## Abstract

We investigated an Laguerre-Gaussian (LG) beam that can carry an orbital angular momentum and has a doughnut-shaped intensity pattern. We developed a multilevel spiral phase plate (SPP) that generates an LG beam by applying the wave surface of a spiral structure directly to a Gaussian beam for application to microscopic laser material processing. We experimentally demonstrate, for the first time, that it is possible to generate an LG beam with the multilevel SPP that allows the use in high intensity laser pulses.

©2004 Optical Society of America

## 1. Introduction

A Laguerre-Gaussian (LG) beam that can carry an orbital angular momentum about the beam axis associated with helical wavefronts has attracted much interest in recent years. The orbital angular momentum in the propagation direction has a discrete value of *lh̄* per photon for beams with helical phase fronts, characterized by an exp(-*ilϕ*) azimuthal phase dependence[1], where *l* is the azimuthal mode index, and can be converted into rotational movements of macroscopic objects. This has been observed as rotations of particles trapped in LG beams[2] and the orbital angular momentum spectrum[3]. An LG beam has also the features of beam intensity distribution. Because the radiation intensity of the LG beam must be zero at the beam axis for a spiral phase structure, the intensity pattern results in a doughnut-shaped pattern. A number of studies based on these concepts constitute fields of research such as the trapping and rotating of micro particles[4, 5], non-linear optics[6], and atom-light interaction[7].

We have studied the interaction of a LG beam and matter, especially laser material processing at length scales ranging from nanometers to micrometers as a new LG beam application, for example, needle-shaped microprocessing utilizing a doughnut-shaped intensity pattern. Several LG beam generation methods have been proposed, including: conversion from the Hermite-Gaussian (HG) mode using a cylindrical lens[8, 9], a spiral phase plate[10, 11](SPP), and a phase hologram element[5, 12, 13]. The output beam mode-converted by a cylindrical lens for transforming HG into LG modes gives a pure LG mode. The computer hologram method can generate a beam with a high order of phase singular points and obtain high conversion efficiency. However, these methods are not suitable for the short-pulse high intensity laser because of laser damage, difficulties in fabricating a large diameter device, and the energy loss by passing through many optical components. We employed a large diameter optical device which has multilevel spiral phase distribution in consideration of its application to ultrashort high intensity laser pulses. The LG beam is produced by inserting the SPP, which has an optical thickness that increases with the azimuthal angle, just in front the focusing optics. The multilevel SPP is fabricated on silica glass in a multi-stage vapor deposition process to form multi phase levels; therefore, having the advantages of a high laser damage threshold and being able to relatively easily fabricate a large diameter device.

In this paper, we report an LG beam generated with a multilevel SPP for application to high intensity laser pulses. We theoretically mode-analyze the influence of the multilevel discrete phase steps on the produced LG beam. Using the fabricated SPP, we experimentally demonstrate, for the first time, that it is possible to generate an LG beam with a multilevel SPP that allows its use in an ultrashort pulse intense laser.

## 2. Multilevel spiral phase plate

A SPP is an optical device with an optical thickness that increases with the azimuthal angle so that an incident beam emerges with a helical phase front. Ideally, the phase changes smoothly and continuously, which the presently available vapor deposition technology does not permit. In addition, high damage threshold against a laser beam is required. Therefore we discretized the phase distribution to multilevel steps. The LG beam produced by the SPP is not a pure mode, but an infinite superposition of LG modes. The conversion efficiency from HG_{00} to LG_{10} mode was calculated to be 78%[11]. The multilevel SPP might reduce this conversion efficiency or not transform to the LG modes. We theoretically mode-analyzed the influence of the multilevel discrete phase steps on the produced LG beam. In discussing the mode analysis we limit the case of *l*=1. The complex amplitude of an LG beam which propagates along the z axis can be given by

$$\times \mathrm{exp}[-i\left(n-m\right)\varphi ]{\left(-1\right)}^{min(n,m)}{\left(\frac{r\sqrt{2}}{w}\right)}^{\mid n-m\mid}{L}_{min(n,m)}^{\mid n-m\mid}\left(\frac{2{r}^{2}}{{w}^{2}}\right)$$

with

where ${L}_{p}^{l}$
(*x*) is the generalized Laguerre polynomial, *k* is the wave number, *ρ* is the distance from the z axis, *ϕ* is the azimuthal coordinate, and *z*_{R}
is the Rayleigh range of the mode. ${C}_{\mathit{\text{nm}}}^{\mathit{\text{LG}}}$
is the normalization constant determined by $\int {\mid u\mid}^{2}drd\varphi =1:{C}_{\mathit{nm}}^{\mathit{LG}}={\left(\frac{2}{\pi n!m!}\right)}^{\frac{1}{2}}$!. The mode decomposition of a mode ${u}_{\mathit{\text{nm}}}^{\mathit{\text{LG}}}$
whose phase distribution has been modified by exp(-*iϕ*) is defined by the expansion coefficients

where the brackets denote integration in the transverse plane (*ρ,ϕ*). The relative weight of the modes is given by *I*_{nm,st}
=|*a*_{nm,st}
|^{2}.[11]

We substitute *ϕ*′ discretized to *N* steps for *ϕ* in Eq. (3) and then calculated the coefficients *I*_{nm,st}
by numerical integrals. *N* is the number of the discrete phase step. Table 1 shows the dependence of the number of the discrete step *N* on the conversion efficiency when LG_{00}, that is TEM_{00} mode (Gaussian beam), converts to LG_{10} mode through the SPP. The values in Table

1 represent the conversion efficiencies from the LG_{00} to the LG_{10} mode, that is, *I*
_{00,10}. Table 1 indicates the tendency that the efficiency reduces with a decrease in the number of the discrete step *N*. However, a multilevel SPP with *N*=32 or *N*=16 can maintain efficiency compared with that with a continuous phase (*N*=∞). These results reveal that a multilevel SPP can carry out functions similar to a continuous SPP as a mode converter satisfactorily. We discretized the substrate to 16 phase steps in consideration of the fabrication process.

