## Abstract

We report a focus shaping technique using generalized cylindrical vector beams. A generalized cylindrical vector beam can be decomposed into radially polarized and azimuthally polarized components. Such a generalized cylindrical beam can be generated from a radially polarized or an azimuthally polarized light using a two-half-wave-plate polarization rotator. The intensity pattern at the focus can be tailored by appropriately adjusting the rotation angle. Peak-centered, donut and flattop focal shapes can be obtained using this technique.

©2002 Optical Society of America

## 1. Introduction

Recently, there is an increasing interest in laser beams with cylindrical symmetry in polarization. These so-called cylindrical vector beams can be generated by active or passive methods [1–7] and have been the topic of numerous recent theoretical and experimental investigations [8–20]. Applications of such beams include microscopy, lithography [8], frequency shifting [15], electron acceleration [16], optical trapping and manipulating [17, 18], material processing [19] and high-resolution metrology [20]. Among these applications, particular interest has been given to the high numerical aperture (NA) focusing property of these beams and their application as a high-resolution probe. Due to the symmetry of the polarization, the electric field at the focus of a cylindrical vector beam has unique polarization properties. For example, it has been shown that the longitudinal component of the focus from such a cylindrical beam is much stronger than the transversal component, and the size of the longitudinal focus is much smaller than the transversal focus [9, 10]. This property could find applications in high-resolution microscopy, microlithography, metrology and nonlinear optics, etc.

To the best of our knowledge, previous research has dealt with the high NA focusing of either radially polarized or azimuthally polarized light. In this paper, we study the focusing property of a generalized cylindrical vector beam. A generalized cylindrical vector beam can be decomposed into a linear superposition of radially polarized and azimuthally polarized components. A simple polarization rotator consisting of two half-wave plates can be used to convert a radially polarized beam or azimuthally polarized beam into such a generalized cylindrical vector beam.

In section 2, we present the mathematical expression of a generalized cylindrical vector beam and the methods of generating such a beam. We will then present the high NA focusing property of this type of beam using the Richards and Wolf vectorial diffraction method in section 3. We will show that a flattop focus can be generated for a particular generalized cylindrical vector beam. In section 4, we will briefly discuss the applications of this technique.

## 2. Generating a generalized cylindrical vector beam

Figure 1 shows the polarization pattern of a generalized cylindrical vector beam. Instead of a radial polarization or an azimuthal polarization, each point of the beam has a polarization rotated by ϕ_{0} from its radial direction. The electrical field of this beam can be expressed in a cylindrical coordinate system as

where *e*⃗_{r} is the unit vector in the radial direction and *e*⃗_{φ} is the unit vector in the azimuthal direction. P is the pupil apodization function denoting the relative amplitude of the field, which only depends on radial position. Thus, a generalized cylindrical vector beam is just a linear superposition of a cylindrically symmetric radial polarization and a cylindrically symmetric azimuthal polarization.

Many techniques to generate radially polarized beams or azimuthally polarized beams have been reported. Some of them, e.g. a space-variant liquid crystal cell [4], may be used to create generalized cylindrical vector beams. However, usually the polarization pattern created by these methods is fixed and there is less flexibility of alternating the polarization pattern. In this paper, we propose a simple and flexible technique that can convert a radial polarization or azimuthal polarization into a generalized cylindrical polarization. A simple polarization rotator consisting of two half-wave plates can be used to perform such a conversion [ 19, 20]. Figure 2 shows the proposed polarization rotator. The Jones matrix of this polarization rotator can be shown as [20]

Unlike the rotation from a single half-wave plate, this rotation operation is independent of the initial polarization. The amount of rotation is determined by the angle ∆ϕ between the fast axes of the two half-wave plates. When ∆ϕ=ϕ_{0}/2, a generalized cylindrical vector beam illustrated in figure 1 can be generated from a radially polarized beam. By simply rotating one of the half-wave plates, we can vary ∆ϕ and control the angle ϕ_{0}, thus generating different cylindrical vector beams. Such flexibility is very important to the focus shaping technique we will discuss in the next section.

