Producing focused extreme ultraviolet vortex with Fermat-spiral photon sieves

Abstract

Extreme ultraviolet (EUV) light is difficult to focus due to strong absorption of most materials. Photon sieves (PS), rather than Fresnel zone plates (FZP), can focus EUV to smaller spot and suppress the higher orders of secondary maxima by several orders of magnitude. The number of pinholes used in PS is far more than that of transparent rings used in FZP, providing a great flexibility to manipulate structured focusing in EUV. In this work we investigate the Fermat-spiral PS to produce focused vortices with different topological charges. Experiment at the wavelength of 46.9 nm is carried out and multi-planar coherent diffractive imaging is used to retrieve the phase map of the focused EUV vortices. These results show the enormous potential of PS for manipulating EUV light. This study not only provides a compact, affordable substitute to focusing vortices where transmissive optics materials are unavailable, but also provides a route of converting various complex light manipulation ranging from visible light to EUV and soft x-ray.

Introduction

Extreme ultraviolet (EUV) radiation has a wavelength ranging from 10 nm to 121 nm, corresponding to photon energies of 124ev to 10.2ev respectively. Shorter wavelength means higher resolution in focusing and imaging. Thus EUV radiation is attractive for high-resolution imaging and nanometer lithography [1]. Meanwhile, EUV radiation has higher photon energies and deeper penetration depths than electrons, allowing for a nano-scale view into intricate three-dimensional structures, which are important for materials science [2, 3], atto-second metrology [4], and semiconductor characterization. Fortunately, a variety of EUV sources, such as high-harmonic generation (HHG) [5], free-electron lasers (FELs) [6], capillary discharge lasers [7, 8] and solid-state EUV source, have been developed during the past few decades, opening up the possibility of light-matter interactions on the nanometer scale by means of structured EUV illumination [9,10,11]. Unfortunately, focusing and imaging remains challenging due to the strong absorption of most materials in the EUV regime.

Over the past decade, EUV focusing has undergone a transformation, ranging from reflective and refractive optics to concentrators in the form of diffractive devices [12]. Strong absorption makes it difficult to deploy complicated fine refractive lenses under practical conditions. Diffractive elements, by contrast, become more attractive when the incident wavelengths range from EUV radiation to soft X-rays. To date, focused EUV beams have been achieved primarily with a specialized Fresnel zone plate (FZP). Contrarily, photon sieves (PS) perform better than FZP in terms of producing smaller focal spots and suppression of secondary maxima and higher diffraction orders [13]. Following these works, several theoretical and experimental studies were carried out in visible light to determine the parameters of individual nanostructured holes that generate complex structured light fields at a targeted plane.

One of the most relevant structured light beams are those carrying orbital angular momentum (OAM), also known as vortex beams, which are appealing for application such as in quantum optical communications, particle manipulation [14], lithography [15], super-resolution imaging and so on [16]. Recent advances in high harmonic generation (HHG) have been reported to produce OAM beams in the EUV radiation or x-rays [17,18,19,20]. Spurred by these exciting technologies, parallel interest in the ability to focus and manipulate the OAM of ultrafast light pulses has also emerged. However, for the same reason, all fields that use EUV radiation are encumbered by handing problems that arise from the strong absorption. Until now, there is a lack of reports on producing EUV radiation focus, let alone focused EUV field carrying complex OAM. The capability of producing focused EUV vortex optical field with arbitrary topological charges may spur promising applications including the inspection of dynamics of non-homogeneous molecular systems at the nanometric, optical trapping and super-resolution imaging.

Inspired by the prior works in the visible regime [21,22,23,24,25,26,27], here we for the first time, to the best of our knowledge, present the Fermat-spiral PS to realize focused vortices with different topological charges. Focusing is critical for a host of applications in imaging and spectroscopy, as well as for enhancing our ability to optically manipulate macro- to nano-scale objects such as particles, molecules, atoms and electrons. The Fermat-spiral PS with focusing vortices was successfully obtained by genetic optimization algorithm [28], and then its focusing properties, including intensity and phase maps, were measured by means of multi-planar phase retrieval algorithm [29]. The results show that for the first two focused vortices with topological charges of -1 and +3 has the full width at half maximum (FWHM) of the dark ring of 260 nm and 742 nm, respectively. They are all in good agreement with the theory. This research offers a solution to the challenging problem encountered with EUV radiation, which is the lack of suitable optical devices for focusing and controlling light fields. We believe the current work is an important step forward for focusing and manipulating extreme short-wave optical fields, which may have great potential for applications in material science and lithography.

