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The conventional means of generating Bessel–Gauss beams by axicons in the laboratory results in the distinct disadvantage of an abrupt change in intensity at the boundary of the non–diffracting region. We outline theoretically and then demonstrate experimentally a concept for the creation of Bessel–like beams that have a z–dependent cone angle, thereby allowing for a far greater quasi non–diffracting propagation region.
©2010 Optical Society of America
1. Introduction
The zero–order Bessel beam was first introduced by Durnin [1], and has been studied extensive due to its non–diffracting and self–reconstructing properties [2, 3]. Laboratory demonstration of such beams has been possible through a range of optical techniques [4–7], including the now ubiquitous axicon, or conical lens, where the Bessel beam may be studied within a well defined region of validity. The length of this region is inversely proportional to the cone angle γ of the propagating waves. Recently it has also been shown that optical fields with a tunable propagation constant may be created using amplitude holograms placed in the Fourier plane [8]. In general, the non–diffracting nature of these beams changes abruptly at the boundary of the non–diffracting region from a Bessel function (near–field profile) into a conical field with the characteristic ring–shaped intensity distribution (far–field profile). The significant difference between the near–field and the far–field intensity pattern can be considered a disadvantage of such beams, in contrast to Gaussian beams which preserve their profile while propagating in free space.
There is an elegant possibility to eliminate this abrupt transformation by allowing the cone angle γ to decrease during propagation so that as z→∞ so γ→0. We refer to such beams as z–dependent Bessel–like beams (BLBs). Previously such BLBs have been generated by incorporating spherical aberration into the optical design (lens and axicon pair) [9,10]. The central idea in such set–ups is to create cone–like propagation, and have been investigated mainly with the aim of obtaining a uniform on–axis profile, a constant diameter central spot size, as well as to minimize astigmatism – an aberration that is typically large for conical optics [11]. In a similar manner [12,13], it has been shown that it is possible to achieve a high transverse resolution at large distances when a set–up comprising a defocused Galilean–type telescope with negative spherical aberration is used. However, even in these schemes there has not been a detailed investigation of the transverse structure in the near–field and far–field of such BLBs, nor the possibility of managing the axial intensity of such BLBs through a suitable parameter choice in the implementation method.
In this paper we outline an optical design for producing BLBs with a z–dependent cone angle through the use of conventional optical elements, without the need for deliberate aberrations to be included. The design concept is shown graphically in Fig. 1 , and consists of two axicons, Ax 1 and Ax 2, and a spherical lens of focal length f. The Bessel beams generated by these axicons are characterized by cone angles of γ 1 and γ 2 respectively. The fact that aspheric elements with custom aberrations are not required in this design makes the practical implementation of the set–up less complex and less costly.
Fig. 1 The optical set–up for the formation of a Bessel–like beam with a cone angle that reduces with propagation distance z.
A simplified analysis of the design can be made using geometrical optics: An incoming Gaussian beam is transformed by the spherical lens and axicon Ax 1 into a ring field of radius R 1 = f tanγ 1 near the focal plane of the lens; the ring width is known to be inversely proportion to the diameter of the incoming Gaussian beam, while the angular divergence is directly proportional to the diameter. Thus, if an input Gaussian beam is well collimated, the second axicon Ax 2 will be illuminated with a ring–like beam with a high divergence angle, θ. Besides, this annular beam belongs to the conical type with average cone angle γ 1. The second axicon then refracts the incoming field in the direction of the optical axis. Thus after passing through the axicon Ax 2, this beam is transformed so that its carrier spatial frequency is decreased but the divergence angle is fixed. As a result, the region behind the axicon Ax 2 will be illuminated with an imaginary ring–like light source (IB in Fig. 1). This source may also be located after the axicon Ax 2 if z 1 < f. The pertinent property of this scheme is that the crossing angle of a light ray on the optical axis decreases with increasing distance z, yet remains the same at any distance ρ from the optical axis for a given z. These properties suggest that the output field from such an optical system will be a Bessel–like beam with a z–dependent cone angle. In the limit that γ 1+ θ ≈γ 2, the maximum crossing angle converges to zero as z→∞.
2. Theory
A more rigorous analysis using the method of stationary phase may be used to show that the field at the input plane of the second axicon is an annular ring, whose cross–section is enveloped with a Gaussian function (i.e., an off–centered Gaussian), as predicted by the geometrical optics analysis.
