Telephoto axicon ABSTRACT

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1 Telephoto axicon Anna Burvall, Alexander Goncharov, and Chris Dainty Applied Optics, Department of Experimental Physics National University of Ireland, Galway, Ireland ABSTRACT The axicon is an optical element which creates a narrow focal line along the optical axis, unlike the single focal point produced by a lens. The long and precisely defined axicon focal line is used e.g. in alignment, or to extend the depth of focus of existing methods such as optical coherence tomography or light sectioning. Axicons are generally manufactured as refractive cones or diffractive circular gratings. They are also made as lens systems or doublet lenses, which are easier to produce. We present a design in the form of a reflectiverefractive single-element device with annular aperture. This very compact system has only two surfaces, which can be spherical or aspheric depending on the quality required of the focal line. Both surfaces have reflective coatings at specific zones, providing an annular beam suitable for generating extended focal lines. One draw-back of a normal axicon is its sensitivity to the angle of illumination. Even for relatively small angles, astigmatism will broaden the focus and give it an asteroid shape. For our design, with spherical surfaces concentric about the center of the entrance pupil, the focal line remains unchanged in off-axis illumination. Keywords: Axicons, diffraction 1. INTRODUCTION Axicons 1 4 are optical elements that create focal lines along the optical axis, instead of the focal point produced by a normal lens. The extended depth of focus along with the extremely narrow focal line is useful e.g. in alignment, 5 for extending the focal depth of methods such as light sectioning 6 or optical coherence tomography, 7 and for x D 2 /2 Intensity ρ D 1 /2 y d 1 d 2 z d 1 d 2 z Figure 1. Geometry of an axicon. Corresponding author A. Burvall: anna.burvall@nuigalway.ie, telephone

2 particle trapping and guiding. 8 The focal line forms a Bessel beam, sometimes referred to as non-diffracting since it retains its shape on propagation. The principle of the axicon is illustrated in Fig. 1. Rays entering the axicon are refracted towards the optical axis by approximately the same angle, independently of height in the axicon. This creates a conical wave which produces the focal line. One way of accomplishing this is to use a glass cone, mostly referred to as a refractive axicon. Another option is a grating made up of concentric circles, a diffractive axicon. The possibility of creating the axicon as a combination of lenses, utilizing the spherical aberration to produce the extended optical line, has been demonstrated in several publications. 9, 1 The surfaces of the lens or lenses can be spherical 11 or aspherical. 12 Spherical surfaces have the advantages of being easy and relatively inexpensive to produce, while the disadvantage is the difficulties of obtaining a good quality focal line. The most inexpensive element would be a singlet lens, but since it has positive spherical aberration the width of the focal line will vary significantly along the optical axis. 11 For this reason, the more advanced lens axicon systems are either doublets, defocused telescopes, or aspherics. We suggest that if some reflective coatings are used, the singlet lens can be used in a double-pass configuration resembling photographic objectives. This is schematically shown in Fig. 2. This way, even a singlet lens with spherical surfaces can produce negative spherical aberration and give a Bessel beam of good quality. The loss of light in the blocked part of the beam represents no draw-back, since the central part of the axicon aperture is habitually blocked to improve the quality of the focal line. Furthermore, a carefully positioned aperture stop will improve the off-axis properties far beyond those of an ordinary axicon. The outline of this paper is as follows: section 2 covers the specifications to be considered when designing an axicon, while section 3 outlines the design process. Two examples are given and tested by numerical calculations. R 1 R 2 z d 1 d 2 Figure 2. Principle of a singlet-lens reflective-refractive axicon. The thick lines indicate reflective coatings. 2. DESIGN SPECIFICATIONS The main specifications of the axicon are the diameters D 1 and D 2 of the annular aperture stop, and the beginning and end points d 1 and d 2 of the focal line. They correspond more or less to specifying focal length and F-number of a lens. Then two main properties of the focal line have to be considered: the on-axis intensity and the width, which might both vary along the optical axis. For a linear axicon, e.g. a refractive cone, the convergence angle and line width are constant while the axial intensity grows linearly with z. In this case, it is enough to specify D 1, D 2, and the convergence angle. A logarithmic axicon has constant intensity, which is achieved by slight adjustments of the convergence angles, but this in turn leads to variations in line width. The differences are realized by changes in the phase function, e.g. slight changes to the cone shape for a refractive