Focused EUV vortex with Fermat-spiral PS

To achieve the focused Vortex in EUV, a Fermat-spiral PS serves as the primary optical component that controls the amplitude and phase of the incident EUV light. The implementation of this concept is shown in Fig. 1a. The incident plane wave is modulated by Fermat-spiral PS and then propagates to the focal plane. Actually, the diffraction field at the focal plane is the coherent superposition of individual diffraction field of each pinhole, which can be calculated by Fresnel-Kirchhoff diffraction formula [13].

$$\begin{aligned} \text {E}(x,y) = -i\frac{A}{2\lambda } \int \nolimits _{holes} \frac{e^{ik(r + s)}}{rs} (\cos \delta r + \cos \delta s)dS \end{aligned}$$

(1)

where E(x,y) is the diffraction field of any given pinhole at the focal plane, A is the amplitude at unit distance from the source and the integral is performed over the pinholes, $\lambda$ is the incident wavelength, r is the distance from the light source to the given pinhole, s is the distance from the given pinhole to the focal spot, $\delta r$ is the incident angle between the incident light and the normal of the given pinhole, $\delta s$ is the diffracted angle between the diffracted light and the normal of the given pinhole. Figure 1b and c indicate the SEM images of Fermat-spiral PS with topological charges of -1 and +3, respectively.

Fig. 1

a Sketch of focused OAM. b and c SEM images of Fermat-spiral PS with topological charge of -1 and +3, respectively. d Flowchart of genetic algorithm to optimize Fermat-spiral PS (see (e)) so as to produce focused OAM

Obviously, the intensity and phase maps of the focused Vortex is determined by the diameter and the position coordinate of each pinhole. Here the Fermat-spiral PS achieves OAM by varying the radial distances of the pinholes with a constant azimuthal increment. In order to determine exactly the position coordinates and the sizes of those transparent pinholes, genetic algorithm is applied to optimize the Fermat-spiral PS [28]. During the optimization process, the unknown parameters $R_n$, $R_n$ means the radius of the nth ring, and its corresponding pinhole radius $a_n$. The pinhole number $M_n$ in the ring $R_n$ can be evaluated by $Mn\mathrm{{ = }}\frac{{2\pi Rn}}{{ln}}$, where $l_n$ is the arc length between the centers of two neighboring pinholes. Taking into account structured vortex beam in the center surrounded by a series of azimuthally splitting focal spots, in this case a standard Fermat spiral can be chosen as the initial structure to speed up the convergence and improve the design precision of the optical structure. Figure 1d shows the flow chart of the genetic algorithm and Fig. 1e presents the SEM image of Fermat-spiral PS with focused OAM.

Experimental results and discussions

The experimental setup of Fermat-spiral PS focusing is shown in Fig. 2a. The EUV radiation is a capillary discharge 46.9 nm laser with an output energy of 500$\mu$J operating at the single-shot mode [30, 31]. The recording medium, a 200nm thick layer of PMMA (polymethyl methacrylate, 950000 molecular weight) coated on a silicon wafer, is used to record the focal spots and is fixed on a two-dimensional stage to realize z-scan along the beam propagation. Under the condition of single-shot exposure, the energy of the single pulse laser that deposited on PMMA surface is below 1$\mu$J. Each position was exposed by single laser pulse with a dose in the region of the linear response of PMMA. Consequently, the pattern height printed in PMMA surface was equivalent to the focusing light intensity. After the exposure, PMMA was developed by 9 minutes and stopped by 30 seconds, then dried using nitrogen. The processed PMMA targets are detected by atomic force microscopy (AFM).

Fig. 2

Sketch of the experiment setup for focused OAM and recorded by PMMA (a), detection of AFM (b), (c) and (d) the intensity and phase distribution of the focused OAM with topological charge of -1; e diagonal intensity distribution of experimental results versus the theory; f, g the intensity and phase distribution of the focused OAM with topological charge of +3; h diagonal intensity distribution of experimental results versus the theory

