as κ⊥ = C/a after averaging over interference oscillations [5] [12], with C = 1.07 pN/mW for φ = 0 (and C = 1.09 pN/mW for φ = π/2) in our numerical example, in good agreement with an independent GO calculation. In the intermediate range a ∼ λ1 , the crossover between Rayleigh (κ⊥ ∼ a3 ) and GO (κ⊥ ∼ 1/a) regimes is marked by a peak, whose position is (slightly) polarization-dependent as shown in Fig. 3. The peak position and height are very sensitive to additional perturbations. In Refs. [3] and [4], MDSA theory with circular polarization was compared to the experimental data for linear polarization with φ = 0 shown in Fig. 3. The figure shows that the effect of polarization is relatively small as far as stiffness is concerned. The overall agreement is not significantly improved by taking the correct polarization
Figure 3: Transverse trap stiffness (divided by the local power) as a function of sphere radius for linear polarization with φ = 0 (solid line) and φ = π/2 (dashed line). Experimental points with φ = 0 from Ref. [4] are represented with their error bars. The geometrical optics values are virtually independent of polarization (dotted line). Inset: transverse force −Qρ in the paraxial focal plane as a function of the lateral displacement ρ for (a) a = 0.26 µm and (b) a = 1.0 µm. The values for circular polarization are given by the average of the results for the two orthogonal linear polarizations. Laser wavelength λ0 = 1.064 µm. into account, indicating that additional effects, such as objective aberrations in the infrared, need to be included when analyzing the peak region. This is currently being investigated. In the inset to Figure 3, we plot the transverse force −Qρ as a function of the radial coordinate ρ in the paraxial focal plane for (a) a = 0.26 µm and (b) a = 1.0 µm. In both cases, the maximum transverse force is larger along the direction perpendicular to the input polarization (φ = π/2), specially for the smaller sphere. For larger spheres, the asymmetry of the optical force field becomes more pronounced for distances2 ρ > ∼ a. As discussed below, such asymmetry is closely connected with Mie resonances. We take parameters corresponding to the experiment reported in Ref. [7], with a = 4.5 µm, NA=1.25 and beam waist radius w = 5 mm. We select the wavelength λ0 = 0.789 µm, such that the corresponding size parameter β = 2πn1 a/λ0 = 47.7 is very close to the real part of a pole of the electric multipole Mie scattering matrix SjE = 1 − 2aj with j = 51. This corresponds to a Transverse Magnetic (TM) Mie resonance [13]. As shown in Fig. 4, the transverse force is nearly isotropic up to ρ/a ∼ 0.5. From this point on, the force along the direction of polarization (φ = 0) becomes gradually larger. Geometrical optics (GO) is an excellent approximation in this example up to ρ ∼ a. Beyond this point, the proportion of light rays within the beam focal region passing outside of the sphere becomes increasingly larger, and then the GO results decrease to zero, as shown by the dotted (φ = 0) and dash-dotted (φ = π/2) lines, since those above-edge rays do not contribute to the optical force in the GO approximation. In the wave-optical MDSA theory, the sums over light rays are replaced by sums over angular momenta j. In the short-wavelength (semiclassical) limit (λ1 ¿ a), the localization principle [8] provides a direct connection between the two pictures, through the correspondence j + 1/2 = k1 b where b is the impact parameter of a given ray (distance of closest approach). Above-edge rays correspond to j + 1/2 > β = k1 a. They are responsible for the resonance effects, giving rise to considerable deviation from the GO result for ρ > ∼ a (Fig. 4). The resonance effect introduces a strong cylindrical asymmetry: the transverse force along the polarization direction (φ = 0) becomes much larger, oscillating well above the GO curve. On the other 2 The optical force exerted by the trapping beam holding the microsphere is not easily measurable for ρ ∼ a. When a lateral external force (for instance the drag force exerted by a moving fluid) is applied, the sphere is also displaced along the z-axis to a new equilibrium position, and it is lost from the trap as ρ approaches a [5]. However, the optical force exerted by an additional weak ‘perturbing’ beam has been measured for ρ ∼ a, with ρ representing the distance between the perturbing beam axis and the sphere, which is held by a stronger trapping beam [7].
hand, the force along the orthogonal direction (φ = π/2) oscillates around the GO value and is suppressed at some positions. In fact, coupling with the selected TM resonance is favoured by displacing the sphere along the polarization direction, so as to place the sphere edge in a region where the electric field is approximately radial. For a TE resonance (not shown), the situation is reversed, and the force becomes larger along the orthogonal direction in the range 1.07 < ρ/a < 1.21.