3 axicon, or variation in the distances between the concentric circles for the diffractive version. The width of the focal line is actually narrower than that of the corresponding lens, and can be calculated through stationary-phase approximation of the Fresnel diffraction integral: 1 w(z) = 2.448λ z(ρ) 2π ρ (1) where λ is the wavelength, ρ the radius at which the ray leaves the axicon, and z(ρ) the position where it crosses the optical axis. This implies that constant line width is achieved if z(ρ) = c 1 ρ. The axial intensity is, by the same method, 1 I(z) = 4π 2 ρ(z) 2 λz 1 zφ (ρ) (2) where kφ(ρ) is the phase function, and φ (ρ) = ρ/z(ρ). Using the chain rule to get the second derivative and inserting it into Eq. (2) yields the expression I(z) = 4π2 ρ λ z (ρ). (3) So in order to have constant intensity, we should instead have z(ρ) = c + c 2 ρ 2. In the case of a lens axicon, the design is often adjusted to give the best compromise between constant intensity and constant line width. In off-axis illumination, aberrations become a serious problem. Astigmatism, which through diffraction effects destroy the focal line, shows up at much smaller angles than for an ordinary lens. For a linear axicon, the halfwidth of the asteroid-shaped distorted focus is given by 13 w(z) = αz(1 1/cos 2 ω) (4) where the phase of the axicon is kαρ, and ω is the off-axis angle. In particular for axicons with short focal lines placed close tho the axicon, these effects show up even for very small angles. Lens axicons, however, offer the possibility of compensating for the amount of astigmatism that appears, e.g. by using aberration-free surfaces. In our design, the surfaces are concentric or nearly concentric about the entrance pupil, thus eliminating coma, astigmatism, and distortion. Of the third- and higher-order aberrations, only two remain: spherical aberration, which is used to form the focal line, and field curvature. The field curvature makes the focal plane concentric about the entrance pupil, but will not affect the quality of the focal line. 3. EXAMPLE DESIGNS In this section, two examples are shown. For the first design, both surfaces are concentric about the chief ray. The gains of changing refractive index are very small, so the glass type were assumed to be BK7. This left only two parameters to vary: the distance from the aperture stop to the first surface, and the thickness of the lens. Approximate values were inserted into the lens design software Synopsys, and then changed by hand until the angles of the rays leaving the axicon looked relatively constant. This very simple optimization process still yields acceptable results. For the second design, the surfaces were no longer required to be perfectly concentric, but some variations were allowed. The lens was optimized in Zemax Concentric surfaces If both surfaces are concentric about the center of the entrance pupil, most of the skew aberrations are removed and the axicon will have excellent off-axis properties. This constraint, however, also implies that other properties such as axial intensity distribution or line width will not be optimal. The parameters for such a design are given in Table 1, and some of the resulting convergence angles for different ray heights in Table 2. We note that the convergence angles (tan α.2) are fairly steep for an axicon - often axicons have angles closer to tanα.2. As a consequence, the focal line will be short and narrow, and appear close to the last surface of the lens. Figure 3, showing the line width and the axial intensity distribution, confirms this. The numerical calculations