To better understand the focused vortex, two Fermat-spiral PSs with topological charge of -1 and +3 are presented and verified by EUV focusing experiment. The corresponding intensity and phase maps of the two foci are separately displayed in the left and middle columns in Fig. 2. Figure 2d and g show the recovered phase map by use of multi-planar coherent diffractive imaging (CDI) [29]. Here, in our experiment configuration, once the three recording distances of diffractogram are obtained, multi-planar CDI is adopted to retrieve the phase information of light field. Concretely, to speed up the convergence rate, the theoretical phase map at the focal plane can be set as the initial guess phase. Therefore, dozens of iterations are enough to reconstruct the wavefront of the object. The details of phase reconstruction process also can be seen in the Supplementary document. Obviously, there is a helical phase in the center of the focused OAM. Figure 2e and h depict the diagonal intensity distributions corresponding to the experimental and theoretical results, respectively. For focused vortices with topological charge of -1 and +3, the experimental results show that the full width at half maximum (FWHM) of the dark ring is 260 nm and 742 nm, respectively. The experimental results indicate that the Fermat-spiral PS can be used to focus the EUV light. The above experimental results demonstrate that our light manipulation on EUV radiation provides opportunities for applications such as nanoscale structure focusing, extreme ultraviolet phase shifters, nano-imaging, and of course, extreme ultraviolet interferometry and diffractometry.

Furthermore, the use of photon sieve offers extremely high degree of freedom in the design to produce more customized focal fields, which may be highly desirable in applications such as nanometer structured lithography, interaction between complex structured light and matter, multichannel particle acceleration and so on. Here, as an example, we demonstrate an anamorphic Fermat spiral to generate focused EUV vortex field dressed with surrounding petals. In this case, we investigated the advantage of using genetic algorithm to optimize the Fermat-spiral PS to produce structured-focusing EUV OAM. Figure 3a-c present three frames of diffractograms from pre- to post-focus position. In order to obtain the real record position in the experiment, correlation coefficient can be calculated via the comparison between the recorded diffractogram and a series of theoretical diffractograms, as shown in Fig. 3d. The centroid corresponding to the maximum correlation coefficient is considered as the real centroid. Thus, the maximum correlation coefficient is 0.94 and the recording distance is plus 5.2 $\mu$m derived from the focal plane. Similarly, the maximum correlation coefficient is 0.92 and the recording distance is minus 24 $\mu$m derived from the focal plane for Fig. 3a. For Fig. 3c, the maximum correlation coefficient is 0.93 and the recording distance is plus 32 $\mu$m derived from the focal plane. Once the record distances are found, multi-planar CDI can be performed to recover the distribution of phase information. Figure 3f shows the recovered phase map corresponding to the square block diagram in Fig. 3e. Obviously, there is also a helical phase in the center. For Fig. 3e, the base width of the dark ring is 467 nm and the base width of the doughnut is about 1.091 $\mu$m, while the base width of the dark ring is 428 nm and the base width of the doughnut is 1.013 $\mu$m in theory, as shown in Fig. 3g. All details can be seen in the Supplementary ducument.

Fig. 3

Experiment on the focused EUV OAM with high degree of freedom, a prior to focus, b near focus, c after focus, d Correlation coefficient curve between the (a)-(c) and a series of theoretical diffractograms from minus 40 $\mu$m to plus 40 $\mu$m derived from the focal plane, the red line, green line and blue lines correspond to the correlation coefficients curves in (a), (b) and (c), respectively. f Recovered phase map corresponds to the square block diagram in (e). g Diagonal intensity distribution of experimental results versus the theory

Conclusions

In summary, we present the design, fabrication and testing of Fermat-spiral PS for producing focused EUV vortices. The experimental results are in good agreements with the theoretical analysis. The phase distributions of focused EUV vortices with different topological charges are successfully recovered by multi-planar CDI and the central helical phase structures are revealed. Compared with FZP, PS offers several other distinctive advantages. Firstly, PS device can overcome the limitation of spatial resolution determined by the outmost width of the zone. Secondly, compared with the mono-focal FZP, PS with hundreds of thousands of transparent pinholes with different sizes, shapes and position coordinates, offers potential to realize much more complex structured focus such as multi-foci, which may be applied for interferometry and diffractometry in the region of EUV and soft x-ray. Thirdly, the free-standing PS is self-supporting and its fabrication is much easier than supported FZP. Note that as one kind of amplitude-only diffractive element, the diffraction efficiency of the optimized PS is approximately 7$\%$. To further improve the diffraction efficiency, metasurface device [12] may be employed to couple the EUV light to the multi-functional PS. PS with hundreds of thousands of transparent pinholes whose sizes, shapes and position coordinates offers large number of degree of freedom in the design that can be optimized to meet the requirement of various light manipulations ranging from the visible light to EUV and X-rays.