## Abstract

A double tweezers setup was employed to perform ultra sensitive force measurements and to obtain the full optical force curve as a function of radial position and wavelength. The light polarization was used to select either the transverse electric (TE), or transverse magnetic (TM), or both, modes excitation. Analytical solution for optical trapping force on a spherical dielectric particle for an arbitrary positioned focused beam is presented in a generalized Lorenz-Mie diffraction theory. The theoretical prediction of the theory agrees well with the experimental results. The algorithm presented here can be easily extended to other beam geometries and scattering particles.

©2006 Optical Society of America

## 1. Introduction

One important contribution of the optical tweezers technique is the ability to extract the missing mechanical measurements in the world of microorganisms and cells that could be correlated to biochemical information. The importance of understanding the optical forces in dielectric beads under different incident beam conditions comes from the fact that they are the natural transducer for force measurements with the optical tweezers technique [1]. Geometrical optics has been used when the microsphere dimensions are much greater than the light wavelength and Rayleigh scattering theory for the opposite. However, most of the trapping experiments are performed with particles at the intermediate size regime, with diameters up to ten wavelengths where Mie resonances appear strongly, and uses nonparaxial very high numerical aperture beam with diffraction fringes. These conditions demand the use of the Generalized Lorentz-Mie Theory (GMLT). The main difficulty with GLMT optical force models has been the partial wave decomposition of an arbitrarily located high numerical aperture incident beam with any polarization. Approximations and circular polarization beams have been used to overcome the GMLT mathematical difficulties [2–9]. The linear polarization is important for the selective coupling of the light to the TE, the TM or both TE and TM microsphere modes, as we have shown in previous work [10, 11]. We also recently showed that an exact partial beam decomposition of a general optical beam at an arbitrary location and polarization is possible [12, 13].

The best experimental validation for the theoretical descriptions would be the one that observe the whole optical force curve as a function of position, wavelength and other beam parameters instead of punctual numerical force evaluation. In this paper we use the double optical tweezers setup described before [10], to obtain the dielectric microsphere optical force curve as a function of radial beam position and polarization, and compared the results with the calculations performed with the beam shape coefficients (BSC) obtained from the previous paper [12], shortly described in section 2 of this paper. Our results show how careful one has to be to validate theoretical models with experimental measurements. Morphology-Dependent Resonances (MDR) can change the force values by more than 30–50% depending on the wavelength/particle radius ratio, beam position and polarization. Azimuthal symmetry in the horizontal plane, usually assumed, is not valid because the beam polarization breaks this symmetry. Finally, diffraction fringes appear when the objective is overfilled with observable effects, especially when the beam is positioned at the particle's edges.

## 2. Theory

To determine the optical forces rigorously on a dielectric sphere of an arbitrary size, the electromagnetic (EM) equations must be solved for the appropriate boundary conditions. Instead of using the standard 5th order corrected Davis-Barton beam approximation of the highly focused beam [14, 15], we used the Angular Spectrum Representation [16]. For the purpose of comparison with other results, we used the following convention for the partial wave decomposition:

$${\mathbf{H}}_{\mathit{inc}}=\frac{{E}_{0}}{Z}\sum _{n,m}\left[{G}_{n\phantom{\rule{.2em}{0ex}}m}^{\mathit{TM}}{j}_{n}\left(kr\right){\mathbf{X}}_{n\phantom{\rule{.2em}{0ex}}m}(\theta ,\varphi )-\frac{i}{k}{G}_{n\phantom{\rule{.2em}{0ex}}m}^{\mathit{TE}}\nabla \times {j}_{n}\left(kr\right){\mathbf{X}}_{n\phantom{\rule{.2em}{0ex}}m}(\theta ,\varphi )\right]$$

Where
${\mathbf{X}}_{n\phantom{\rule{.2em}{0ex}}m}(\theta ,\varphi )=\frac{\mathbf{L}Y}{\sqrt{n\left(n+1\right)}}$
, is the vector spherical harmonic, *j*_{n}
(*kr*) are spherical Bessel functions and
$Z=\sqrt{\frac{\mu}{\epsilon}}$
is the medium impedance. The BSCs obtained from our previous work [12] for the case of an incident linear *x*-polarized truncated TEM_{00} Gaussian beam focused at the (*ρ*_{o}
, *ϕ*_{o}
, *z*_{o}
) cylindrical coordinates point, is given by the expression:

