Double Optical Tweezers for 3D Photonic Force ...

Double Optical Tweezers for 3D Photonic Force Measurements of Mie Scatterers Antônio A. R. Neves*, Adriana Fontes, Wendel L. Moreira, André A. de Thomaz, Diogo Burigo de Almeida, Luiz C. Barbosa and Carlos L. Cesar Quantum Electronics Department, Gleb Wataghin Physics Institute, State University of Campinas, Campinas, Brazil, 13083-970. ABSTRACT The ability to observe quantitatively mechanical events in real time of biological phenomena is an important contribution of the Optical Tweezers technique for life sciences. The measurements of any mechanical property involves force measurements, usually performed using a microsphere as the force transducer. This makes the understanding of the photonic force theory critical. Only very sensitive and precise experimental 3D photonic force measurements for any particle size will be able to discriminate between different theoretical models. In particular it is important to obtain the whole photonic force curve as a function of the beam position instead of isolate particular points. We used a dual trap in an upright standard optical microscope, one to keep the particle at the equilibrium position and the other to disturb it. With this system we have been able to obtain these force curves as a function of x, y and z position, incident beam polarization and wavelength. We investigated the optical forces for wavelengths in and out of Mie resonances of dielectric microspherical cavities for both TM and TE modes and compared the experimental results with the calculations performed with different models for the optical force. Keywords: optical tweezers, optical force, optical trapping, Mie resonances, microsphere, resonances, dielectric, laser beam

1. INTRODUCTION Optical tweezers have become an important tool for biological manipulations and cell mechanical properties measurements.1,2 These measurements use the displacement from equilibrium position of a microsphere as the force transducer. Therefore, the calibration procedure requires the use of good models for the optical force in microspheres. The main difficulty arises in the proper description of an optical beam near the focus of a highly focused optical system were the paraxial limit fails. This becomes more complex when it involves diffraction, compatible with optical setup of a high numeric objective where the back objective aperture is generally overfilled. When diffraction is taken into account, so does the polarization since it is now widely known that a linearly polarized highly focused beam exhibits all three polarization in the focus, the basic field we have to work with is the electromagnetic field and not just the intensity. Another wishful aspect of a good formalism for obtaining Beam Shape Coefficients (BSC) is to describe it in terms of easily obtainable experimental parameters, in the case of Optical Tweezers (OT) the highly focused beam is generally located in an immersion medium with sub-micron beam waist, it is thus difficult to obtain the proper value of beamwaist parameter with standard methods of knife-edge or Ronchi-ruling for the OT setup. Most formalism begin treating the problem with an approximation due to the size regime of the individual scatterer, Rayleigh for particles smaller than the wavelength and geometrical optics for sizes greater than wavelength, which is inappropriate when the scatterer is of the order of wavelength and since it doesn’t account for diffraction resonance effects are not taken in account. It is thus necessary to start with the general Mie regime taking into account all possible scattering sizes. Recently a new model3 has been proposed to take into account all these effects which the incident beam is subjected thanks to an analytical solution of one important integral4 and here we compare the optical force theory with experimental results. These observations match well with numerical solutions based on the proposed model. *[email protected]

Optical Trapping and Optical Micromanipulation III, edited by Kishan Dholakia, Gabriel C. Spalding, Proc. of SPIE Vol. 6326, 63260L, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.680859

Proc. of SPIE Vol. 6326 63260L-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/10/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

2. THEORY OF OPTICAL FORCES Light-matter interactions that dictate the dynamic of matter at the microscopic level can be described by Maxwell Equations. In OT our model comprises of a highly focused laser beam due to the use of a high numerical aperture to employ trapping in all three dimensions. This electromagnetic field holds transparent dielectric spheres of diameter d in a host medium of lower refractive index, n1 , than the sphere, n2 . To start applying the continuity of the electromagnetic field on the surface of the sphere, these fields must be known and conveniently written in terms of partial wave. As presented in our earlier work the incident be can be written as,3

⎡i ⎤ TE Einc = E0 ∑ ⎢ GnTM m ∇ × jn ( kr ) X n m (θ , φ ) + Gn m jn ( kr ) X n m (θ , φ ) ⎥ k ⎦ n, m ⎣ H inc

E = 0 Z

i ⎡ ⎤ ∑ ⎢GnTMm jn (kr ) X n m (θ , φ ) − k GnTEm∇ × jn (kr ) X n m (θ , φ )⎥ ⎦ n, m ⎣

where X n m (θ , φ ) is the vector spherical harmonic, jn (kr ) are spherical Bessel functions and Z =

(1)

µ ε is the

medium impedance. The incident field has a time varying harmonic component exp(−iω t ) that has beam omitted. The BSC for a linear x-polarized Gaussian TEM0,0 laser beam are,

⎡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

cos α exp(− f 2 sin 2 α ωb2 ) exp(−ikzo cos α )

