Open Access
Add to BibSonomy
Abstract
The nano-focusing performance of adiabatically designed Cartesian oval refocusing lenses is compared with other well known compound refractive lenses with parabolic profiles (both simple concave and kinoform types). Using beam propagation method (BPM) simulation, it is shown that our design based on oval lenses does significantly improve the focusing properties compared to other parabolic lens based designs, e.g. adiabatically focusing lenses (AFLs), which doesn’t take into account of the refocusing effect. This points to the importance of optimizing complex lens design in improving nano-focusing lens performance.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microanalysis requires small and intensive x-ray micro or nano size beams, thus generally optical devices for micro or nano focusing are needed. The common methods to achieve hard X-ray focusing are reflection [1], refraction [2–4], and diffraction [5,6]. Among them, the refractive lens has received wide attention due to its many excellent performances [7]. Compound refractive lenses (CRLs) are the standard means of generating small X-ray focal spots, the first CRL was fabricated with a circular lens profile in 1996 [8]. X-ray is focused by arranging a series of lenses that have a weak refractive function in a row to achieve a short focal length. Then CRL with a parabolic profile was proposed in 1999 [9], after that the planar nanofocusing parabolic refractive lenses (NFLs) were fabricated and the focal size of tens of nanometers was achieved [10,11]. X-ray refractive planar parabolic lenses with minimized absorption were fabricated in 2000 [12], then the elliptical profile was used to fabricate planar single-element elliptical kinoform lenses in 2003 [13]. It is shown that for an incoming monochromatic plane wave incident from the plane side of the lenses, the ellipse profile is the ideal shape for a single-element refractive lens.
Adiabtically nanofocusing lens was proposed in 2005 [2] and it has been realized [4]. This lens design outperforms the NFLs. But the aberration issue exists since most of the lenses are focusing on already converging X-ray rays. When the number of lenses is big, the effect on final focusing is non-neglectable.
Sanchez del Rio and Alianelli summed up some conclusions about aberration-free surfaces for focusing a parallel X-ray beam: from the material into the air, an elliptical surface is applicable; and from the air to the material, a hyperbolic surface is applicable [14]. But for CRLs, the surface profiles should be able to focus X-ray from point to point to achieve an aberration-free focusing for the already converging beam. This kind of ‘ideal cartesian oval’ shape was proposed in 2017 and was used to design compound refractive lenses [15,16]. Based on this shape we optimize the design of the compound refractive lenses which could focus x-ray precisely with no aberration of geometrical optics, we call this kind of compound lenses ‘OVAL’ for short in this article.
The simulation method chosen in this paper is beam propagation method (BPM) [17]. BPM is widely used to analyze the edge diffraction effect of X-ray optical elements [18]. It naturally takes into account the diffraction, refraction and absorption effect of X-ray passing substances and through their interfaces, and can simply simulate optical elements with complex shapes through the distribution of susceptibility in space. It is suitable to calculate the wave field in x-ray optics with an arbitrary structure in the forward propagation scheme usually in case of small numerical aperture (NA) or under paraxial approximation. BPM is a numerical method for solving Helmholtz equation, and its computational complexity is much lower than other numerical solutions. BPM is widely used in the simulation of optical waveguides [19], other photonic devices and even photonic chips (such as BeamPRO developed by Synopsys). The actual process parameters can be easily added to the BPM simulation, which makes it easy to estimate the effects of roughness, profile errors, and so on the actual light propagation. Hanfei conducts a comparative study on short and long kinoform lenses (KLs) using the geometrical theory, the dynamical diffraction theory, and the beam-propagation method [20]. The applicability and limitations of each method are discussed. He showed that the result predicted by the geometrical theory deviates quickly from the one calculated by the more rigorous fullwave theories as the NA increases.
KLs received increasing attention for x-ray nanofocusing due to their ability of bending incoming photons with minimum absorption [12,21,22]. Short KL usually outperforms long KL (LKL) in terms of efficiency and focal size because they are less affected by the edge diffraction effect. However, LKL has only one focus, a great advantage compared to their short counterparts [20]. Due to the existence of edge diffraction in a LKL, the wavefront distorts, this can be reduced by changing the shape design of the lens. A smaller focus size and higher intensity gain can thus be achieved.
In this article, we simulate the focusing effect of adiabatically designed OVAL by BPM, the results show that the intensity gain and full width at half maximum (FWHM) of the focus improved significantly, and the position of the focus matches the geometrical design very well.