## 3. Focus shaping using generalized cylindrical vector beams

The basic focusing property of highly focused polarized beams can be analyzed with the Richards and Wolf vectorial diffraction method [21, 22]. This method has been extensively used to study cases in which the illumination on the pupil has no spatial variation in polarization [23]. This method has also been used to calculate the electric fields in the vicinity of the focal spot for radial and azimuthal polarization [11, 12]. In this paper, we further extend this calculation for a generalized cylindrical vector beam illumination.

#### 3.1 Geometry and mathematics

The geometry of the problem is shown in figure 3. The illumination is a generalized cylindrical vector beam described by equation (1), which assumes a planar wavefront over the pupil. An aplanatic lens produces a spherical wave converging to the focus of the lens. Since this beam can be expressed as a linear superposition of radial polarization and azimuthal polarization, the field near the focus can be expressed as linear combination of the focal fields of radial polarization and azimuthal polarization. Adopting the expressions for the radial polarization and azimuthal polarization developed by Youngworth and Brown [11], the focal field of a generalized cylindrical beam can be written as

where *e*⃗_{z} is the unit vector in the z direction. E_{r}, E_{z} and E_{φ}, are the amplitudes of the three orthogonal components that can be written as

where θ_{max} is the maximal angle determined by the numerical aperture of the objective lens, P(θ) is the pupil apodization function, k is the wave number and J_{n}(x) is the Bessel function of the first kind with order n. Note that all components are independent of φ, which means the field maintains cylindrical symmetry. With these equations, we can calculate the intensity and amplitude distributions corresponding to different components as well as the total field in the vicinity of focus.

For all the examples in this paper, we choose a simple annulus pupil apodization function:

The NA of the lens is chosen to be 0.8 and the length unit is normalized to wavelength, therefore, λ=1.

#### 3.2 Focusing of radially and azimuthally polarized beams

Fig. 4 shows the calculated results for ϕ_{0}=0°, which corresponds to a radially polarized incident beam. The two dimensional intensity distributions at the vicinity of the focus are shown in Fig. 5. The transversal intensity is the sum of the azimuthal intensity and radial intensity. In this case, the azimuthal component disappears and only the radial and longitudinal components are present. Thus the transversal field is purely radially polarized. One observation is that the longitudinal component is stronger than the transversal component and has only half of the width (approximately) of the total intensity distribution. If certain physical mechanisms can be applied such that the object at focal plane only responds to the longitudinal component, the resolution of the optical system will be increased by a factor of two [10].

ϕ_{0}=90° corresponds to an azimuthally polarized incident beam. In this case, only the azimuthal component is present at the focus (see Fig. 6). The two dimensional distribution is shown in Fig. 7. This donut shape focus has been reported previously [11].

From Fig. 4 and Fig. 6, we noticed that the radial and azimuthal polarization components of the foci resemble donuts that have dark centers. The sizes of the donut shapes of these two components are similar. For the longitudinal polarization component, however, there is only one bright sharp peak at the center. The size of this peak is similar to the size of the dark center of the donut shape. These observations indicate that we can obtain a flattop total intensity distribution at the focus by adjusting the weightings of the three field components through controlling ϕ_{0}. This control of ϕ_{0} can be achieved using the pure polarization rotators described previously.

#### 3.3Flattop focusing

For our particular calculation setup, the flattop condition is found to be at ϕ_{0}=24°. The intensity distributions at the focal plane are plotted in Fig. 8 and the corresponding two dimensional total intensity through the focus (in r-z plane) is shown in Fig. 9. A flattop total intensity distribution in the focal plane is obtained.