4 Table 1. Parameters for a lens axicon with concentric surfaces. Radii of curvature R mm R mm Distance to stop d Thickness d Refractive index n Inner stop dia. D 1 1. mm Outer stop dia. D 2 2. mm x 1 3 Intensity [a.u.] 1 5 line width [mm] z [mm] (a) z [mm] (b) Figure 3. (a) Axial intensity produced by the axicon in Table 1. It was calculated using finite raytracing, with the Fresnel diffraction integral applied to the field at the last surface of the lens. (b) Line width of the same axicon, calculated from the stationary-phase approximation in Eq. 1. were done in Matlab using finite raytracing to obtain the optical path, position and angle of a ray as it leaves the last surface. The intensity was then calculated using the Fresnel diffraction integral at a plane in the same place as the last surface, with the optical path giving the phase function. The line width was obtained from the approximation in Eq. 1. The intensity at the end of the line drops to about 25 percent of the intensity at the beginning of the line, and the angle varies by around 4 percent. While still far from an ideal linear or logarithmic axicon, these properties are fully comparable to or better than those of other single-element lens axicons with spherical surfaces presented in the literature. 11 Table 2. Angles for rays leaving the last surface. Entrance height [mm] Exit height [mm] Angle 1.4 o 12.4 o 14.2 o 16. o 17.4 o 18.2 o The main advantage of this design is the off-axis properties. For ordinary diffractive or refractive axicons, illuminating the axicon at even a very small angle will have significant effects on the focal line. For our new design the resulting Bessel beam remains unchanged even for angles of 2 or 3 degrees. In Fig. 4, the resulting transverse intensity distributions are shown, as and 3 degrees incidence for our new design and at and 2 degrees for an ordinary axicon. The intensities are evaluated using the full Fresnel-Kirchoff diffraction integral on data generated by raytracing, since the Fresnel approximation is inaccurate for off-axis calculations. 13 The

5 x (a) x x 1 3 (b) x (c) x (d) Figure 4. (a) Transverse intensity produced by the lens axicon defined in Table 1, at z = 8 mm on the optical axis. It was calculated using finite raytracing, with the Fresnel-Kirchoff diffraction integral applied to the field at the last surface of the lens. (b) Same as in (a), but at 3 o off-axis angle. (c) Transverse intensity produced by a linear axicon of phase function.3kρ, D 1 = 1 mm, and D 2 = 2 mm at z = 25 mm on the optical axis. (d) Same as in (c), but at 2 o off-axis angle. The scale has been altered, compared to the other three graphs, to show all of the astigmatic focus. axicon has been chosen to have the same convergence angle as the new design, and also the sma initial aperture size. Since the telephoto effect places teh focalline closer to the lens axicon than to an ordinary axicon, the distance z has been adjusted to be in the middle of the line for both cases. The new design is much less sensitive to misalignment. Its performance in wide-angle applications is still limited, however, since vignetting will cause problems at wider angles Nearly concentric surfaces If the surfaces are still spherical, but the constraint of keeping them concentric is removed, the axial properties of the line can be improved. Both intensity and line width can be made more uniform with an additional degree of freedom. The trade-off is the off-axis aberrations now introduced. However, they can be kept fairly small if the first surface is kept concentric about entrance pupil centre, and the second nearly so. An example of this is given in Table 3. The resulting axial intensity and line width are shown in Fig. 5. The methods of calculation are identical to those in Fig. 3. As can be seen from the figures, the focal line is now longer, wider and placed further from the axicon, to show that different kinds of lines can be generated using this scheme. (In particular for the nonconcentric designs - for concentric surfaces the focusing angle tends to get steep, in order to avoid vignetting.) The intensity is now more uniform, dropping by 5 percent form the beginning to the end of the line. The convergence angle is also more uniform, varying around 3 percent. For a single-element system constructed from spherical surfaces, this is as far as we know the best axial intensity presented.

