## Abstract

A design of an optical resonator for generation of a doughnutlike laser beam in the far field is proposed. The resonator consists of a toric mirror, a flat output coupler, and a w-axicon with a movable center axicon. Two-dimensional vector electric field simulation has shown that any one of the Laguerre–Gaussian modes can be selected by sliding the center axicon. Therefore this resonator is capable of generating doughnut-like laser beams, whose dark spot size can be controlled in real time. This feature of the proposed resonator is advantageous for atom trapping and optical tweezers.

©2004 Optical Society of America

## 1. Introduction

In recent years, doughnut-like laser beams have been of interest for trapping cold atoms and for optical tweezers. Several methods, such as cancelling out the central part of a laser beam by destructive interference [1], transforming the beam with an axicon lens [2], use of a hologram [3], and fast steering of the beam circularly [4], have been demonstrated. However, the existing methods have limitations depending on their principle. Some methods could make only a limited region of a doughnut-like optical field, and some methods lack flexibility in terms of changing beam parameter, e.g., changing the dark spot size of the doughnut arbitrarily.

In the course of our study of a w-axicon optical resonator for efficient energy extraction from annular media, we found that it could be oscillated at any one of the high-order Laguerre–Gaussian L${\mathrm{G}}_{0}^{\pm l}$ modes depending on the shape of the w-axicon. The L${\mathrm{G}}_{0}^{\pm l}$ modes have doughnut shapes, and the the dark spot size depends on the azimuthal order of the mode. The shape of the w-axicon can be varied easily if one splits the w-axicon with the outer part and the center part as in Fig. 1(a), and slides the center axicon along the optical axis.

Because any L${\mathrm{G}}_{0}^{\pm l}$ are a pure “mode” of the electromagnetic wave, the doughnut-like shape is unchanged during its propagation or spherical lens transformation. Therefore a very long “optical funnel” can be produced when the beam is focused by a lens with a long focal length.

In this paper, we report on the feasibility of the doughnut-like beam generator based on this novel w-axicon resonator. This resonator should be useful for atom trapping and optical tweezers, since the dark spot size could be varied in real time.

## 2. Theory

Figure 1(a) shows the schematic drawing of a w-axicon resonator with a movable center axicon. The resonator can be considered to be a combination of an annular resonator and a solid resonator connected by a w-axicon. In the outer part, the self-reproduction condition is fulfilled with toric Gaussian modes as long as the thickness of the ring is much smaller than its radius. We consider the fundamental mode oscillation in the radial direction by placing an aperture with an appropriate width. The electric field on the toric mirror is then expressed to

where *l* is the azimuthal mode index, *r*
_{0} is the radius of the mirror vertex, and *w*
_{0} is the Gaussian beam width [5].

It is readily seen that all the high-order azimuthal modes of the toric resonator have the same spatial distribution in the radial direction, and therefore azimuthal mode discrimination is quite poor.

On the other hand, resonance modes of the inner part can be expanded by the well-known Laguerre–Gaussian modes. The electric field on the output coupler is expressed as

where *p,l* are the radial and azimuthal mode indices, *w*
_{0} is the spot size (radius of the fundamental mode), and ${L}_{p}^{l}$
denotes the Laguerre polynomial [6].

The outer part and inner part are coupled by the w-axicon as shown in Fig. 1(b). The unique mode selection mechanism of this resonator is explained as follows; Each L${\mathrm{G}}_{0}^{\pm l}$ mode (*l* ≫ 1) of the inner part is approximated to a thin ring whose radius is proportional to |*l*|^{1/2} . The w-axicon connects those modes to the ring part geometrically as shown in the figure, and only one of the L${\mathrm{G}}_{0}^{\pm l}$ modes could couple with the ring part with minimum diffraction loss. The radius of the fundamental mode, *w*
_{0} in Eq. (2), is dominated by the Gaussian width of the outer part, and the order *l* is chosen so that the radius of the L${\mathrm{G}}_{0}^{\pm l}$ mode equals *w*
_{1}(*P*) in Fig. 1(b). On the other hand, the azimuthal order of the outer part is dominated by the order of the inner part. In this way, a single-mode oscillation is accomplished.

Because an axicon is a no-paraxial device, linear polarization is not preserved by the reflection from a w-axicon. This phenomenon is known as “polarization scrambling effect” [7]. Still, in the case that the complex reflectivity of the mirror surface could be approximated to (*ϕ*_{p}
- *ϕ*_{s}
) = *π* and (|*r*_{p}
| = |*r*_{s}
|), the round-trip propagation cancel outs the polarization scrambling and the electric field of the w-axicon resonator could be treated with a scalar electric field model [8]. In reality, i.e., the polarization-dependent reflectivity and nonideal phase delay between *s* and *p* polarizations of the mirror surface at an incident angle of *π*/4, the vector electric field treatment is indispensable for accurate modeling of the w-axicon resonator.

## 3. Simulation model

We developed an optical resonator simulation code with Cartesian coordinate, paraxial, and vector complex electric fields. The vector electric field is expressed by two scalar components normal to the propagation, *E*_{x}
(*x,y*) and *E*_{y}
(*x,y*).