We found that OVAL can have a higher intensity gain and smaller FWHM than other CRLs at similar working distances, but the focal depths of the two are similar. Through LKL processing of OVAL, we find that the OVAL LKL designed in this paper reduces the wave field distortion and gets smaller focus and focal depth, and the focal plane position is consistent with the theoretical position.
2. Kinoform lenses
2.1 Beam propagation method
As X-rays travel through the material, the refractive index is
where δ is the difference of the refractive index from one and β corresponds to absorption. Magnetic susceptibility can be written as [20]Therefore ${\chi / 2} = n - 1$. For a plane-concave lens with focal length f, if ignoring the surface refraction, the relationship between the thickness of the lens y and the radius of the lens hole x can be expressed as
When dealing with the theoretical studies of KLs, the framework of geometric optics is not applicable, because this framework does not take into account the diffraction effect inside the optics, and it will fail when the thickness of the lens is thick or the diffraction element of the lens is small. Therefore, it is necessary to use the full wave theory to carry out theoretical studies on KLs. Hanfei Yan shows that the result predicted by the geometrical theory deviates quickly from the one calculated by the more rigorous full wave theories as the NA increases [20]. BPM is based on wave theory and can deal with the diffraction effect of optical components very well.
When the BPM is used, the paraxial approximation shall be satisfied, and then the space shall be sliced. The thickness of the slice shall be as small as possible. The incident wave propagates from one slice to the next slice and finally goes through the complete slicing. Think of the lens space as composed of many thin slices with small thickness Δz. The incident wave on the surface before the slice is $\psi ({x,y,z} )$, and then the output wave $\psi ({x,y,z + \Delta z} )$ on the surface after the slice can be calculated by using BPM.
2.2 Kinoform lenses with different m
The ideal length of the Kinoform lens (KL) is written as
where N is the number of steps, m is an integer, m*2π is the phase shift X-ray undergoes when passing the lens compared to air. For fixed X-ray energy, the length of the kinoform is decided by the product of m and N. Keeping the lens length fixed while reducing m to as small as possible obviously helps reduce the absorption and increase the valid aperture of the lens and helps improve the focus (Fig. 1).For example, we use the BPM to investigate the characteristic of a LKL of Si with a focal length of 10 mm and a radius of 8.11 µm. It is operated at the energy of 9.04 KeV and has m equal to 1, 2, 3, 4, 6, 8 and 12, respectively. And at the same time, the corresponding N is equal to 24, 12, 8, 6, 4, 3 and 2, so that each lens has a thickness of 554.33 µm along the optical axis.
Figure 1 shows the simulation results, we plot the intensity distribution of the focal plane when m is equal to 1, 2, 3, 4, 6, 8 and 12. At the focal plane position, we can see that intensity decreases and FWHM increases as m increases. This means that if we want a smaller focus with a higher intensity gain, we should choose a smaller value of m. Figure 2 shows two typical kinoform lens profiles in this case.
The simulation results of LKL with m = 1 and 6 are shown in Fig. 3 with the spatial distribution of intensity after LKL plotted (the position of the entrance plane of the lens is set as z = 0). Compared with short KL, all elements of LKL are not located in the same plane, so the wave propagation distance is longer and the wave distortion is more serious. The transmitted wave will converge to a different position from that of f, resulting in larger aberration and offset of its focus and the formation of a larger focus. Figure 3 shows that there is only one focus of LKL, and the focus is not on f, but on the position before f. This is because the wave at the step edge of LKL will interfere, resulting in distortion of the wave propagating further. We can see visually that the intensity of m = 1 is higher than that of m = 6 and the FWHM of m = 1 is less than that of m = 6. This is because the absorption of LKL decreases as m decreases.
So in lens design, one needs to decrease m number to achieve a smaller beam size with a higher intensity when the length of the lens is fixed. But the difficulty of lens fabrication is increased when the m value is reduced. For our silicon lens design, the step height is set to be greater than 1 µm and the minimum line width of the lens be greater than 100 nm.