$$\underset{0}{\overset{{\alpha}_{max}}{\int}}d\alpha \sqrt{\mathrm{cos}\alpha}\mathrm{exp}\left({-f}^{2}{\mathrm{sin}}^{2}\frac{\alpha}{{\omega}_{a}^{2}}\right)\mathrm{exp}\left(-ik{z}_{o}\mathrm{cos}\alpha \right)\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\{\left[{m}^{2}\frac{{J}_{m}\left(k{\rho}_{o}\mathrm{sin}\alpha \right)}{k{\rho}_{o}\mathrm{sin}\alpha}{P}_{n}^{m}\left(\mathrm{cos}\alpha \right)-{\mathrm{sin}}^{2}\alpha {J}_{m}^{\prime}\left(k{\rho}_{o}\mathrm{sin}\alpha \right){P}_{n}^{\prime m}\left(\mathrm{cos}\alpha \right)\right]\mathrm{cos}{\varphi}_{o}$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+im\left[m{J}_{m}^{\prime}\left(k{\rho}_{o}\mathrm{sin}\alpha \right){P}_{n}^{m}\left(\mathrm{cos}\alpha \right)-{\mathrm{sin}}^{2}\alpha \frac{{J}_{m}\left(k{\rho}_{o}\mathrm{sin}\alpha \right)}{k{\rho}_{o}\mathrm{sin}\alpha}{P}_{n}^{\prime m}\left(\mathrm{cos}\alpha \right)\right]\mathrm{sin}{\varphi}_{o}\}$$

These are exact BSCs for the incident optical beam EM fields with respect to an arbitrary origin, polarization and amplitude in terms of experimental parameters and not an approximation. Because we obtained an analytical expression for several integrals and only one numerical integration has to be performed, these BSCs are also numerically efficient, requiring under one minute CPU time (on a Pentium 4 and Mathematica v5.2) to evaluate all the ${G}_{n\mathit{,}m}^{\mathit{\text{TM}}\mathit{,}\mathit{\text{TE}}}$
coefficients for *n* up to 30 and *-n*≤*m*≤*n* for each beam position. The software is made available upon request to the author. These BSCs can than be used to calculate radial force by adjusting Gouesbet *et al* expression [2–4] with our convention from [17]:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}[\left({a}_{n+1}+{a}_{n}^{*}-2{a}_{n+1}{a}_{n}^{*}\right){G}_{n+1,-\left(m+1\right)}^{\mathit{TM}}{{G}_{n,-m}^{\mathit{TM}}}^{*}+\left({a}_{n}+{a}_{n+1}^{*}-2{a}_{n}{a}_{n+1}^{*}\right){G}_{\mathit{nm}}^{\mathit{TM}}{{G}_{n+1,m+1}^{\mathit{TM}}}^{*}+$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\left({b}_{n+1}+{b}_{n}^{*}-2{b}_{n+1}{b}_{n}^{*}\right){G}_{n+1,-\left(m+1\right)}^{\mathit{TE}}{{G}_{n,-m}^{\mathit{TE}}}^{*}+\left({b}_{n}+{b}_{n+1}^{*}-2{b}_{n}{b}_{n+1}^{*}\right){G}_{\mathit{nm}}^{\mathit{TE}}{{G}_{n+1,m+1}^{\mathit{TE}}}^{*}]$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}-\frac{1}{n}\sum _{m=-n}^{n}\sqrt{\left(n+m+2\right)\left(n+m+1\right)}\sqrt{\left(n-m\right)\left(n+m+1\right)}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\left[\left({a}_{n}+{b}_{n}^{*}-2{a}_{n}{b}_{n}^{*}\right){G}_{\mathit{nm}}^{\mathit{TM}}{{G}_{n,m+1}^{\mathit{TE}}}^{*}-\left({b}_{n}+{a}_{n}^{*}-2{b}_{n}{a}_{n}^{*}\right){G}_{\mathit{nm}}^{\mathit{TE}}{{G}_{n,m+1}^{\mathit{TM}}}^{*}\right]\}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}$$

Where *a*_{n}
and *b*_{n}
are the usual Lorenz-Mie coefficients that presents the observed Mie resonances.

## 3. Materials and methods

We performed the force spectroscopy experiment trapping a 3, 6 and 9 µm polystyrene microsphere diluted in water with a double optical tweezers setup. One beam from a Nd:YAG continuous-wave laser (cw), denoted as a trapping beam, is used to keep the particle trapped. A second beam from a tunable Ti:Sapphire cw laser, which is modulated (10 Hz) and highly attenuated, denoted as a perturbing beam, is used to perturb the particle from its equilibrium position. Both beams are Gaussian TEM_{00} laser beam brought to diffraction limited focal spot with a large NA microscope objective (1.25NA 100x oil). We used the same oil immersion objective lens for focusing the trapping beam, the perturbing beam, and collecting the backscattered light [10]. This setup allowed: (1) to obtain a full optical force curve in the axial and radial directions as a function of beam position and wavelength for arbitrary polarization and (2) to observe the MDR resonances of the TE and TM modes of the microsphere by the selective coupling the beam to any one of the modes. Figure 1 shows a schematic diagram of the optical tweezers configuration used in this work.

