Numerical aperture invariant focus shaping using spirally polarized beams

Abstract

The concept of spiral polarization is proposed as an extension of the generalized cylindrical vector beam. The focusing properties of this spatially variant polarization under high NA are studied. It can be shown that with one such polarization, the focus maintains a flat-top intensity shape independent of NA from NA = 0.82 up to NA = 0.95.

Introduction

Beam shaping using spatially variant polarization has been of an increasing research interest in recent years. In dealing with two-dimensional configurations, radial and azimuthal polarizations are usually chosen as the polarization basis states due to symmetry considerations. Using this basis, the generalized cylindrical vector beam [1] has been proposed as a linear combination of radial and azimuthal components as shown in Fig. 1. Mathematically, the generalized cylindrical vector beam can be expressed as [1]$\overrightarrow{E}(r\text{,}\varphi )=P[\cos {\phi }_0\overrightarrow{e_r}+\sin {\phi }_0\overrightarrow{e_{\varphi }}]$, where $\overrightarrow{e_r}$ and $\overrightarrow{e_{\varphi }}$ are the radial and azimuthal basis polarizations, respectively. The relative strength of the radial and azimuthal components is determined by the rotation angle ${\phi }_0$ from the radial direction.

This generalized cylindrical vector beam is shown to be quite useful in beam shaping [1] under high numerical aperture (NA), which is due to the special focusing properties of radial and azimuthal polarization. (Here NA is defined as $\text{NA}=\sin \theta$, where focusing in the air is assumed.) For a given (high) NA, the focal field intensity distribution of the radially polarized light has a peak at the center, contributed by the strong z-component [1], [2]; the focal pattern of the azimuthally polarized light has a null at the center, with only the transverse component and no z-component in place. So by changing the angle ${\phi }_0$ of the generalized cylindrical vector beam, we can balance the z-component and the transverse components to yield a flat-top focus. For example, in Ref. [1], Zhan and Leger have shown that for NA = 0.8, with the rotation angle ${\phi }_0$ of 24°, the focal intensity shape is flat-top, as is shown in Fig. 1b. Such uniform irradiance is very important in a variety of applications, such as laser micro machining [3], laser-assisted thermal annealing [4], optical recording [5] etc. Other beam shapes can also be realized by appropriate polarization modulation. For example, the annular-shaped spot resulting from the transverse-only polarization component has certain advantages in materials processing and micro-welding [6], [7]. One inconvenient aspect of this polarization beam shaping scheme involves adjusting the angle ${\phi }_0$ for different NA’s. Fig. 2 shows the NA dependence of the rotation angle ${\phi }_0$ for the generalized cylindrical vector beam to produce a flat-top focal pattern. It can be clearly seen in Fig. 2 that as the NA increases, the characteristic angle ${\phi }_0$ that achieves flat-top beam shaping also becomes larger. In practice, the changing of the angle ${\phi }_0$ is usually performed by mechanically rotating a wave-plate to some prescribed orientation.

In this paper, we propose another polarization beam shaping method based on the use of “spiral polarization”, which is an extension of the generalized cylindrical vector beam concept. By numerical simulation, we show that one such polarization family is capable of delivering the flat-top focal pattern invariant to NA, from NA = 0.82 up to NA = 0.95.