## 3. Fabrication of a Multilayer SPP

Figure 1 shows the manufacturing process of an SPP. Multilayer coating was carried out on each division by electron beam deposition of SiO_{2}. Phase difference (optical path length) was varied by controlling vapor deposition film thickness. The same number of masks ((b)–(e) in Fig. 1) are prepared as the vapor deposition counts derived from the number of steps (log _{2}
*n*; where *n* is the number of steps), and vapor deposition is repeatedly performed. At this time, the optical path length in each step should be adjusted so that the following equation is satisfied:

where *λ*=789 nm (wavelength of Ti sapphire laser) and *n*: the refractive index of SiO _{2}. The vapor deposition procedure of a SPP of 16 steps is as follows: (1) 928 (=116×8) nm deposition using mask (b), (2) 464 (=116×4) nm deposition using mask (c), (3) 232 (=116×2) nm deposition using mask (d), (4) 116 (=116×1) nm deposition using mask (e).

This vapor deposition procedure features a lower vapor deposition count compared to the number of steps. Figure 2 presents the measurement results of the phase distribution of an SPP manufactured with an interferometer, where the vertical axis represents the phase difference. The diameter of the fabricated SPP is 8 centimeters. The figure indicates that the phase is distributed as a spiral that is designed to have 16 steps. Furthermore, a phase difference between the maximum and minimum of 2 *π* suggests that the SPP has been manufactured as designed.

Next, to verify the performance and fabricating accuracy of the fabricated SPP, the interference pattern of the LG beam generated through the SPP and a reference wave (a plane wave and a spherical wave) were measured with a Michelson interferometer. The interference pattern between the LG beam and the spherical reference wave are shown in Fig. 3. The results of the experimental observations (Fig. 3 (a)) are compared with the numerically calculated intensity distribution (Fig. 3 (b)), which is plotted in the case of the azimuthal mode index *l*=2. As shown in Fig. 3, the interference pattern observed between the LG beam and a spherical wave indicates a characteristic vortex shape and corresponds well with that in theory. Although the SPP was designed as *l*=1, the interference pattern indicated *l*=2, because light passed through the SPP twice because of the installation of the measuring instrument for the interference pattern measurement. Therefore, its phase shift is observed not as 2*π* (*l*=1), but as 4*π* (*l*=2). Fig.4 shows the interference pattern between a LG beam and the plane reference wave. The results of the experimental observations (Fig. 4 (a)) are compared with the numerically calculated intensity distribution (Fig. 4 (b)), which is plotted in the case of the azimuthal mode index *l*=2. An interference fringe between the LG beam and the plane wave results in a characteristic fork-like structure. As shown in Fig. 4, because the number of branches at the singular point is two also in this pattern, it is suggested that the order of the phase singular point is two. This fact agrees well with the result in Fig. 3.

## 4. Intensity Distribution Measurement in a Far-Field

We measured the far-field intensity distribution of a LG beam generated by the SPP using a semiconductor laser beam (*ϕ*=6cm) with a wavelength of 780 nm. The beam pattern focused with a 50 mm focal length plano-convex lens was expanded with an object lens of a microscope of 10-times magnification and a plano-convex lens of a focal length of 1,000 mm, and then observed using a CCD camera. The far-field pattern of the LG beam that passes through an SPP inserted in front of the focusing lens is shown in Fig. 5 (a). Figure 5(b) presents the horizontal linear profile of the intensity distribution in Fig. 5(a), wherein the solid, broken, and dotted lines represent the experimental LG mode with a SPP inserted, with no SPP inserted, and that theoretical calculated with the same parameters substituted into Eq. (1) as the experimental condition, respectively. As shown in Fig. 5(b), an intensity distribution of the experimentally generated LG beam corresponds almost to that obtained theoretically. The integration value of this profile in a radial direction yielded a conversion efficiency from HG to LG mode of 55%. Beam conversion with an SPP has better efficiency than generation by the superposition of the higher order HG mode using a beam splitter, or generation using an amplitude hologram. Moreover, because this element is made only of quartz, it has a high laser damage threshold against ultrashort high intensity laser pulses. In fact, no damage was observed even with continuous irradiation of a Ti sapphire laser (800-nm wavelength) of an energy of 200 mJ with a 130-fs pulse duration for 20 min.

## 5. Conclusion

The authors have developed a large diameter multilevel SPP that generates an LG beam by applying the wave surface of a spiral structure directly to a Gaussian beam for application to ultrashort high intensity laser pulses. A mode analysis revealed that a multilevel SPP can carry out functions similar to a continuous SPP as a mode converter satisfactorily. We fabricated a multilevel SPP using a multi-stage vapor deposition process and confirmed the accurate control of the phase distribution by measurement of the interference pattern.

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