3. Design for the compound refractive lens of ideal cartesian oval lens shape
3.1 Ideal aperture design
Under the condition of parallel X-rays incidence, if we ignore refraction at the incident interface, the ideal first lens profile can be written as
where f1 is the focal length of lens 1.John P. Sutter et al. (2017) have derived the analytical expression of lens surface function which can focus an already convergent beam without aberration [15]. In this paper, we apply the same cubic X-ray approximation solution to the ideal lens surface for design and use the same function name yxr(x) to name the surface function. As the beam propagation direction in this paper is opposite to Ref. [15], a negative sign is added in front of yxr(x). The introduction of yxr(x) is given in Supplement 1. As Fig. 4 shows, the beam originally focused to point (0, q1) is focused to point (0, q2) through surface yxr(x) with aberration-free. When the lens material is fixed, the shape of yxr(x) only depends on q1 and q2.
Fig. 4. Schematic drawing of the exact surface yxr(x) for focusing a focused beam. (a) From the medium into the air. (b) From the air into the medium. The blue part is the medium, and the white part is air (or vacuum), n is the refractive index of the incident medium, and n’ is the refractive index of the outgoing medium. The X-ray beam runs along the positive direction of the Y-axis as the red line show. Point (0, q1) and (0, q2) are, respectively, the focal points of the beam before and after being focused (q1 and q2 are greater than zero).
The schematic diagram of lens design is shown in Fig. 5, Li is set artificially so we think Li is a known number. So for function yxr(x) of surface i (except surface 1), if Ai, Li and q1i are known, q2i can be derived. The calculation starts from the second lens, q21 = f1, q12 = q21 – d. According to the triangle approximation principle, A2 can be expressed as:
with a known A2, q22 can be derived by yxr(x), q13 = q22 – L2 – L3. Surface 3 focuses the x-ray focused by surface 2, from Fig. 5, so with known q13 , L3 and A3, The aperture of surface 4 can be calculated like surface 2,Repeat the process above, in this way, the parameters of each surface can be deduced. Summed up in a general formula, for even lens:
Fig. 5. Schematic drawing of the aperture precisely fits the outermost beam. The blue part of the image is the medium (Si for example), and the white part is air or vacuum (air for example). Surface 1 is the profile focusing X-ray from air into Si. q1i, q2i, Ai, Li are, respectively, object distance, image distance, aperture size, and length of surface i. The solid line represents the actual propagation path of the X-ray, different colors represent the path after refraction. The dotted line with the same color as the solid line represents the path before refraction. d is the distance between two lenses vertexes which depends on the limit of machining accuracy.
For odd lens (except A1),
3.2 Other designs to compare with
The profile yxr(x) is the solution that neglects terms beyond the cubic of the ideal lens surface equation. So there might be some error in focusing. To get a better focusing effect, we tried two other aperture designs. Call ideal aperture design D1 for short, design 2 as D2 for short:
Compared to D1, in D2, Aeven and Aeven-1(odd) are not equal anymore, Aeven in D2 is larger than Aeven in D1, and Aodd in D2 is smaller than Aodd in D1; in D3, all Ai are larger than D1. Detailed parameters of the OVAL lenses designed by these three designs are given in Table S1-S3 of Supplement 1, lenses were made from Si for energy 20 KeV.
Schroer and Lengeler proposed a kind of lens whose aperture is gradually (adiabatically) adapted to the size of the beam as it converges to the focus called AFL [2]. We designed an AFL that works at the same distance, energy and incident aperture to compare with OVAL. Table 1 shows a brief parameter comparison of the two lenses (the detailed parameters of AFL are given in Table S4 of Supplement 1). The back surface of the AFL forms a mirror image of the front one, while the OVALs have different q1 and q2 values. Here the back surface is set to have a smaller aperture than the front surface in OVAL. A schematic of two kinds of lenses is given in Fig. 6.
Fig. 6. Schematic of (a) AFL and (b) OVAL-D2. Ai, Li are respectively, the diameter and the length of surface i, d is the minimal lens thickness, f is the working distance.
Table 1. Brief parameter comparison of AFL and OVAL. a
3.3 Simulation by BPM
BPM was used to simulate the focus effect of OVAL and AFL with the same working distance and incidence aperture, the incident light is a plane wave with an intensity of 1, focus intensity distribution and profiles simulation are shown in Fig. 7. It can be seen that the focal position of the OVAL lens is the same as the focal position of the theoretical design, and OVAL-D2 performs best. The focal depth of OVAL lenses is smaller than that of AFL lens due to aberration-free focusing, this leads to more concentrated light near the focal point of the OVAL lens, which increases the intensity of the focal point.