## 4. Applications and discussions

#### 4.1Aplication to optical tweezers

Optical tweezers is an optical tool that uses a tightly focus laser beam to trap and manipulate small particles, such as atoms, molecules, cells, etc. It has wide applications in many areas, such as DNA sequencing, genes transplant, micro-assembly and micro-machining. The origin of the trapping force is the gradient of field intensity. Generally, this gradient force can be expressed as [24]

where R is the size of the particle, c is the speed of light in vacuum, ∇I is the gradient of the intensity , ε_{1} is the dielectric constant of the ambient and ε_{2} is the dielectric constant of the particle. From this equation, we can see that the trapping performance depends on the dielectric constants of the particle and the ambient. If ε_{1}<ε_{2}, the gradient force tends to pull and trap the particle to the highest intensity region of the focused beam. On the contrary, this gradient force tends to pull and trap the particle to the lowest intensity region if ε_{1}>ε_{2}. Most existing optical tweezers use a focused Gaussian beam, which has the highest intensity at the center. Thus they are only suitable for trapping and manipulating particles with a dielectric constant higher than the ambient. For particles with a dielectric constant lower than the ambient, a specifically designed laser mode such as a donut mode needs to be applied. Using our focus tailoring method, we can easily change the focal intensity distribution from a donut shape to a peak-centered shape by adjusting the amount of rotation from the two-half-wave-plate polarization rotator, thus enabling trapping and manipulating a wide variety of particles in the same optical system.

#### 4.2 Other applications

The flattop focus obtained above may also find other applications such as improved printing filling factor, improved uniformity and quality in materials processing and micro-lithography, and so on. In our calculations, we have used a very simple pupil apodization function. However, the pupil apodization provides another degree of freedom to shape the focus. It is possible to improve the quality of the flattop focus, such as the edge abruptness, by using a more complex pupil apodization function.

## 5. Conclusions

We have described a unique far-field beam shaping technique - focus shaping using generalized cylindrical vector beams. A simple polarization rotator setup is proposed for the generation and modification of generalized cylindrical vector beams, which, in turn, can be used to modify the focal intensity shape. At a particular condition, a flattop focus can be generated. The focus shaping technique may find wide applications, such as optical tweezers, laser printing and material processing.

## Acknowledgement

This work is supported by the CyberOptics Corp. The authors are thankful for their support.

## References and links

**1. **D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE_{01},” Appl. Phys. Lett. **20**, 266–267 (1972). [CrossRef]

**2. **M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astignmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**, 123–132 (1993). [CrossRef]

**3. **M. W. Beijerbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beam produced with a spiral phaseplate,” Opt. Commun. **112**. 321–327 (1994). [CrossRef]

**4. **M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. **21**, 1948–1950 (1996). [CrossRef] [PubMed]

**5. **R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. **182**, 205–208 (2000). [CrossRef]

**6. **R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. **77**, 3322–3324 (2000). [CrossRef]

**7. **K. Schuster, “Radial polarization-rotating optical arrangement and microlithographic projection exposure system incorporating said arrangement,” US patent 6191880 B1 (2001).

**8. **L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens system,” Opt. Commun. **191**, 161–172 (2001). [CrossRef]

**9. **S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light- theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B **72**, 109–113 (2001). [CrossRef]

**10. **S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. **179**, 1–7 (2000). [CrossRef]

**11. **K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Optics Express **7**, 77–87 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77. [CrossRef] [PubMed]

**12. **D. P. Biss and T. G. brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express **9**, 490–497 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-10-490. [CrossRef] [PubMed]

**13. **H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B. **15**, 1381–1386 (1998). [CrossRef]

**14. **H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

**15. **J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. **81**, 4828–4830 (1998). [CrossRef]

**16. **B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E **55**, 3539–3545 (1997). [CrossRef]

**17. **T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. **78**, 4713–4716 (1997). [CrossRef]

**18. **S. Sato, Y. Harada, and Y. Waseda, “Optical trapping of microscopic metal particles,” Opt. Lett. **19**, 1807–1809 (1994). [CrossRef] [PubMed]

**19. **V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D. **32**, 1455–1461 (1999). [CrossRef]

**20. **Qiwen Zhan and James R. Leger, “Microellipsometer with radial symmetry,” submitted to Appl. Opt.

**21. **E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Ser. A **253**, pp. 349–357 (1959). [CrossRef]

**22. **B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A **253**, pp. 358–379 (1959). [CrossRef]

**23. **Min Gu (editor), *Advanced optical imaging theory*, 75 (Springer-Verlag, New York, 1999)

**24. **Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh cattering regime,” *Opt. Commun*. **124**, 529–541 (1996).