First, the field at the input plane of the second axicon is calculated from the Fresnel integral as:
where , w 0 is the half–width of the input Gaussian beam and k 0 is the wavenumber of the field. Applying the method of stationary phase to Eq. (1) yields:where and only the positive stationary solution is valid (ρ ≥ γ 1 z 1). As is seen from Eq. (2), the field incident on the second axicon appears to be annular, but modulated with an enveloping function. Moreover, for typical experimental values of the parameters, the z 1/z 0 term contributes little as compared to the z 1/f – 1 term and may be neglected if the beam size is large enough. One also notes that the wavefront curvature is positive when z 1 > f and negative when z 1 < f.Similarly to the above, the field at some distance z after the second axicon is then found from:
where and we have definedUsing the known asymptotical form for the zeroth order Bessel function of the first kind, , we arrive at:The amplitude function g ±(ρ,z) has only a weak dependence on the transverse co–ordinate, ρ, and thus the dominant amplitude envelop is the zeroth order Bessel function. The dependence of the amplitude function g ±(ρ,z) on the longitudinal coordinate z is essential here and will be investigated in the section to follow. The dependence of cone angle γ on the longitudinal co–ordinate z is evident from Eq. (4), from which we note that the cone angle always decreases with increasing z. Thus we have verified theoretically that our approach, as outlined in the previous section, results in the creation of a Bessel–like field with a z–dependent cone angle.
For the particular case when γ 1 = 0 the second axicon will be illuminated not by an annular field but by a divergent (z 1 > f) or convergent (z 1 < f) spherical wave. This scenario has been studied previously and does not result in the desired reshaping of the far field [14]. When γ 2 = 0 our scheme is reduced to a doublet comprising a positive lens and an axicon. Such a setup has been investigated previously [15], where it has been shown that the resulting Bessel–like field cannot be tailored to have a higher on–axis intensity than the input Gaussian, and furthermore a z–dependent cone angle could only be achieved when the lens carried spherical aberration. Thus our scheme overcomes these disadvantages by incorporating the second axicon.
3. Numerical results
The study of the transverse intensity distribution, as well as of the angular spectrum of the generated beam, is of a great interest. Equations (1) and (3) were solved numerically to find the propagating field after the first and second axicon respectively. Unless otherwise indicated, the following parameters were used in all calculations: wavelength λ = 630 nm; cone angles γ 1 = 0.5°; γ 2 = 0.92°; w 0 = 1 mm and z 1 = 1.9f, with f the focal length of the lens. The numerically calculated field and its angular spectrum are shown in Fig. 2 (a) and (b) . The angular spectrum was calculated by propagating the field through a lens and considering the resulting field in the Fourier plane. We note that the profiles of the intensity distribution and the angular spectrum have an oscillating character close to a Bessel function, as suggested by Eq. (5). This counter intuitive result is due to the fact that the Fourier transform of a Bessel function containing a quadratic phase multiplier is itself a Bessel function.
Fig. 2 Intensity distribution (a) and angular spectrum (b) of the beam after the second axicon at z = 15 m and f = 0.18 m.
Further analysis shows that as z increases, so the shape of the angular spectrum is retained, whereas the oscillation frequency in the intensity distribution diminishes.
This proves the key property of our set–up: a changing cone angle of the Bessel field during propagation. As the cone angle changes, so the radial wavevector also changes, resulting in a changing period of oscillation across the Bessel field. The novelty here is that the angular spectrum (far–field pattern) is not annular in shape but rather is Bessel–like in shape, signifying a significant departure from the standard Bessel–like fields generated using axicons. The price to pay for this is that the field is no longer perfectly non–diffracting but rather a quasi non–diffracting field (see the discussion later).
The transition from our Bessel–like angular spectrum to that of a convention annular one may be achieved by steadily increasing the focal length of the lens. An intermediate state is shown in Fig. 3(a) , where a central high intensity Bessel field (Fig. 3(b)) is enclosed by a low intensity annular ring. The annular component of the field is characterized here by a large width, which is the result of a relatively high divergence of the annular field incident upon the second axicon. Note that in these calculations the distance between axicons was taken as 1.9f, which results in the divergence of the beam incident upon the second axicon decreasing as f increases. At large f the divergence becomes very small so that the field can be thought of as propagating like a plane wave, and so the field behind the axicon approaches the conventional Bessel beam, which in turn leads to the domination of the annular component in the angular spectrum of the output beam. Thus, when the focal length f and correspondingly the distance z 1 in Fig. 1 increases, the far–field generated by this scheme changes from a Bessel–like field to the conventional annular field.
It is also possible to control the on–axis intensity of the resulting field by judicious choice of parameters (e.g., cone angles and focal lengths), and in fact we will show that it is possible to produce a Bessel–like beam with an on–axis intensity that is higher at larger distances from the axicon than the input Gaussian field. Figures 4 (a) and (b) illustrate the change in the on–axis intensity with propagation distance z, but at relatively short distances from the axicon. As is seen, it is a one–peaked curve typical for Bessel beams, but with much smaller intensity oscillations than usually observed with a Bessel beam generated with a single axicon. Further, it is shown that a decrease in the focal length f leads to an increase of the region with a high intensity. As z increases up to several meters or more, there occurs a slow monotonic decrease in the peak on–axis intensity with propagation distance as is typical of Gaussian beams.