The Gimbal mount changes the angle of incidence at the back aperture of the objective lens and moves the incident (perturbing) beam focus position laterally (for radial trapping force) using a step motor. The telescope changes the focus depth [18]. A lens system guarantees that the steered beam is pivoted on the objective aperture to avoid power loss. The power transmitted through the objective, measured with an integration sphere as a function of the radial and axial position, was constant. The calibration of the radial beam position was performed by moving a calibrated mark on the Neubauer chamber with the microscope translating stage and using the steering mechanism of the Gimbal mount to position the laser beam at the mark, creating a direct correspondence between position in microns and the beam steering step counts. The radial optical force on the microsphere as a function of beam position was measured holding the microsphere by the trapping beam while the perturbing beam was moved in the radial direction by using the Gimbal mount.

A signal proportional to the displacement was measured using the backscattering of a He-Ne laser (after passing through two short pass filters to reject the Nd:YAG and Ti:Sapphire laser beams) and detected with a photomultiplier tube (PMT) coupled to the eyepiece of the microscope and a lock-in amplifier. The forces were observed by monitoring the amplitude of the displacements while changing the position of the Ti:Sapphire laser. Polarization was controlled by a λ/2 waveplate.

The MDR were observed by focusing the perturbing beam just outside the surface of the sphere (where the coupling was strongest) and changing the wavelength of the perturbing beam by rotating the birefringent plate inside the laser cavity with a rotating step motor. The wavelength was calibrated with a monochromator. The fact that the microsphere is suspended in the fluid by the optical tweezers is very important since any near surfaces could ruin the boundary conditions.

## 4. Results and discussion

The microsphere trapped by a highly focused beam will experience a force moving it to a stationary position on the optical axis, generally just behind the focal point. This distance was determined by changing the telescope until the axial force is zero. We used the expressions presented in Section 2 to simulate the optical forces in the radial direction for the zero axial force condition for different polarizations, wavelengths and microsphere sizes. The numerical convergence is determined by the maximum angular momentum chosen in the expansion, *n*
_{max}=*kr*_{o}
[12].

Exact fitting with theory was not possible due to unknown exact values of the microsphere size, refractive indexes and the objective overfilling degree. Small changes in these values affect the diffraction rings around the beam in the focal region that contribute to the optical force. Diffraction effects can be seen as the oscillations at the end of Fig. 2 and Fig. 3 for both polarizations and reproduced adequately in the theory. The MDR, Fig. 4(a), were observed with different beam polarizations to excite the TM or TE mode individually, as it was demonstrated previously [10]. This can be intuitively understood by noticing that only the radial component of the electric field will excite the TM mode, and only the radial component of the magnetic field will excite the TE modes. It can also be noticed that for the resonant wavelength the radial optical force extends further outside the microsphere (dotted line). The MDR resonances enhance optical forces by more than 30–50%, just outside to the sphere radius, where coupling to resonance mode is more efficient [10]. Figure 4(b) shows the full radial force curve, normalized with respect to the resonant peak for each polarization, on a 9 µm sphere with respect to the beam movement for wavelength in resonance condition with the perpendicular polarization. From Fig. 4(b) one can observe that the radial force for the parallel polarization is different from the radial force for the perpendicular polarization, proving that the usually assumed azimuthal symmetry in the horizontal plane no longer holds. The anisotropy of the radial optical force is observed for all examined microspheres.

## 5. Conclusions

We acquired the curves for the optical force as a function of radial distance, wavelength and beam polarization for three microsphere sizes with a double tweezers setup where the radial position can be precisely controlled (±10 nm steps). This setup can be used to monitor fine shifts in morphology dependent resonances and to test optical force theories. The experiment shows the necessity to take into account Morphology Dependent Resonances (MDR), diffraction effects and polarization selection rules for a good description of optical forces.

We used a formalism based on rigorous solution of Maxwell equations for a proper description of the optical forces, without any limitations for the size of the scatterer that can be applied for all, Rayleigh, Mie and Geometrical Optics regimes. The optical forces are described in terms of experimental parameters such as the beam profile, objective filling factors, beam polarization and dielectric properties of the medium. The theoretical results agree with the experimental ones and demonstrate how careful one has to be when using optical force models for mechanical properties measurements. The framework of the formalism presented here can also be easily applied to other beams such as Laguerre beams, laser sheets, doughnut and top-hat beams.

## Acknowledgments

This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) through the Optics and Photonics Research Center (CePOF).

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