Inverse-phase composite zone plate providing deeper focus than the normal diffraction-limited depth of X-ray microbeams

A novel type of zone plate, i.e. an inverse-phase composite zone plate, is proposed and examined with the aim of achieving deeper focus with little reduction in spatial resolution.


Introduction
Fresnel zone plates (ZPs) are major optical elements in X-ray microscopes. There are several derivations such as sputteredsliced ZPs (Rudolph et al., 1982;Koyama et al., 2012), multilayer Laue lenses (Maser et al., 2004;Koyama et al., 2008) and total reflection ZPs (Takano et al., 2010). There are other types of optical elements such as grazing-incidence total reflection mirrors of Kirkpatrick-Baez mirrors (Mimura et al., 2007) and Wolter type I mirrors (Aoki et al., 1992), compound refractive lenses (Schroer et al., 2005) and Bragg-Fresnel lenses (Erko et al., 1994). The development of these optical elements has made sub-100 nm-spatial resolution easily available in X-ray microscopes. As the spatial resolution becomes higher, the demand for localized analysis of practical samples is rapidly increasing. However, according to the physical principle of the diffraction limit, the higher the spatial resolution (the smaller the focused beam size), the shallower the depth of focus in a focusing optical system. Optical elements have to date been unable to evade the diffraction limit. The diffraction limit in optics, which corresponds to the uncertainty principle in quantum mechanics, is usually expressed as being that the product of the spatial resolution (Á res ) and the numerical aperture (NA) cannot be smaller than the wavelength, namely, Á res Â NA ! . Because the depth of focus (DoF) is also determined by NA such that DoF / 1/NA 2 , improved spatial resolution and deeper DoF are incompatible. This can be interpreted as an alternative expression of the diffraction limit. This limit restricts the thickness of samples to be analyzed and is an impediment to high-spatial-resolution micro-analysis of practical samples using X-ray microscopes. In particular, the DoF is a critical parameter for X-ray micro-CT (computed tomography) with X-ray microscopes. The DoF of a ZP was well studied both with simulations and experiments aiming at high-spatial-resolution X-ray micro-CT (Wang et al., 2000).
A novel type of ZP has been proposed, named an inversephase composite ZP (IP-CZP), in order to relax this limit, with an initial solution that facilitates a deeper DoF with little reduction of the spatial resolution (Kagoshima & Takayama, 2018). This paper reports on the focusing properties of the IP-CZP from the point of view of both DoF and spatial resolution.
It should be noted here that another way to produce a blurred focal spot with an extended DoF has been recently reported using multiple zone plate stacking with misalignment along the optical axis (Li & Jacobsen, 2018 There are two types of ZPs, positive and negative, as shown in Figs. 1(a) and 1(b). Both ZPs function as thin lenses. They have identical focal intensity distributions in the focal plane and along the optical axis, that determine Á res and DoF, respectively. On the other hand, the phase of the complex amplitude of focusing waves is opposite to each other. A composite ZP was proposed, and its optical properties were well studied (Michette, 1986). If the inner ZP with a first-order focal length of f is surrounded by the outer ZP with a thirdorder focal length also of f, the effective NA becomes larger as shown in Fig. 1(c). This leads to higher spatial resolution (smaller focused beam size) and an increase of the focused intensity.
In order to deepen the DoF without reducing the spatial resolution, the focal intensity distribution along the optical axis should be broadened, whereas the focusing beam size in the focal plane should be as unchanged as possible. Our strategy to accomplish this is as follows. We previously proposed that the phase of the inner ZP (iZP) and outer ZP (oZP) are inversely composed as shown in Fig. 1(d) (Kagoshima & Takayama, 2018). Because the aperture of the oZP is annular, the focused beam size produced only by the oZP becomes narrower than that of a circular aperture with the same NA. As the effective number of zones of the oZP becomes smaller, the monochromaticity require-ment for the oZP becomes looser, which leads to the broadening in the focal intensity distribution of the oZP along the optical axis. The diffraction efficiency of ZPs depends on the zones' material and thickness. If opaque zones act as phase material, ZPs work as phase ZPs (Kirz, 1974). By tuning the thicknesses of the iZP and oZP independently, the DoF and Á res of the IP-CZP can be controlled because the complex amplitude of the IP-CZP is the coherent sum of each complex amplitude of the iZP and oZP. Therefore, if the parameters of the IP-CZP are appropriately chosen, the DoF could be deepened with little reduction of the spatial resolution.