6 Table 3. Parameters for a lens axicon with nearly concentric surfaces. Radii of curvature R mm R mm Distance to stop d 14.8 mm Thickness d 1. mm Refractive index n Inner stop dia. D 1 1. mm Outer stop dia. D 2 2. mm 25 x 1 3 Intesnity [a.u.] line width [mm] z [mm] (a) z [mm] (b) Figure 5. (a) Axial intensity and (b) line width for the axicon in Table 3. The methods of calculation are the same as in Fig. 3. The off-axis result is still acceptable, at least at the angles that can be reached without significant vignetting. The resulting transverse intensity distribution is shown in Fig. 6, for the lens axicon and for an ordinary axicon at and 2 degrees incidence angles. The focus produced by the lens still resembles a Bessel beam, whereas for the ordinary axicon it is beginning to take on an asteroid shape. We note, though, that the lens axicon suffers from distortion: the shape of the focus remains the same, but its position is shifted. 4. CONCLUSIONS We have described a general idea for a lens axicon, which despite being a singlet with spherical surfaces will have negative spherical aberration. This is possible by allowing several reflections inside the lens. We have also suggested that by making the surfaces concentric about the chief ray, we can remove the off-axis aberrations and produce narrow focal lines also for oblique illumination. Two actual designs have been presented: one with non-concentric spherical surfaces for the best axial line properties, and one with concentric surfaces for superior off-axis properties. The designs have been evaluated using numerical calculations, based on geometric optics and diffraction analysis. The evaluation shows their intensity distributions to be more uniform, and their off-axis focal lines narrower, than other single-element lens axicons with spherical surfaces. ACKNOWLEDGMENTS This research was supported by Science Foundation Ireland under Grant No. SFI/1/PI.2/B39C.

7 (a) (b) (c) (d) Figure 6. (a) Transverse intensity produced by the lens axicon defined in Table 3, at z = 55 mm on the optical axis. It was calculated using finite raytracing, with the Fresnel diffraction integral applied to the field at the last surface of the lens. (b) Same as (a), but at 2 o off-axis angle. (c) Transverse intensity produced by a linear axicon of phase function.8kρ, D 1 = 1 mm, and D 2 = 2 mm at z = 1 mm on the optical axis. (d) Same as (c), but at 2 o off-axis angle. REFERENCES 1. Z. Jaroszewicz, Axicons: design and properties, Research and Development Treatises, Vol. 5, SPIE Polish Chapter, Warsaw, J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, Non-paraxial design of generalized axicons, Appl. Opt. 31, pp , L. M. Soroko, Axicons and meso-optical imaging devices, in Progress in Optics, Vol. XXVII, E. Wolf, ed., pp , North-Holland, Amsterdam, J. H. McLeod, The axicon: a new type of optical element, J. Opt. Soc. Am. 44, pp , R. B. Gwynn and D. A. Christensen, Method for accurate optical alignment using diffraction rings from lenses with spherical aberration, Appl. Opt. 32, pp , G. Hausler and W. Heckel, Light sectioning with large depth and high resolution, Appl. Opt. 27, pp , Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, High-resolution optical coherence tomography over a large depth range with an axicon lens, Opt. Lett. 27, pp , J. Arlt, T. Hitomi, and K. Dholakia, Atom guiding along laguerre-gaussian and bessel light beams, Appl. Phys. B 71, pp , Z. Jaroszewicz and J. Morales, Lens axicons: systems composed of a diverging aberrated lens and a converging aberrated lens, J. Opt. Soc. Am. A 16, pp , 1999.

8 1. W. H. Steel, Axicons with spherical surfaces, in Optics in metrology, P. Mollet, ed., Colloquia of the International Commission for Optics, pp , A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, A simple lens axicon, Appl. Opt. 43, pp , M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, Refracting system for annular gaussian-to-bessel beam transformation, Appl. Opt. 37, pp , A. Thaning, Z. Jaroscewicz, and A. T. Friberg, Diffractive axicons in oblique illumination: analysis and experiments and comparison with elliptical axicons, Appl. Opt. 42, pp. 9 17, 23.

Lecture 2: Geometrical Optics. Geometrical Approximation. Lenses. Mirrors. Optical Systems. Images and Pupils. Aberrations.

Lecture 2: Geometrical Optics. Geometrical Approximation. Lenses. Mirrors. Optical Systems. Images and Pupils. Aberrations. Lecture 2: Geometrical Optics Outline 1 Geometrical Approximation 2 Lenses 3 Mirrors 4 Optical Systems 5 Images and Pupils 6 Aberrations Christoph U. Keller, Leiden Observatory, keller@strw.leidenuniv.nl