**1. Introduction**

Radiation pressure force (RPF) indeced by a focused laser beam has bean widely utlized for the manipulation of small particles, and has found more and more applications in various fields including physics [1], biology [2], and optofludics [3, 4]. Accurate prediction of optical force exerted on particles enables better understanding of the physical mechanicsm, and is of great help for the design and improvement of optical tweezers.

Many researches have been devoted to the prediction of radiation pressure force (RPF), and different approaches have been developed for the theoretical calculation of RPF exerted on a homogeneous sphere. The geometrical optics [5-7] and Rayleigh theory [8] are respectively considered for the particles much larger and smaller than the wavelength of incident beam. Since geometrical optics and Rayleigh theory are both approximation theories, rigorous theories based on Maxwell's theory have been considered [9-13]. Generalized Lorenz-Mie Theory (GLMT) [14] has been used to investigate the RPFs exerted on some regular parti‐ cles[10-13, 15, 16] induced by a Gaussian beam. GLMT can rigorously calculate RPF induced by any beam. To isolate the contribution of various scattering process to RPF, Debye series is introduced [17, 18].

Traditional optical tweezers use Gaussian beams as trapping light sources. This approach works well for the manipulation of microscopic spheres. However, the deveopment of science and technology brings new challenges to optical tweezers, and several approaches have been developed. Holographic methods have been used to increase the strength and dexterity of optical trap [19]. Another approch is the employment of non-Gaussian beam including Laguerre-Gaussian beams [20] and Bessel beams [28]. Laguerre-Gaussian beams have zero on-

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axis intensity, and can increase the strength of optical trap. Bessel beams consist of a series of concentric rings of decreasing intensity, and have characteristics of non-diffraction and selfreconstruction. A single Bessel beam can be used to simultaneously trap and manipulate, accelerate, rotate, or guide many particles. Bessel beams can trap and manipulate both highindex and low-index particles.

In addition to Laguerre-Gaussian and Bessel beams, there is a speical class of beams which have cylindrical symmetry in both amplitude and polarization, hence the name Cylindrical Vector Beams (CVBs) [29-32]. CVBs are solutions of vector wave equation in the paraxial limit. The special features of CVBs have attracted considerable interest for a variety of novel applications, including lithography, particle acceleration, material processing, high-resolution metrology, atom guiding, optical trapping and manipulation. The most interesting features for optical trapping arise from the focusing properties of CVBs. A radially polarized beam focused by a high numerical aperture objective has a peak at the focus, and can trap a highindex particle. On the contrary, an azimuthally polarized beam has null central intensity, and can trap low-index particle. These two kinds of beams can be experimentally switched.

CVBs can be generated by many methods, which are categorized as active or passive depend‐ ing on whether amplifying media is used. The simplest mothod is to convert an incident Gaussian beam to a radially polarized beam using a radial polariser. However this method does not produce very high purity tansverse modes. Moer efficient methods use interferom‐ etry. Since a CVB can be expressed as the linear superpostion of two Hermite-Gaussian or Laguerre-Gaussian beams with different orientations of polarization. Another efficient method is based on optical fiber [33]. This technique takes advantage of the similarity between the poarization propeties of the modes that propagate inside a step-index optical fiber and CVBs. When TE01 or TM01 is excited in the fibre, it excites a CVB in free space. Fiber-generated CVB, taking Bessel-Gaussian as example, and its applicaions top optical manipulations will be discussed in this chapter.

