Download Electromagnetic forces for an arbitrary optical trapping of a spherical

Electromagnetic forces for an arbitrary optical
trapping of a spherical dielectric
Antonio A. R. Neves, Adriana Fontes, Liliana de Y. Pozzo, Andre A. de Thomaz, Enver
Chillce, Eugenio Rodriguez, Luiz C. Barbosa and Carlos L. Cesar
CePOF, Instituto de Física, Universidade Estadual de Campinas, Brazil
[email protected]
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
OCIS codes: (170.4520) Optical confinement and manipulation; (140.7010) Trapping;
(180.0180) Microscopy; (260.2110) Electromagnetic theory; (290.4020) Mie theory; (260.1960)
Diffraction theory.
References and links
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Received 10 October 2006; revised 13 December 2006; accepted 14 December 2006
25 December 2006 / Vol. 14, No. 26 / OPTICS EXPRESS 13101
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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:
Einc = E0
H inc
⎡i
⎢
n, m ⎣ k
E
= 0
Z
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∑
∑
⎤
TE
GnTM
m ∇ × jn ( kr ) X n m (θ , φ ) + Gn m jn ( kr ) X n m (θ , φ ) ⎥
⎦
⎡ TM
G j (kr ) X n m (θ , φ ) −
⎢ nm n
n, m ⎣
i TE
⎤
Gn m ∇ × jn (kr ) X n m (θ , φ )⎥
k
⎦
(1)
Received 10 October 2006; revised 13 December 2006; accepted 14 December 2006
25 December 2006 / Vol. 14, No. 26 / OPTICS EXPRESS 13102
Where X n m (θ , φ ) = LY / n(n + 1) , is the vector spherical harmonic, jn (kr ) are
spherical Bessel functions and Z = μ ε is the medium impedance. The BSCs obtained
from our previous work [12] for the case of an incident linear x -polarized truncated TEM00
Gaussian beam focused at the ( ρ o , φo , z o ) cylindrical coordinates point, is given by the
expression:
⎡GnTM
⎤
2n + 1 (n − m )!
m
⎢ TE ⎥ = ±2π ikf exp( −ikf ) i n − m exp( −imφo )
4π n(n + 1) (n + m )!
⎢⎣ Gn m ⎥⎦
α max
∫ dα
0
cos α exp(− f 2 sin 2 α ωa2 ) exp(−ikzo cos α )
(2)
⎧⎪⎡ 2 J m (kρ o sin α ) m
⎤
Pn (cos α ) − sin 2 α J m′ (kρ o sin α ) Pn′m (cos α )⎥ cos φo
⎨⎢ m
kρ o sin α
⎪⎩⎣
⎦
⎫
⎡
⎤
J ( kρ o sin α ) m
⎪
Pn′ (cos α ) ⎥ sin φo ⎬
+ im ⎢ mJ m′ (kρ o sin α ) Pnm (cos α ) − sin 2 α m
k
ρ
sin
α
⎪
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
,TE
coefficients for n up to 30 and − n ≤ m ≤ n for each beam position. The
the GnTM
,m
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]:
n
⎡C x ⎤
1 ⎡ Re⎤
i ⎪⎧
n ( n + 2)
( n + m + 2)(n + m + 1)
⎨
⎢C ⎥ = 2 ⎢ ⎥
⎣ y ⎦ 4k ⎣ Im ⎦ n =1 ( n + 1) ⎪⎩ ( 2n + 3)( 2n + 1) m= − n
∑
∑
*
*
*
TM *
TM TM
⎡( a
+ a n* − 2a n +1 a *n )G nTM
+1, −( m +1) G n , − m + ( a n + a n +1 − 2a n a n +1 )G nm G n +1,m +1 +
⎢⎣ n +1
*
*
TE TE
(bn +1 + bn* − 2bn +1bn* )G nTE+1, −( m +1) G nTE, − m + (bn + bn*+1 − 2bn bn*+1 )G nm
G n +1,m +1 ⎤⎥
⎦
n
1
−
( n + m + 2)( n + m + 1) (n − m)(n + m + 1)
n m=−n
(3)
∑
⎡( a + b * − 2a b * )G TM G TE * − (b + a * − 2b a * )G TE G TM * ⎤ ⎫
n
n n
nm
n, m+1
n
n
n n
nm n,m +1 ⎥ ⎬
⎢⎣ n
⎦⎭
Where an and bn 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
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position. Both beams are Gaussian TEM00 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.
Chopper
Ti:Sapphire
Telescope
ND Filter
HeNe
Waveplate λ/2
Microscope
PMT
Argon
Nd:YAG
CCD
Gimbal mount
Fig. 1. Complete scheme for the double optical tweezers for ultra-sensitive force spectroscopy.
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 HeNe 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
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Received 10 October 2006; revised 13 December 2006; accepted 14 December 2006
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force condition for different polarizations, wavelengths and microsphere sizes. The numerical
convergence is determined by the maximum angular momentum chosen in the expansion,
nmax = kro [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.
Fig. 2. Radial optical forces for parallel polarization with respect to radial beam movement for:
3, 6 and 9 μ m spheres: (red) experimental; (blue) theory.
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Fig. 3. Radial optical forces for perpendicular polarization with respect to radial beam
movement for: 3, 6 and 9 μm spheres: (red) experimental; (blue) theory.
Fig. 4. Radial optical forces enhancement due to MDR for a 9 μm sphere and polarizations as a
function of: (a) wavelength; (b) radial distance.
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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Received 10 October 2006; revised 13 December 2006; accepted 14 December 2006
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