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Abstract
An X-ray parabolic refractive axicon, as novel type of beam-shaping optical element, consisting of a refractive lens with a form of the surface represented as a parabolic cone, is proposed and demonstrated. Under coherent X-ray illumination, the axicon generates Bessel-like beam along the optical axis and ring-shaped beam at the imaging distance. A theoretical analysis of the beams formation by the axicons was carried out and corresponding computer simulations were performed. The proposed parabolic axicon was experimentally tested at ID06 ESRF beamline in the X-ray energy range from 10 to 15 keV. Experimentally recorded images of shaped beams are in a good agreement with computer calculations. Future possible applications of the axicon are discussed.
© 2017 Optical Society of America
1. Introduction
The continuous development of X-ray sources, such as 3rd generation synchrotrons and free electron lasers, has resulted in a dramatic increase of coherence properties of a photon beam especially in the hard X-ray region. The increasing brightness of X-ray sources enabled a new generation of techniques that had revolutionary impact across a broad area of science. The high coherent flux facilitated the development of a number of new experimental methods based on coherence and expanded the application area of existing techniques such as phase contrast imaging, X-ray photon correlation spectroscopy, coherent diffraction imaging [1–3].
The new generation of X-ray optics was developed to fully utilize these beams of unprecedented brightness, amongst which the youngest X-ray refractive optics progressed at a remarkable pace [4,5]. The field of applications of refractive optics is not limited to beam conditioning, but can be extended into the area of Fourier optics, as well as coherent diffraction microscopy and interferometry [6–16].
Due to the dramatic improvement in manufacturing and preparation techniques, all types of X-ray focusing optics: zone plates, refractive lenses and mirrors reached 10 to 50 nm spatial resolution. Nevertheless, until now the design of new optical devices has not proceeded far beyond simple focusing optical elements. It was an attempt to use diffractive optical elements (DOEs) to perform apart from focusing more complex optical functions in the X-ray regime [17,18]. Recently also, based on DOEs beam-shaping condenser lenses for full-field transmission X-ray microscopes were proposed [19,20]. However, since many unknown parameters of the beam have to be defined in advance, the design beam-shaper optical elements is usually challenging and in addition there are some restrictions owing to the fabrication limitations.
It is known in visible light optics that some beam-shaping elements can be performed using refractive lenses having a special shape. To illustrate, in laser optics there is an extensive class of elements having axial symmetry which are able to convert the incident beam with high spatial coherence to a narrow axial straight line segment. These optical elements are called axicones [21–23]. They create a non-diffractive Bessel beam in the near field, and transform the light into a ring in the far field. Axicons are used for variety of applications [24–28]. A focused laser light to an annular-shaped beam, can be used in hole drilling, microscopy and medical applications. The Bessel beams generated by axicon lenses in the near field can be used in applications for guiding atoms or molecules, in laser processing and for generating plasma in linear accelerators.
In this paper we present, for the first time, an X-ray parabolic refractive axicon consisting of a refractive lens with a form of the surface represented as a parabolic cone. Parabolic refractive axicon is a clear demonstration of X-ray beam shaping element, which allow to observe not only the Bessel beam but also obtain a ring-shaped focused beam. The beam propagation through the proposed axicon lens was described theoretically and computer simulations were performed. The optical properties of the axicon were studied experimentally in the X-ray energy range from 10 to 15 keV.
2. Theory
The profile of a parabolic axicon refractive surface is shown in Fig. 1(a). It consists of two parabolic branches with displacement of their apexes relative to the optical axis by a distance b. Both parabolic branches have the same radius of curvature R at their apexes and we will define it as effective radius of curvature of the axicon. The equation of the refractive surface in the cylindrical coordinate system can be written as:
This equation contains a quadratic term that describes the shape of an ordinary parabolic lens cavity with a radius of curvature R, and also a linear term which corresponds to a cone surface. It is obvious that the considered conical parabolic form of refractive surface is characterized by transformations of incident radiation, generated by both the parabolic and the conical surfaces.
Fig. 1 (a) Schematic construction of the profile of a parabolic axicon refractive surface. (b) Cross-section of the single parabolic axicon lens.
We consider an X-ray compound parabolic axicon lens consisting of several single axicons, each of them is a biconcave lens with two conical parabolic cavities [4]. The cross-section of the axicon lens is presented in Fig. 1(b), where d is the distance between apexes of axicon cavities and 2R0 is axicon physical aperture.