### **2. Mathematical description of cylindrical vector beams**

Cylindrical vector beams are solutions of vector wave equation

$$
\nabla \times \nabla \times \vec{E} + k^2 \vec{E} = 0,\tag{1}
$$

where *k=2π/λ* is wavenumber with λ being the wavelength. In the paraxial approximation, the radially and azimuthally polarized vector Bessel-Gaussian beams, two kinds of typical CVBs, can be expressed as

$$\vec{E}\_{rad} = E\_0 \frac{\rho}{\omega\_0} e^{-\frac{\rho^2}{\nu\_0^2}} e^{l(at - kz)} \hat{e}\_\rho \tag{2}$$

Fiber-Based Cylindrical Vector Beams and Its Applications to Optical Manipulation http://dx.doi.org/10.5772/59151 201

$$\vec{E}\_{azl} = E\_0 \frac{\rho}{\omega\_0} e^{-\frac{\rho^2}{\kappa\_0^2}} e^{\iota(\alpha t - kz)} \hat{\mathbf{e}}\_{\phi} \tag{3}$$

where *r* and *ϕ* are respectively the radial and azimuthal coordinates, *e* ^ *<sup>ρ</sup>* and *e* ^ *<sup>ϕ</sup>* are unit vectors in *ρ* and *ϕ* directions, and the subscripts *rad* and *azi* denote the polarization state. *w0* is the width of beam waist, and *E0* is a constant. Fig. 1(a) and (b) respectively give the intensity distribution of radially and azimuthally polarized Bessel-Gaussian beam in the plane *z=0*. Note that the longitudial component of CVB is negligible under the condition of paraxial approxi‐ mation. A general CVB can be considered as a linear superposition of a radially polarized CVB and an azimuthally polarized one.

axis intensity, and can increase the strength of optical trap. Bessel beams consist of a series of concentric rings of decreasing intensity, and have characteristics of non-diffraction and selfreconstruction. A single Bessel beam can be used to simultaneously trap and manipulate, accelerate, rotate, or guide many particles. Bessel beams can trap and manipulate both high-

200 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

In addition to Laguerre-Gaussian and Bessel beams, there is a speical class of beams which have cylindrical symmetry in both amplitude and polarization, hence the name Cylindrical Vector Beams (CVBs) [29-32]. CVBs are solutions of vector wave equation in the paraxial limit. The special features of CVBs have attracted considerable interest for a variety of novel applications, including lithography, particle acceleration, material processing, high-resolution metrology, atom guiding, optical trapping and manipulation. The most interesting features for optical trapping arise from the focusing properties of CVBs. A radially polarized beam focused by a high numerical aperture objective has a peak at the focus, and can trap a highindex particle. On the contrary, an azimuthally polarized beam has null central intensity, and can trap low-index particle. These two kinds of beams can be experimentally switched.

CVBs can be generated by many methods, which are categorized as active or passive depend‐ ing on whether amplifying media is used. The simplest mothod is to convert an incident Gaussian beam to a radially polarized beam using a radial polariser. However this method does not produce very high purity tansverse modes. Moer efficient methods use interferom‐ etry. Since a CVB can be expressed as the linear superpostion of two Hermite-Gaussian or Laguerre-Gaussian beams with different orientations of polarization. Another efficient method is based on optical fiber [33]. This technique takes advantage of the similarity between the poarization propeties of the modes that propagate inside a step-index optical fiber and CVBs. When TE01 or TM01 is excited in the fibre, it excites a CVB in free space. Fiber-generated CVB, taking Bessel-Gaussian as example, and its applicaions top optical manipulations will

**2. Mathematical description of cylindrical vector beams**

<sup>2</sup> Ñ´Ñ´ + = 0, r r

where *k=2π/λ* is wavenumber with λ being the wavelength. In the paraxial approximation, the radially and azimuthally polarized vector Bessel-Gaussian beams, two kinds of typical CVBs,

> 2 2 <sup>0</sup> ( )

w

r

ˆ

r

0 0

r- - <sup>=</sup> <sup>r</sup> *<sup>w</sup> i t kz E E ee e rad <sup>w</sup>*

*E kE* (1)

(2)

Cylindrical vector beams are solutions of vector wave equation

index and low-index particles.

be discussed in this chapter.

can be expressed as

**Figure 1.** Intensity distribution of CVB. The arrows indicate the direction of polarization
