### Theoretical model

Theoretical description of optical forces is simple only for very small particles (so-called Rayleigh particles - RP), whose radius fulfills a<<l/20, where l is the trapping wavelength in the medium. Such a small particle behaves as an induced elementary dipole and the optical forces acting on it can be divided into two components - gradient and scattering forces. The gradient force comes from electrostatic interaction of a particle (dielectrics) with an inhomogeneous electric field and the scattering force results from the scattering of the incident beam by the object.

For particles bigger than l/20, a more complex concept of stress tensor of electromagnetic field surrounding the particle must be generally used. This requires the knowledge of the total field outside the confined particle. The original theory based on the plane wave scattering has been gradually modified so that it can be applied for a spherical or spheroidal particle placed into an arbitrary field distribution. It is commonly referred to as the generalized Lorenz-Mie theory (GLMT).

The GLMT uses the scattering procedure
presented first by Mie who derived expressions for the field
distribution outside a spherical object of arbitrary size placed into a
plane wave. This original method was generalized so that it enables an
expression of the forces acting on the spherical and oblate objects
placed in a general electromagnetic field.
For the description of a focused beam, a modification to the
fundamental Gaussian
beam was presented (so called 5th order corrected Gaussian beam (CGB)).
It uses a field expansion in the beam size parameter *s=1/(kw _{0})*
to the 5th order, achieves better agreement with the wave equation,
and, therefore, provides more precise calculation of the optical forces.

Because we consider the standing wave created by the interference of two counter-propagating focused laser beams, we have easily adapted the above mentioned GLMT formalism to this case. Instead of a single CGB we summed field components of two counter-propagating CGB with overlapped beam waists to get the initial field components of the standing wave. Moreover, we assumed that the spherical object was located on the beam axis and so we could employ the radial symmetry of the problem and simplify the calculation. We wrote the modified code ourselves but we do not present a detailed mathematical description, because the method is well described in literature.

We neglected any electrostatic
interactions between the surface and the particle as well as multiple
scattering of the incident beam. Furthermore we assumed that the beam
waist was placed on the surface with reflectivity equal to 100%. The
axial positions *z _{s}* of the
sphere center, where we calculated the axial forces, satisfy the
inequality

*a <= z*l.

_{s}<= a+Although the adopted simplifications (CGB, absence of spherical aberrations, diffraction, and multiple scattering events) could seem drastic, this model provides at least correct qualitative description of the behavior of dielectric spheres in the GSW and acceptable speed of calculations.

Related publications

P. Jakl, M. Sery, J. Jezek, A.
Jonas, M. Liska, P. Zemanek: **"Behaviour
of an optically trapped probe approaching a dielectric interface"**,

J
Mod Opt 50, 1615-1625, 2003,
ABSTRACT DOWNLOAD

P. Zemanek, A. Jonas, P. Jakl, J.
Jezek, M. Sery, M. Liska: **"Theoretical comparison of optical
traps created by standing wave and single beam"**,

Opt.
Comm. 220, 401-412, 2003,
ABSTRACT DOWNLOAD

P. Zemanek, A. Jonas, M. Liska:
**"Simplified description of optical forces acting on a
nanoparticle in the Gaussian standing wave"**,

JOSA A 19, 1025-1034, 2002,
ABSTRACT DOWNLOAD

A. Jonas, P. Zemanek, E.-L.
Florin:
**"Single-beam trapping in front of reflective surfaces"**,

Opt. Lett. **26**, 1466-1468,
2001,
ABSTRACT DOWNLOAD

P. Zemanek, A. Jonas, L. Sramek,
M. Liska:
**" Optical Trapping of Nanoparticles and Microparticles by a
Gaussian
Standing Wave"**,

Optics Letters, 24, 1448-1450, 1999. ABSTRACT DOWNLOAD

P. Zemanek, A. Jonas, L. Sramek,
M. Liska, : "**Optical
Trapping of Rayleigh Particles Using Gaussian Standing Wave**.",

Optics Communications,
151, 273-285, 1998. ABSTRACT DOWNLOAD

Conference proceedings

P. Jakl, M. Sery, J. Jezek, P. Zemanek:

**"How the size of a particle approaching dielectric interface influences its behavior"**,

Proceedings of SPIE 5514, 636-642, 2004, ABSTRACT DOWNLOAD

M. Sery, P. Jakl, J. Jezek, A. Jonas, M. Liska, P. Zemanek:

**"Influence of weak reflections from dielectric interfaces on properties of optical trap"**,

Proceedings of SPIE 5036, 624-629, 2002.

Jan Jezek, P. Zemanek, Alexandr
Jonas, Mojmir Sery, Pavel Pokorny, Miroslav Liska:
**"Study of the behavior of nanoparticle and microparticle in
the
standing wave trap"**,

Proceedings of SPIE 4356, 318-325, 2001,
ABSTRACT

**"Comparison of the single beam and the standing wave trap stiffnesses"**,

Proceedings of SPIE 4356, 347-352, 2001, ABSTRACT

P. Zemanek, L. Sramek, A. Jonas, Z.Moravcik, R. Janisch, M. Liska:

**"Standing Wave Trap and Single Beam Gradient Optical Trap - Experiments and Biological Applications.",**

Proceedings of SPIE 3820, 401-410, 1999. ABSTRACT

A. Jonas, L. Sramek, M. Liska, P.
Zemanek : "**Efficient
particle trapping using an upward reflected laser beam**.",

Proceedings of SPIE,
3580, 91-101, 1998. ABSTRACT DOWNLOAD

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