In the frame of the geometrical optics approximation the parabolic axicon lens focuses incident parallel X-rays into a ring-shaped beam with a diameter 2b at a distance f. The f value corresponds to the focal length of the parabolic refractive lens with the radius of curvature at parabola apex R. The ray propagation through the parabolic axicon lens is shown in Fig. 2(a). In the focusing position, the width w of the ring is diffraction limited and the angular dependence of the intensity distribution is missing due to the axial symmetry of the considered optical system. The typical pattern of the ring-shaped beam in the focusing position and its cross-section are presented in Fig. 2(b). With the increase of the distance z, the ring diameter is linearly increasing and the ring becomes more blurred.
Fig. 2 Schematic view of the shaped beam produced by X-ray parabolic axicon lens. (a) The beam propagation through the axicon lens and the intensity distribution along the optical axis. Transformed beam forms an interference area in the near field and a ring area in far field. (b) The intensity distribution of the ring-shaped beam in the focusing position f and its cross-section. (c) The typical pattern of the Bessel beam produced at the interference area and its cross-section.
What is more, the parabolic axicon illuminated by coherent beam forms an interference area in the region situated between the lens and its focus position. Therefore let us look closely at the intensity distribution in this area. Consider the plane monochromatic X-ray radiation incident on the conical parabolic lens. To describe the propagation of the X-rays wave in the interference area we use the Fresnel diffraction integral for the axially symmetric case:
where F is the value proportional to the electric field intensity, r is radial coordinate at lens plane, ρ is the radial coordinate of the considered point in the observation plane, z is the distance from the lens to the observation plane, k = 2π / λ is the wave number, where λ is the wave length of monochromatic radiation and J0(x) is the zero order Bessel function of the first kind. The multiplier τ(r) is the transmission function of a compound parabolic axicon lens, and in the paraxial approximation it can be written as follows:where N is the number of single parabolic axicon lenses, δ is the refractive index decrement and µ is the total linear attenuation coefficient.The approximate analytical solution of the integral can be found by using the stationary phase method, and wherein the intensity distribution in the interference area will be as:
where f = R / 2Nδ is the focal length of the compound axicon lens and I0(z) is the intensity distribution along the optical axis. The intensity distribution in the observation plane perpendicular to the optical axis is described by the square of the Bessel function and depends on the coordinate z. The typical pattern in this plain and its cross-section are presented in Fig. 2(c). The estimation of the central spot size can be obtained by equating the argument of the Bessel function from the Eq. (4) to the value of its first minimum which is equal to 2.4:where dρ(z) is the diameter of the central part of the interference pattern on the distance z from the axicon lens. Thus, the parabolic axicon generates a Bessel beam which size is decreasing with the distance z, forming a bright convergent axial straight line segment. Such convergence of the wave front allows us to talk about its “diffraction-free” properties since natural diffraction loss of energy from the central part of the beam compensated by the supply of the distributed lateral radiation.The intensity distribution I0(z) along the optical axis is described by the following equation:
The typical appearance of this curve is shown in Fig. 2(a). The peak position zm depends on the value μb2 / fδ. So if it is equal to one than the position of maximum intensity is half of the focal length f, if the value exceeds one then the peak position shifts to the lens, and while if the value is less than one then the peak position closes to the focus.3. Manufacturing and experiment
The parabolic axicon is made from polycrystalline aluminium by a pressing technique. The pressing tools consist of two convex parabolic cones facing each other and guided in the centring ring. The aluminium blank of 5 mm diameter and 1 mm thick in which the conical parabolic cavities are to be pressed is tightly held into the centring ring. The lens profile is simultaneously pressed into the aluminium from both sides [29]. The sketch of the manufactured axicon lens is presented in Fig. 1(b). The effective radius of curvature R of the axicon lens is 50 μm, and the displacement of parabola branches b is equal to 23 μm. The aperture of the axicon lens 2R0 is 380 μm, the distance between the cavities apexes d is about 100 μm.
The experimental tests of parabolic axicon were carried out at the Micro Optics Test Bench (MOTB) at the ID06 ESRF beamline. The beam was produced by an in-vacuum undulator and desired X-ray energies were selected by a cryogenically cooled Si (111) double crystal fixed exit monochromator. To reduce the influence of the third undulator harmonic, the second crystal of the monochromator was slightly detuned from the Bragg position. The detuning angle was around 10 µrad (2 arc. sec.) to provide the remaining fundamental harmonic flux of about 80%.