0

(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 kρ o sin α ⎪⎭ ⎣ ⎦ where f is the objective focal length, ωb is the incident beam waist before the objective aperture, α max is the numerical aperture half-angle. The beam is positioned at the cylindrical coordinate ( ρ o , φo , zo ) in relation to the center of the sphere at the origin. The electromagnetic fields due to the sphere are given by Mie theory for the internal and scattered region as,

⎡i ⎤ E scat = E0 ∑ ⎢ anm∇ × hn(1) (k1r ) X n, m (θ , φ ) + bnm hn(1) (k1r ) X n, m (θ , φ )⎥ n , m ⎣ k1 ⎦ H scat

E = 0 Z1

⎡ ⎤ i ∑ ⎢anm hn(1) (k1r ) X n, m (θ , φ ) − k bnm∇ × hn(1) (k1r ) X n, m (θ , φ )⎥ 1 ⎦ n, m ⎣

⎡ i ⎤ Eint = E0 ∑ ⎢ cnm∇ × jn ( k 2 r ) X n, m (θ , φ ) + d nm jn (k 2 r ) X n, m (θ , φ )⎥ n, m ⎣ k 2 ⎦ H int

E = 0 Z2

⎡ ⎤ i ∑ ⎢cnm jn (k2 r ) X n, m (θ , φ ) − k d nm∇ × jn (k2 r )X n, m (θ , φ )⎥ 2 ⎦ n, m ⎣

(3)

(4)

where the subscript ‘1’ refers to the surrounding medium and ‘2’ refers to the medium of the sphere. These expansion coefficients are determined satisfying the continuity of the fields inside and outside the sphere, in this case only the first two are of interest,


− an =

anm Mψ n ( Mx)ψ ' n ( x) −ψ 'n ( Mx)ψ n ( x) = TM ψ 'n ( Mx)ξ n ( x) − Mψ n ( Mx)ξ 'n ( x) Gnm

(5)

b Mψ 'n ( Mx)ψ n ( x) −ψ n ( Mx)ψ ' n ( x) − bn = nm = TE Gnm ψ n ( Mx)ξ ' n ( x) − Mψ 'n ( Mx)ξ n ( x)

These coefficients are called Mie coefficients and don’t depend on the incident beam. The minus sign has been placed to maintain the use of standard notation. For transverse magnetic (TM) modes we have a n and for transverse electric (TE) modes we have bn . The Mie coefficients are written in term of Riccati-Bessel functions and in terms of the size parameter x = n1π d λ and relative refractive index M = n2 n1 . Mie resonance condition occurs for either TM or TE modes when the denominators become very small, these resonant modes are also termed Morphology Dependent Resonances (MDR). Once all expansion coefficients are known we can proceed to determining the optical forces with the use of Maxwell stress tensor defined as,

Fi = ∫ Tij n j dA =

(

)

1 1 Re ∫ [ε Ei E *j + µ H i H *j − ε E ⋅ E* + µ H ⋅ H* δ ij ] n j dA 2 2

(6)

integrating this over a sphere where kr → ∞ and using the ortonormality and vectorial properties of the vector spherical harmonic we obtain the longitudinal optical force cross section as, Cz =

1 2k

1

2

∑ (n + 1)

n =1

n ⎧⎪ n(n + 2) TM TM * (n + m + 1)(n − m + 1){Re[i[(an + an* +1 − 2an an* +1 )Gnm Gn +1, m ] ⎨ ∑ ⎪⎩ (2n + 1)(2n + 3) m = − n TE TE * Gn +1, m ]+ + Re[i (bn + bn*+1 − 2bn bn*+1 )Gnm

(7)

1 n TM TE * ⎫ m[Re[i (an + bn* − 2an bn* )Gnm Gnm ]⎬ ∑ n m = −n ⎭

and the transverse optical force cross section as, 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 [(an +1 + an* − 2an +1an* )GnTM +1, − ( m +1) Gn , − m + ( an + an +1 − 2a n an +1 )Gnm Gn +1, m +1 + *

*

TE TE (bn +1 + bn* − 2bn +1bn* )GnTE+1, − ( m +1) GnTE, − m + (bn + bn* +1 − 2bn bn* +1 )Gnm Gn +1, m +1 ]

−

(8)

1 n ∑ (n + m + 2)(n + m + 1) (n − m)(n + m + 1) n m = −n

TM TE * TE TM * ⎫ [(an + bn* − 2an bn* )Gnm Gn, m +1 − (bn + an* − 2bn an* )Gnm G n , m +1 ] ⎬ ⎭

These last two relations along with the BSC and Mie coefficients model fully the optical forces incorporating the highly focusing effect of optical beams and diffraction on Mie scatterers, with the incident beam being placed in an arbitrary position in relation to the sphere.