Fig. 7. Focus intensity distribution and focus profiles simulation by BPM. (a) AFL (b) OVAL-D1 (c) OVAL-D2 (d) OVAL-D3. The intensity values from (a) to (d) are 313.3, 440.1, 453.7 and 395.4, and the FWHMs from (a) to (d) are 29.80, 21.30, 21.00 and 24.40 nm.
As shown in Table S1-S3 of Supplement 1, the total length of the OVALs can be sorted from smallest to largest: D2, D1 and D3, and the order of focusing effects from good to bad are the same. Under aberration-free focusing, D1 achieves precise focusing of all beams, D2 may have lost focus on some beams in the outer layer because of the smaller odd number of apertures, but under the premise of reducing the total length of the lens, the material absorbs less beam so effective aperture is increased to get a higher light intensity of the focal point. The larger aperture D3 compared to D1 might focus more beams which are missed in D1, but more beams are a negative improvement in contrast to more absorption.
To further enhance the light intensity of the focus, kinoform designs for AFL and OVAL are proposed. As an example, it is necessary to consider the condition that the minimum line width is more than 100 nm and the minimum thickness between the two adjacent steps should be greater than 1 µm. LKLX (X is a positive integer) means that we only used kinoform profile for the first X lenses, and the rest of the lenses are of the regular parabolic profile, considering the actual manufacturing accuracy limitation. AFL LKL72 and OVAL LKL66 of D1/2/3 are designed, the kinoform diagram of the first nine lenses of OVAL-D2 is shown in Fig. 8. Considering the actual machining accuracy. For AFL LKL72, m of the first 70 lenses are equal to 1 and m for the 71th and 72th lenses are equal to 2. For OVAL LKL66, m of the first 64 lenses are equal to 1 and m for the 65th and 66th lenses are equal to 2.
Fig. 8. The kinoform diagram of the first nine lenses of OVAL-D2. Ai, Li are respectively, the diameter and the length of surface i, d is the minimal lens thickness.
Focus intensity distribution and focus profiles simulation of kinoform design is given in Fig. 9. The focused light intensity of OVAL LKL66 of D2 is the largest and the FWHM is the smallest. The FWHMs of OVAL LKL66 of D1 and D2 are basically the same, and the light intensity of D1 is weaker, the OVAL LKL66 of D3 has the worst focusing effect. The light intensity gain of OVAL LKL66 of D2 is close to 60% being the highest and the FWHMs of it are reduced by up to 35% compared to uncarved one. The OVAL LKL66 has the same focal plane position as the design, with a smaller FWHM width and higher intensity gain, it is proved that the LKL needs adaptive lens design to reduce wavelength distortion to get a better focusing effect [20]. Therefore, LKL OVAL can focus better by reducing the lens's X-ray absorption and thereby increasing the effective aperture.
Fig. 9. Focus intensity distribution and focus profiles simulation by BPM. (a) LKL AFL (b) LKL OVAL-D1 (c) LKL OVAL-D2 (d) LKL OVAL-D3. The intensity values from (a) to (d) are 375.7, 722.6, 723.8, 613.2, and the FWHMs from (a) to (d) are 24.80, 13.62, 13.62 and 15.90 nm.
4. Discussion
In this paper, a compound focusing refraction lens (OVAL) design that can adiabatically focus X-rays accurately without geometric aberration is proposed. The compound refraction lens is designed to satisfy the paraxial condition. BPM simulation shows that significant improvement is achieved compared to other previously published lens designs, which is of great importance to the extreme nanofocusing. In the case of the diamond lens, the performance will be further improved compared to the silicon lens we used here. Also, note that design including X-rays propagate with wide off-axis angles can be further explored to increase the NA. BPM is used here under paraxial conditions since it can produce large errors for extreme nanofocusing or other special cases when X-rays propagate in wide angles off-axis. This can be improved by the Padé Approximation. Padé Approximation was initially widely used in wide-angle simulation in the field of acoustics and was later used in visible light and microwave simulation. Introducing Padé Approximation into X-ray optics design and simulation will help a lot in the pursuit of extreme focusing. The design of a wide-angle/large NA lens system will thus be introduced in our next article.
Funding
National Key Research and Development Program of China (2017YFA0403801); National Natural Science Foundation of China (U1732120).
Acknowledgments
We thank Hanfei Yan for an enlightening discussion on BPM.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Supplement 1.
Supplemental document
See Supplement 1 for supporting content.