Fig. 4 Dependence of the on–axis BLB intensity with propagation distance z (for short distances from the axicon): (a) standard parameters and f = 0.18 m, (b) γ 1 = 0.5°; γ 2 = 0.6° and f = 0.5 m.
By tailoring the ratio of the cone angles, it is possible to create a Bessel–like beam with this set–up that shows a higher on–axis intensity (at large distances from the axicon) as compared to the input Gaussian beam, as illustrated in Fig. 5 . In order to allow a comparison, we assume the Gaussian beam continues to propagate without the impact of the optical elements within the set–up, i.e., free space propagation.
Fig. 5 Dependence of the on–axis BLB and equivalent Gaussian intensity with propagation distance z (for long distances from the axicon): (a) standard parameters and f = 0.18 m, (b) γ 1 = 0.5°; γ 2 = 0.6° and f = 0.5 m.
Firstly, we note (see for example, Fig. 4) that the intensity of the Bessel–like beam near the axicon is significantly higher than the intensity of the input Gaussian; this is typical even for single axicon set–ups for creating Bessel–Gauss beams. But far from the second axicon the on–axis intensity of Bessel–like beam may be either smaller (Fig. 5(a)) or larger (Fig. 5(b)) than that of the Gaussian beam depending on the choice of optical parameters. This differs considerably from traditional schemes for creating Bessel beams, and is deserving of further study. As pointed out earlier, for the optical scheme with one axicon and a lens, the on–axis intensity of the Bessel–like beam did not exceed that of the Gaussian beam. Thus the use of the second axicon in our design allows one to create long–range Bessel–like beams with high on–axis intensity. At small distances (~ 1 m), which are realized in laboratory conditions, the divergence of such BLBs is rather small, and in this sense one can regard them as quasi non–diffracting.
4. Experimental results
The set–up depicted in Fig. 1 was constructed experimentally. Two axicons with cone angles of γ 1 = 0.89° and γ 2 = 0.98° where separated by z 1 = 0.75 m; the spherical lens had a focal length of f = 0.5 m. An input Gaussian beam of half–width w 0 = 2.5 mm (from a spatially filtered He–Ne laser, λ = 632.8 nm) was used to illuminated the set–up. The angular spectrum was measured at the Fourier plane of a lens of focal length 0.5 m. All beam intensities were captured with a CCD–camera (Photometrics, 1392×1040 pixels of dimension 6.4×6.4 μm2). The evolution of the spatial beam profile with distance from the second axicon is shown in Fig. 6 . It is seen that the beam as a whole contains both axial and annular components (Fig. 6(a)). As the propagation distance increases so the contribution of the annular component diminishes until only a z–dependent Bessel–like beam structure remains (Fig. 6(b)). In this case in the central region of the beam is a Bessel field (Fig. 6(c)) whose cone angle decreases with distance. The angular spectrum of the beam as a whole (Fig. 6(d)) is described with high accuracy by a Bessel function, contrary to conventional schemes where an annular ring would be observed. The experimental results are in very good agreement with the theory presented earlier. These findings lend support to the view that the central region of z–dependent Bessel–like beams, with a high degree of accuracy, is the usual J 0 – Bessel beam.
Fig. 6 Experimentally measured transverse intensity distribution of the beam as a whole at different distances from the second axicon: (a) z = 0.9 m, (b) z = 6.4 m, (c) the central part of the beam at z = 6.4 m, and (d) the Fourier transform of the field. The outer diameter of the annular field is approximately 30 mm in (а) and 60 mm in (b).
The on–axis intensities of the z–dependent Bessel–like beam and input Gaussian beam were measured at a distance of 25 m from the second axicon. The measurement has shown that the axial intensity of the Gaussian beam is ~2.5× smaller, while our theory predicts a factor of 2.7× . The results are in general agreement, with the discrepancy likely to be due to the inaccuracy of the quoted cone angles of the axicons.
5. Conclusion
In this paper a new design was proposed for the generation of z–dependent Bessel–like beams, without the need for special aspheric elements. In particular we show that this method generates BLBs whose cone angle decreases monotonically with propagation distance. Adjustment of the focal length of the lens and the cone angles of the axicons in the design allows one to reshape the transverse distribution in both the near and the far field, and in particular for a smooth transition from a field with a single radial wavevector to a changing radial wavevector with propagation distance, and hence from an annular ring intensity profile to a Bessel–like intensity profile in the far field. Therefore the generated light beams belong to an intermediate class with an angular spectrum that includes both axial and annular components.
The results presented here have indicated a Bessel structure that exists for several tens of meters, whereas the propagation distance of the Bessel field from a single axicon alone would be in the order of tens of centimeters (using our axicon values). Indeed, since the far field is also a Bessel–like structure, the greatly increased quasi non–diffracting propagation distance of such beams is self–evident. Such fields may readily find application for long distance ranging and communication systems, or in the emerging field of decelerating Bessel pulses [16,17].
Acknowledgements
We gratefully acknowledge the support from the South African National Research Foundation (Grant number 67432).
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