The typical source size at the ID06 beamline is 40 μm vertically and 900 μm horizontally. To minimize the blurring of the beam generated by parabolic axicon, occurring due to the large horizontal source size, at a distance of 30 meters upstream from the lenses 30 μm horizontal slit was placed. Two parabolic axicon lenses (4 refractive surfaces) were located on the stage with all necessary translations and rotations at 58 m distance from the source (see Fig. 3). To record images of the Bessel beam formed by the axicon lens in the field of interference, we used high-resolution CCD camera equipped with the fluorescence screen and optical objective providing a spatial resolution of about 1.5 μm (pixel size of 0.73 μm). To observe the images in the ring area, we utilized the sensitive FReLoN CCD camera with a pixel size of 1.5 μm. Moving cameras along the optical axis, we recorded images representing the beam intensity distribution at different distances from the lenses ranging from 0.5 to 5 m covering the energy range from 10 to 15keV.
Fig. 3 Experimental layout for the test of X-ray parabolic axicon lenses. The axicon lenses are illuminated by synchrotron radiation and generate shaped beam registered by CCD detector.
To demonstrate the behaviour of the Bessel beams in the interference area we present experimental data taken at 10 keV incident X-ray radiation. This selected energy allowed us to observe the most interesting area of the Bessel beam. The length of the interference area l (see Fig. 2(a)), was calculated using the Eq. (6), and it was 2 meters. A series of Bessel beam images were experimentally recorded at a distance varying from 0.5 to 1.7 meters from the axicon lense. Figure 4 depicts the image of the Bessel beam in the interference area obtained at a distance of 1.4 meters from the axicon. The transverse intensity distribution of the Bessel beam is a set of concentric rings Fig. 4(a). The vertical cross-section of the intensity distribution across the beam is presented in Fig. 4(b). The radial intensity profile is well described by the Eq. (4), which considering the distance L from the source to the lenses can be expressed as follows:
where A = 1 + fz / L(f – z) is a correction factor that describes the scaled increase of the interference pattern that arises when the source approaching the lens. The calculated intensity variation is shown in Fig. 4(b), and it is nicely coincides with the experimental one. Taking into account the correction factor A, the theoretical size of the central part of the Bessel beam dρ was calculated from the Eq. (5) and was 3.9 μm.Fig. 4 The Bessel beam generated by X-ray parabolic axicon lense in interference area, recorded with 10 keV X-rays at 1.4 m from the lens. (a) Transverse intensity distribution of the Bessel beam (inverted grayscale). (b) Intensity profile obtained for the vertical cross-section through the centre of the Bessel beam pattern.
All other images of the Bessel beam obtained in the interference area are consistent with the theoretical description given by Eq. (7). Increasing the distance between the camera and parabolic axicon leads to compression of the interference rings. A maximum intensity of the central part of the beam is observed at a distance of 1.6 meters, which is very close to the theoretical value zm of 1.62 meters calculated by using the Eq. (6). The diameter of the central part of the Bessel beam at this distance was 3 μm. We would like to stress that central part was not exceeded the value of 8 μm during the entire observation area. As for the energy distribution, for all experimentally obtained Bessel beam patterns, the intensity of the central peak is more than six times greater than the intensity of the first fringe. This ratio is in good agreement with the theoretical intensity distribution. It should be noted that the energy density of the central peak contains of 65% of the total intensity.
To reveal the formation of the annular intensity distribution, the measurements at X-ray energy of 12.4 keV are presented. Images of the ring-shaped beams were recorded at distances between the detector and axicon lens ranging from 3.5 to 4.0 meters. The ring shape focusing was observed at a distance of 3.76 meters, and Fig. 5(a) presents an image of the focusing ring. This distance corresponds to the theoretical value for the axicon imaging distance, calculated according to the thin lens formula. The elliptical shape of the ring is due to the use of a horizontal slit forming a secondary source, which was located at a distance of about 30 meters from the lenses. Obviously, this disturbs the axial symmetry of the optical setup and leads to the defocusing of the ring image in the horizontal direction, also making nonuniform angular intensity distribution. The cross-section of the intensity distribution in vertical and horizontal directions is shown in Fig. 5(b) and 5(c) respectively. The ellipse size in the vertical axis was 48.5 ± 2.5 μm, which is consistent with the corresponding theoretical value of 49 μm. The width of the focusing ring w was 5.5 ± 1 μm (FWHM) that is 1.5 times larger than the theoretical value. The broadening of the width of the ring-shaped beam, as well as the low residual intensity, can be attributed to the slightly aberrated profile of the lens and the scattering by the inhomogeneities of the material of the axicon lens. Closing both horizontal and vertical slits up to 30 μm, we were able to get a sharp image of the ring with a more uniform angular structure that is depicted in Fig. 5(d). We would like to note that the observed ellipse is slightly inclined, which is due to the tilt of the electron beam in the storage ring. This was proven by numerical simulations of the focusing ring at different inclination angles, and in addition it was estimated that beam tilt is around 8°.