Figure 4: Transverse force −Qρ in the paraxial focal plane as a function of the lateral displacement ρ (in units of the sphere radius) for a = 4.5 µm and λ0 = 0.789 µm, with φ = 0 (solid line) and φ = π/2 (dashed line). The geometrical optics results for these two cases (dotted and dash-dotted lines, respectively) are also shown. In order to highlight the role of above-edge rays in the resonance effect, we plot in Fig. 5 the Pj (j) (j) growth curve [14] of the partial sum −Qρ = j 0 =1 (...) (with Qρ = limj→∞ Qρ ) as a function of its upper limit j, for ρ/a = 1.14 and λ0 = 0.789 µm (circles). We also show the values for the nonresonant wavelength λ0 = 0.785 µm (crosses). Since these values are very close, so are the below-edge contributions (j +1/2 < β = 47.7). However, the above-edge j = 50 and j = 51 terms provide large contributions in the resonant case only, thus resulting in a much larger force, as expected since we selected a j = 51 resonance. The key contribution of above-edge rays can be simply understood by describing Mie scattering in terms of an effective potential [8]. In this framework, above-edge rays tunnel through the centrifugal barrier and excite whispering-gallery resonant modes inside the sphere.
(j)
Figure 5: Partial sum −Qρ up to angular momentum j as a function of j, for ρ/a = 1.14.
4
Conclusion
We have presented a detailed theoretical derivation of the optical force exerted by a tightly focused beam, assumed to be linearly polarized at the entrance port of the objective. This is an important extension of MDSA theory, previously derived in the theoretically simpler case of circular polarization, since most setups employ linearly-polarized laser beams. The direction of linear polarization breaks the cylindrical symmetry of the optical force field (that holds for circular polarization). The beam polarization has a relatively small effect on the transverse trap stiffness. For large spheres (a > λ0 ), stiffness is virtually independent of polarization and can be obtained safely (with much less labour) from geometrical optics. In this range, the optical force field is approximately cylindrically symmetric in the neighbourhood of the axis. The cylindrical asymmetry becomes more apparent at distances ρ > ∼ a from the axis, especially if a (Mie) resonant wavelength is selected. Around resonance, above-edge rays provide the major contribution and deviations from geometrical optics become very large. We thank Alexander Mazolli for providing routines for Mie scattering numerical calculations. This work was supported by the Brazilian agencies Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq), Projeto Milˆenio de Nanotecnologia, Projeto Milˆenio de Informa¸c˜ao Quˆantica, Funda¸c˜ao de Amparo `a Pesquisa do Rio de Janeiro (FAPERJ) and Funda¸c˜ao Universit´aria Jos´e Bonif´acio (FUJB). R.S.D. also acknowledges support by Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES).
References [1] Ashkin A, Dziedzic J M, Bjorkholm J E and Chu S 1986 Opt. Lett. 11, 288-90 [2] Grier D G 2003 Nature 424 810-6 Neuman K C and Block S M 2004 Rev. Scient. Instr. 75 2787-809 [3] Viana N B, Rocha M S, Mesquita O N, Mazolli A, Maia Neto P A and Nussenzveig H M, 2006 Appl. Phys. Lett. 88 131110 [4] Viana N B, Rocha M S, Mesquita O N, Mazolli A, Maia Neto P A and Nussenzveig H M 2006 Towards absolute calibration of optical tweezers Preprint physics/0612252 [5] Mazolli A, Maia Neto P A and Nussenzveig H M 2003 Proc. R. Soc. London A 459 3021-41 [6] Rohrbach A 2005 Phys. Rev. Lett. 95 168102 [7] Fontes A, Neves A A R, Moreira W L, de Thomaz A A, Barbosa L C and Cesar C L 2005 Appl. Phys. Lett. 87 221109 [8] Nussenzveig H M 1992 Diffraction Effects in Semiclassical Scattering (Cambridge: University Press Cambridge) [9] Richards B and Wolf E 1959 Proc. R. Soc. Lond. A 253 358-79 [10] Viana N B, Rocha M S, Mesquita O N, Mazolli A and Maia Neto P A 2006 Appl. Optics 45 4263-9 [11] Tor¨ok P, Varga P, Laczik Z and Booker G R 1995 J. Opt. Soc. Am. A 12 325-32 [12] Maia Neto P A and Nussenzveig H M 2000 Europhys. Lett. 50 702-8. [13] Guimar˜aes L G and Nussenzveig H M 1994 J. Mod. Opt. 41 625-47 [14] Bryant H C and Cox A J 1966 J. Opt. Soc. Am. 56 1529-32
