Elsevier

Optics Communications

Volume 281, Issue 17, 1 September 2008, Pages 4240-4244
Optics Communications

Intensity distribution around the focal regions of real axicons

https://doi.org/10.1016/j.optcom.2008.05.027Get rights and content

Abstract

We present a theoretical and experimental study of the intensity distribution of a laser beam, after its propagation through a real axicon. We show that, while an ideal axicon generates Bessel-like radial intensity profile and smooth on-axis intensity around the focal region, in practice, the bluntness of the tip of the axicon causes significant deviations from these profiles. In particular, strong oscillations occur on the on-axis intensity. The input beam size also plays a strong role; for small beams the axicon acts more like a conventional lens, while for the large ones, it exhibits two focal regions. We also propose scheme for apodization. In all cases, the experiments match closely with the calculations. Our results show that axicons cannot be assumed to always generate smooth on-axis intensities and the structures coming from the real profiles should be taken into account for most applications.

Introduction

Axicons have found an increasing use in optical applications, since their invention by McLeod [1]. Laser beams propagating through axicons have mainly two significant properties. First, they generate a line focus, where the on-axis intensity stays high over much longer distances compared to focusing by conventional lenses. Second, they generate ring-like intensity profiles in the far field. Both of these properties proved useful in many applications. The long focal region, for example, is used to increase the depth range in optical coherence tomography [2], as well as to guide atoms as optical tweezers [3]. The ring profile in the far field is exploited to machine precision holes at increased speed [4], [5], and to trap atoms in the dark central region [6]. The high on-axis intensities obtained around the focal region makes the axicons attractive for non-linear optical application, as well [7], where increased interactions lengths are desired, such as to generate long laser-induced plasma channels [8], and to study non-linear propagation in air [7], [9], [10].

The effect of an ideal axicon on an incoming beam is simply introduction of a phase retardation, which is linear with the transverse position. If plane waves are incident on such an axicon, they form radial intensity profiles described by zeroth-order Bessel functions [11], forming “non-diffracting beams” [12]. Such waves, however, possess infinite energy, and hence are unrealistic. This problem can be resolved by using Gaussian input beam profile, which yield Bessel-like radial intensity profile, and a long transverse region where the beam intensity stays high and nearly constant. The transverse position, where maximum intensity occurs can be defined as the focal length of the axicon. Some distance after the focus, the beam forms a ring and propagates in this form henceforth.

While the descriptions above yield smooth on-axis intensity profiles, practical issues cause deviations from these expectations. Tanaka et.al. for example, calculated effects of the coma and astigmatism in case of axicons [13]. The effect of a hard aperture at the edges and a stop in the center is also studied theoretically in case of plane waves [14], [15], and in the case of Gaussian beams [16], where the aperture is shown to cause oscillations of the on-axis intensity, and a beam stop in the center can be used to minimize these oscillations. Apart from aberrations resulting from misalignments, and diffraction effects from hard edges, there is another very important practical consideration, regarding the region around the tip of the axicon: an ideal axicon would have a perfectly sharp point tip. This, however, is practically impossible to realize, and in reality, the tip of an axicon is rather blunt. Depret et al. have shown that the blunt-tip region of an axicon can be described by a hyperbola [17], and they have analyzed the beam profile in the far field, where rings are formed, and show that the bluntness causes non-zero intensity in the central regions. To the best of our knowledge, the effect of bluntness of the axicon on the intensity distribution around the focal region was never studied.

For many applications of the axicons, it is crucial to understand the behavior of the element. For example, when an axicon is used in imaging [2], [18] and for non-linear propagation [9], [19] experiments, separation of the contributions from the effects under study and from the axicon aberrations must be carefully distinguished. In this work, we present our theoretical and experimental study of intensity distribution in the focal region of a blunt-tip axicon. We found that the hyperbolic approximation of the axicon tip, given by Depret et al. [17], yields a very good match between our calculations and experiments. Most importantly, we show that the bluntness causes significant deviations from the otherwise smooth on-axis intensity. The beam size plays a determining role on the profile. In fact, we show that for small beams, the axicon acts more like a conventional lens. Our results demonstrate that the shape of the axicon cannot be assumed ideal, and the blunt profile is essential to take into account for most applications, especially those requiring high, smooth and continuous on-axis intensities. We present our theoretical and experimental results of intensity profiles in Section 2. We propose and demonstrate approach to apodize the profile in Section 3.

Section snippets

Intensity profiles generated by real axicons: simulations and experiments

We evaluate the electric field distribution of light after an axicon, by numerically solving the Fresnel–Kirschoff integral. The input beam is assumed to be Gaussian. The system possesses cylindrical symmetry; hence the field will have only r and z dependence. In cylindrical coordinates, the Fresnel–Kirschoff integral can be written asE(r1,z)=1iλzexpikz+r122z0expik2zr02J0(kr0r1/z)E(r0)r0dr0where E(r1, z) is the light electric field at transverse position z and radius r1, E(r0) is the input

Apodization of the on-axis intensity

The results of the previous section show that for a real axicon, the on axis-intensity is not smooth, but rather oscillatory. These oscillations would be unwanted and biasing for the applications of axicons. As a result, in this section, we propose ways to reduce these oscillations and apodize the intensity profile.

Since the source of the oscillations is mainly interference from the tip and surroundings, the method we propose is to put a beam stop in the center of the axicon, the radius of

Conclusions

In conclusion, we have studied the intensity distribution of light passing through a real axicon. We show that, the bluntness of the axicon causes significant deviations from the ideal case. The on-axis intensity distribution is strongly dependent on the beam size. For small beams, the axicon acts like a regular lens, while for large ones, it exhibits two principal peaks, one coming from the lens-like center, and the other coming from the axicon. The overall profile shows strong oscillations.

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