The free-space propagation is calculated by the numerical Fresnel–Kirchhoff integration, whereas the w-axicon transformation is performed by the analytical formula,

$${r}_{2}={L}_{\mathit{cc}}-{r}_{1}$$

$${E}_{p}({r}_{1},\phi )={E}_{x}({x}_{1},{y}_{1})\mathrm{cos}\phi +{E}_{y}({x}_{1},{y}_{1})\mathrm{sin}\phi $$

where *r*
_{1} and *r*
_{2} are the distance of points *P*
_{1} and *P*
_{2} from the optical axis; *r*_{p}
and *r*_{s}
are the complex reflectivity of the mirror surface for *p* and *s* polarization, respectively; *k* is the wavenumber; and *L*_{cc}
is the w-axicon parameter [8]. The geometric approximation is justified because of the large Fresnel number of the w-axicon transformation.

The initial electric field set before the iterative calculation is very important for stable resonator simulation. When the initial electric field is set improperly, the convergence may fall into the local minimum, which leads to an incorrect conclusion. This problem has been discussed by Bhowmik [9], who proposed a partially coherent input field in the space-frequency domain,

where *ψ*
_{0} is a constant magnitude and 0 ≤ *R*(*x,y*) < 1 is the computer-generated random number, instead of plane or random distribution of initial *E*(*x,y*). Because of this carefully selected power distribution in the space-frequency domain, all modes that could be oscillated in a given resonator are stimulated equally, like with a real resonator. Therefore, the oscillation nature of the simulated resonator copies the real one, and a realistic result is given even if the resonator oscillates in a very large number of modes simultaneously. The result of optical resonator simulation with partially coherent input field is seen in our previous work [10].

Dimensions of the simulated resonator are shown in Fig. 3. There are four stations (S1–S4 in the figure) where the optical field is defined. Two different sheets are used at the w-axicon to represent the center axicon and outer axicon separately. A gain station is set on plane S1.A long distance between the toric mirror and output coupler is intended for placing some wavelength selection devices; however, we assume free free propagation for this section here.

For all stations, the optical field is defined on a sheet of 50 mm × 50 mm, discretized to 1024 × 1024.

The output coupler is assumed to be flat and to have 1.0% transmittance. All the reflecting surfaces are assumed to have 0.1% power loss. Complex reflectivity of the axicon surface is assumed to be |*r*_{s}
| = |*r*_{p}
| = 0.9995 and (Arg[*r*_{p}
] - Arg[*r*_{s}
]) = *π*. The position of the center axicon is varied in the range of 0.0 ≤ *P* ≤= 2.0 mm, where *w*
_{1}(0.0) is defined to be 0.0 mm.

The gain sheet assumed on plane S1 is homogeneously broadened, Γ = 0.1/pass, and *I*_{s}
= 1.0 × 10^{5}[W/m^{2}]. The oscillation wavelength is set at 0.78*μ*m.

Calculation starts with setting the electric field for *E*_{x}
and *E*_{y}
using Eq. (7) with no correlation between *E*_{x}
and *E*_{y}
; then iterative propagation is done until convergence is met.

## 4. Results and discussion

First, we checked the mode selection capability of the proposed resonator by giving it a L${\mathrm{G}}_{0}^{l}$ mode, and calculated the round-trip loss of the bare cavity. Figure 4 shows the result. It is clearly seen that for every axicon position, only a few modes have the minimum diffraction losses, and the order of the lowest loss mode can be selected by sliding the cone.

Next, the loaded cavity calculations were conducted. Figure 5 shows the far-field pattern of the output beam for several axicon positions. It is seen that the resonator outputs are the doughnut-like beams whose dark spot size depends on the axicon position *P*. In all the cases, two degenerated, counterrotating L${\mathrm{G}}_{0}^{+l}$ and L${\mathrm{G}}_{0}^{-l}$ modes are not phase locked, and ripples resulting from the interference are small. This is a good feature for an optical trap; however, mode locking should appear in a real situation because of the small asymmetricity of the resonator, as observed in Ref. [8].

Alignment sensitivity of the proposed resonator was checked. Figure 6 shows the output power of the resonator versus the tilt angle of the toric mirror. Calculations for three cone positions are shown. The output power at perfect alignment depends on the order of the mode, because of the difference in the diffraction losses (see Fig. 4). The misalignment sensitivity depends on the order of the mode, and the higher-order mode is more robust than the lower-order mode. It can be said that the misalignment tolerance of the resonator is more or less 5 *μ*rad.

Free-space propagation of the output beam was simulated. The axicon position *P* was set at 1.5 mm, and the output from the resonator was numerically propagated through a virtual lens of *f* = 1.0 m. Figure 7 shows the intensity distribution as a function of the distance from the lens. It can be seen that the diameter of the optical tube is almost unchanged in the ±500-mm range from the focal point. We emphasize that the dark spot size of this funnel could be varied in real time by simply sliding the center axicon. This feature is very favorable for the condensation of the trapped atom.

## 5. Summary

A unique optical resonator for generation of a doughnut-like beam was presented. The resonator consists of a toric mirror, a w-axicon, and a flat output coupler. The center axicon of the w-axicon is movable for controlling the oscillation mode. The proposed method is superior to the existing methods because the dark spot size of the optical field could be easily controlled in real time.

A two-dimensional vector electric field simulation was conducted to show the feasibility of the proposed resonator. The calculation has shown that the resonator could oscillate any one of the L${\mathrm{G}}_{0}^{\pm l}$ modes. Also, the calculation showed that the output beam could generate a tube-like optical field over a 1000-mm range.

## Acknowledgment

This research was supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), 14740253, 2002.

## References and links

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