References
1. K. Yamauchi, H. Mimura, T. Kimura, H. Yumoto, S. Handa, S. Matsuyama, K. Arima, Y. Sano, K. Yamamura, K. Inagaki, H. Nakamori, J. Kim, K. Tamasaku, Y. Nishino, M. Yabashi, and T. Ishikawa, “Single-nanometer focusing of hard x-rays by Kirkpatrick-Baez mirrors,” J. Phys.: Condens. Matter 23(39), 394206 (2011). [CrossRef]
2. C. G. Schroer and B. Lengeler, “Focusing hard x rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94(5), 054802 (2005). [CrossRef]
3. F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rodel, M. Wunsche, T. Ullsperger, S. Nolte, J. Rahomaki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect X-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8(1), 14623 (2017). [CrossRef]
4. J. Patommel, S. Klare, R. Hoppe, S. Ritter, D. Samberg, F. Wittwer, A. Jahn, K. Richter, C. Wenzel, J. W. Bartha, M. Scholz, F. Seiboth, U. Boesenberg, G. Falkenberg, and C. G. Schroer, “Focusing hard x rays beyond the critical angle of total reflection by adiabatically focusing lenses,” Appl. Phys. Lett. 110(10), 101103 (2017). [CrossRef]
5. X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, and D. Legnini, “11 nm hard X-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3(1), 3562 (2013). [CrossRef]
6. H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. 92(22), 221114 (2008). [CrossRef]
7. M. Lyubomirskiy and C. G. Schroer, “Refractive Lenses for Microscopy and Nanoanalysis,” Synchrotron Radiation News 29(4), 21–26 (2016). [CrossRef]
8. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996). [CrossRef]
9. B. Lengeler, C. Schroer, M. Richwin, J. Tümmler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard x rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924–3926 (1999). [CrossRef]
10. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, and A. van der Hart, “Hard x-ray nanoprobe based on refractive x-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005). [CrossRef]
11. C. G. Schroer, M. Kuhlmann, U. T. Hunger, T. F. Günzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, and A. Somogyi, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82(9), 1485–1487 (2003). [CrossRef]
12. V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, and M. Hoffmann, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058–4060 (2000). [CrossRef]
13. K. Evans-Lutterodt, J. M. Ablett, A. Stein, C.-C. Kao, D. M. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. L. Gammel, and H. Huggins, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef]
14. M. Sanchez del Rio and L. Alianelli, “Aspherical lens shapes for focusing synchrotron beams,” J. Synchrotron Radiat. 19(3), 366–374 (2012). [CrossRef]
15. J. P. Sutter and L. Alianelli, “Aberration-free aspherical lens shape for shortening the focal distance of an already convergent beam,” J. Synchrotron Radiat. 24(6), 1120–1136 (2017). [CrossRef]
16. J. P. Sutter and L. Alianelli, “Ideal Cartesian oval lens shape for refocusing an already convergent beam,” in AIP Conf. Proc.(AIP Publishing LLC2019), p. 030007.
17. J. Van Roey, J. Van der Donk, and P. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71(7), 803–810 (1981). [CrossRef]
18. Y. Chen, G. Simon, A. Haghiri-Gosnet, F. Carcenac, D. Decanini, F. Rousseaux, and H. Launois, “Edge diffraction enhanced printability in x-ray nanolithography,” J. Vac. Sci. Technol., B: Microelectron. Process. Phenom. 16(6), 3521–3525 (1998). [CrossRef]
19. K. Okamoto, Fundamentals of optical waveguides (Academic press, 2006).
20. H. Yan, “X-ray nanofocusing by kinoform lenses: a comparative study using different modeling approaches,” Phys. Rev. B 81(7), 075402 (2010). [CrossRef]
21. L. Alianelli, K. Sawhney, M. Tiwari, I. Dolbnya, R. Stevens, D. Jenkins, I. Loader, M. Wilson, and A. Malik, “Characterization of germanium linear kinoform lenses at Diamond Light Source,” J. Synchrotron Radiat. 16(3), 325–329 (2009). [CrossRef]
22. A. Isakovic, A. Stein, J. Warren, S. Narayanan, M. Sprung, A. Sandy, and K. Evans-Lutterodt, “Diamond kinoform hard X-ray refractive lenses: design, nanofabrication and testing,” J. Synchrotron Radiat. 16(1), 8–13 (2009). [CrossRef]