Fig. 5 The ring-shaped beam generated by X-ray parabolic axicon lens in the ring area, recorded with 12.4 keV X-rays. (a) Transverse intensity distribution of the beam with elliptical shape obtained while horizontal slit was closed to 30 μm (inverted grayscale). The intensity profiles along the (b) vertical major and (c) horizontal minor axes of the elliptical form of the beam. (d) Transverse intensity distribution of the ring-shaped beam obtained at the focusing position with secondary point-like source generated by horizontal and vertical slits (inverted grayscale).
4. Discussion and conclusions
The X-ray refractive parabolic axicon proposed in this paper is a clear demonstration of beam-shaping elements operating in the high X-ray energy domain. The axicon utilizes the most outstanding properties of synchrotron radiation such as brightness, monochromaticity and coherence. The optical properties of the axicon were studied theoretically and computer simulations were performed. Parabolic refractive axicon lens was manufactured and experimentally tested. It was demonstrated that the X-ray parabolic axicon can generate Bessel-like beam in the interference area along the optical axis and ring-shaped beam at the imaging distance.
The Bessel beam generated by X-ray parabolic axicon form a transverse pattern which is convergent with beam propagation and therefore is non-diffracting. Our analytical calculation reveals that the theoretical transverse dimension of the central part of the Bessel beam is about 1.6 times smaller than the size of the focal spot obtained by the ordinary X-ray parabolic refractive lens with the same numerical aperture. This allows us to consider the parabolic refractive axicon as a focusing optical element with the same focusing properties as the refractive lens, but having a much more extended depth of focus. While the efficiency of the axicon focusing is slightly lower than parabolic refractive lens they probably can be successfully used in areas requiring extended focused beams, for instance in some metrological, diffraction and imaging applications.
In addition, the Bessel beam is “self-healing” due to the conical part of the beam wavefront, meaning that the beam can be obstructed as it moves on beyond the obstacles, but will reform at some point further down the beam axis. Based on this feature, optical schemes with multiple narrow Bessel beams will allow to realize a specific illumination which might be applied, for example, for medical applications.
The tunability of the annular beam generated by the axicon from the sharp focus ring to the defocused far field image might be useful in applications requiring special illumination. For instance, it might substantially simplify the manipulation with beam stops in X-ray small-angle scattering experiments or in X-ray dark-field imaging techniques.
The ring-shaped beam also appears to be effective replacement of an input annular aperture in Zernike phase contrast X-ray microscopy [30]. The traditionally used annular aperture significantly reduces optical throughput and, in addition, generates X-ray diffraction effects. For a hard X-ray regime, the aperture should be fairly thick to absorb incident radiation, which in turn makes it difficult to produce and requires special alignment. The annular illumination produced by axicon will allow maintaining the brightness of the incident high X-ray radiation and avoiding diffraction effects.
It is worth noting, the combination of parabolic axicon lens with other X-ray optical elements can lead to completely new optical schemes. For example, an annular beam can significantly improve the performance of elliptical capillaries for focusing and imaging applications [31]. The axicons can also be used for harmonic rejection, but in contrast to the off-axis illumination scheme proposed in [32], in the case of axicon the full lens aperture can be employed.
It was also experimentally demonstrated that the shape and symmetry of the resulting beam image strongly depends on the astigmatism of the source and collimation of the beam by additional slits. This feature allows us to consider the parabolic axicon as a sensitive tool for source diagnostics and beamline alignment.
The very least that can be said about parabolic axicon is that they will be used to the full extent on the X-ray free electron lasers for ultrafast and nonlinear X-ray optics studies.
Funding
Ministry of Education and Science of the Russian Federation (Nº 14.Y26.31.0002).
Acknowledgment
The authors are very grateful to C. Detlefs and P. Wattecamps for their help and support during the experiments at ID 06 beamline. The authors thank A. Borisov for the helpful discussions.
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