3. EXPERIMENTAL SETUP The optical setup is scheme is presented in figure 1. We performed the force spectroscopy experiment trapping 3, 6 and 9 µm polystyrene microsphere (Polyscience, Inc) diluted in water, with a double optical tweezers setup. One beam from


the Nd:YAG laser (1064 nm, model 3800S, Spectra Physics) was the trapping beam, to keep the microsphere trapped at the origin, while a second beam from a tunable Ti:Sapphire laser (model 3900S, Spectra Physics) with a much lower average power was modulated with an optical chopper and used to perturb the microsphere from its equilibrium position. The waveplate allowed controlling the perturbing beam polarization while the computer controlled telescope and gimbal mount provided the axial and radial focus movement. The detection 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 (Hamamatsu) coupled to the eyepiece of the microscope and a lock-in amplifier (Stanford Research Systems, model SR830 DSP). All laser passed through the same 100x 1.25 NA oil immersion objective. With this system we have been able to obtain these force curves as a function of x, y and z position, incident beam polarization and wavelength.

Chopper Ti:Sapphire Telescope

ND Filter

HeNe Waveplate λ/2

Gimbal mount

Microscope

PMT

Argon

Nd:YAG CCD

Fig. 1. Complete scheme for the double optical tweezers for ultra-sensitive force spectroscopy.

To measure the radial optical force on the microsphere as a function of beam position, the microsphere was held in place by the trapping beam while the perturbing beam moved radially from the origin, using the gimbal mount and acquiring the backscattered signal from the photomultiplier tube. To measure the axial force the same procedure was adopted but in this case the first lens from the telescope moved instead of the gimbal mount sweeping the perturbing beam through the whole sphere. For MDR measurements the perturbing beam was placed just outside the sphere where coupling was strongest, with this beam in place the wavelength of the perturbing beam was incremented in a defined interval where 4 MDR peaks was resolved.

4. RESULTS AND CONCLUSIONS In figure 2 the simulation and experimental data for the radial optical forces versus beam displacement from the center of the sphere are reported for diameters of 3, 6 and 9 µm. It can be seen that the optical force continues even though the beam is just outside the sphere and that the ripple structure on the tail of the force represents the diffraction effects in which the model takes fully into account. Fitting is approximate since no least square fitting was adopted to compare the experimental data with the theoretical ones, the errors here is most probably due to the unknown refractive index that was adopted for the polystyrene microparticles. In figure 3, the first picture illustrates the radial force anisotropy by changing the polarization, the force profile is not the same, with a reasonable gain at the border when a resonant wavelength is adopted. The second picture of figure 3 further emphasis this fact, when the wavelength of the perturbing Ti:Sapphire laser is sweep and the corresponding optical force measured for each polarization, on resonance condition optical forces gains ~50%. The last picture of figure 3, shows the experimental data and modeling on the axial optical force, in this case it is isotropic with respect to polarization and fits relatively well.


Theory

Theory

Experiment

Experiment

Experiment

0

0.5

1

1.5

2

2.5

3

Force (u.a.)

Force (u.a.)

Force (u.a.)

Theory

0

1

2

Displacement (µm)

3

4

5

6

0

1

2

Displacement (µm)

3

4

5

6

7

8

Displacement (µm)

Fig. 2. From left to right, 3, 6 and 9µm polystyrene microsphere, showing in red the experimental data points and in blue the theoretical optical force. Pol.S Resonant

P

Pol.S Non Resonant

S

Theory

Pol.P Resonant

Experiment

Force (u.a.)

Force (u.a)

Force (u.a.)

Pol.P Non Resonant

-20

0

1

2

3

4

5

Displacement (µm)

6

7

8

9

730

740

750 760 Wavelength (nm)

770

-10

0

10

20

30

780 Displacement (µm)

Fig. 3. Optical force on 9µm polystyrene microsphere. (left) Radial force for different polarization and wavelength, (right) Mie resonances for different polarizations as a function of wavelength.

With the dual optical trap, it is possible to realize ultrasensitive force measurement. Our simulations based on the proposed model yields good agreement with the experimental results. The theoretical modeling presented here is expected to play a vital role in optical manipulation in future research, giving a comprehensive understanding the optical tweezers performance is affected by the optical parameters. In this case, of up most importance is the optimization of the numerical code so to be employed in least square fitting of the experimental data.

ACKNOWLEDGEMENTS 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). We thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for financial support of this research.

REFERENCES 1. M. M. Brandao, A. Fontes, M. L. Barjas-Castro, L. C. Barbosa, F. F. Costa, C. L. Cesar, and S. T. O. Saad, “Optical tweezers for measuring red blood cell elasticity: application to the study of drug response in sickle cell disease,” Eur. J. Haematol. 70(4), 207-211 (2003). 2. M. L. Barjas-Castro, M. M .Brandao, A. Fontes, F. F. Costa, C. L. Cesar, S. T. O. Saad, “Elastic properties of irradiated RBCs measured by optical tweezers,” Transfusion 42(9), 1196-1199 (2002). 3. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. Accepted (2006). 4. A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39(18), L293-L296 (2006).