Definitions of design parameters
Definition of the main variable design parameters of the IP-CZP are illustrated in Fig. 2. Those of the iZP are the first zone radius, r 1_in , outer radius, r N_in , outermost zone width, Ár N_in , research papers   which is supposed to be equal to the minimum fabrication width Ár fab , and zone thickness, t in . Those of the oZP are the first zone radius, r 1_ou , outer radius, r N_ou , outermost zone width, Ár N_ou , and zone thickness, t in , where is the thickness ratio. In this paper, it is assumed that 1. r N_ou is identical to an outer radius of IP-CZP and represented by an annular parameter " as r N_ou = r N_in /". The focal length of the firstorder diffraction of iZP, f 1_in , is given by r 2 1 in =.
Since the focal length of the third-order diffraction of the oZP, f 3_ou , is equal to f 1_in , r 1_ou is ffiffi ffi 3 p r 1 in . The local diffraction efficiency is determined by the refractive index of the zone material, n = 1 À þ i, and the zone thickness (Kirz, 1974). The initial phase of oZP, , should also be able to be tuned for the optimization. When = and = 1, the IP-CZP is identical to a CZP. The effective number of zones of the oZP, N ou_eff , is N ou eff = N ou À N in =3:

Spatial resolution (D res ) and DoF
The two main focusing properties of optical elements are the spatial resolution (Á res ) and the DoF. The former can be evaluated from the radial intensity distribution in the focal plane perpendicular to the optical axis. The latter can be evaluated from the intensity distribution along the optical axis neighboring the focal point. Both are defined by the numerical aperture (NA) as follows, where m is the refractive index of the object space and is the objective angular semi-aperture. Equation (2) is well known as the Rayleigh criterion and usually understood to express the diffraction limit in microscopes. Á res is the same as the radius of the first null of the Airy pattern. DoF corresponds to a range within which the on-optical-axis intensity decreases by 20% from the peak intensity (Born & Wolf, 1986). Therefore, the smaller the Á res , the shallower the DoF as given by As described above, equation (4) is another expression of the diffraction limit for a circular aperture. For a usual ZP, Á res and DoF can be given by the outermost (narrowest fabricated) zone width, Ár N , as follows (Attwood, 1999), The achievable Á res depends only on Ár N , whereas DoF depends both on Ár N and . The monochromaticity required to achieve equation (5) where N is the total number of zones. If this condition is not satisfied, both Á res and DoF become larger than equations (5) and (6) corresponding to the degree of monochromaticity. In this paper, hereafter, Á res is defined as a radius initially yielding effective null in the intensity distribution.

Relevant design parameters
High-energy X-rays enable experiments to be conducted under atmospheric conditions, which is advantageous for the structural analysis of practical samples. Considering this, the operation X-ray energy of the IP-CZP is at the relatively high energy of 10 keV. The target Á res is set to be about 100 nm. Since the monochromaticity, /Á, of a silicon (111) doublecrystal monochromator typically used in synchrotron radiation beamlines is of the order of several thousands, the effective total number of zones, N tot eff = 3N ou , must be smaller than /Á. The focal length should not be too small in order to maintain a practical working distance.
In order to be consistent with our previous studies (Ozawa et al., 1997), tantalum is chosen as the zone material. According to the data tables (Henke et al., 1993; Center for X-ray Optics, http://henke.lbl.gov/optical_constants/tgrat2. html), the refractive index of tantalum at 10 keV is = 2.34356557 Â 10 À5 and = 3.89601519 Â 10 À6 . As an initial design step, the zone thickness of the iZP, t in , is chosen to be /2 = 2.645 mm, yielding a phase shift of , which means that the iZP works as an ordinary phase ZP when neglecting absorption.
Considering the above boundary conditions, r 1_in and r 1_ou are set to be 2.50 mm and 4.33 mm, respectively, which gives a focal length f 1_in of 50.41 mm at 10 keV. Ár N_in (= Ár fab ) is set to be 84 nm. Thus, r N_in becomes 37.25 mm with N in = 222. By modifying N ou (equally "), and , the performance of Á res and DoF has been optimized.

Diffraction integration
The three-dimensional complex amplitude distribution neighboring the focal point, A(r, , z), has been calculated according to the diffraction integration with a monochromatic plane-wave illumination of wavelength propagating along the z-axis. The optical system is shown in Fig. 3. If those of the iZP and oZP are denoted independently as A in (r, , z) and A ou (r, , z), respectively, the diffraction integration is written as follows in cylindrical coordinates, In the above, (, ') and (x 0 , y 0 ) are polar and Cartesian coordinates in the IP-CZP plane, and (r, ) and (x, y) are those in the observation plane at a distance z (= f + Áz) from the IP-CZ. l is a distance from a point Q in the IP-CZP plane to a point P in the observation plane. n in ; ' ð Þand n ou ; ' ð Þare the refractive index distributions of the iZP and oZP, respectively. I(r, , z) is the intensity. In the case when 6 ¼ 1 for the oZP, n ou in the thickness direction between t in and t in , shown in Fig. 2(b), is replaced by that of the surrounding environment. This scalar wave simulation is valid for the evaluation of the proposed ICP-ZPs as discussed in Section S1 of the supporting information.

Two promising cases
Diffraction integration has been performed to calculate the two-dimensional complex amplitude distribution (magnitude, phase, real and imaginary parts) neighboring the focal point. One dimension is the radial direction, r, and the other dimension is the optical axis direction, Áz. The intensity is the sum of the squares of the real and imaginary parts.
Two characteristic promising cases were found: a pitintensity focus with the deepest DoF (ZP-A) and a flatintensity focus with deeper DoF (ZP-B) than a usual ZP. Table 1 shows parameters of the two promising cases. The iZPs are identical between the two. The diameter of the oZP of ZP-A is larger than that of ZP-B with " of 0.782 and 0.836, respectively. The thickness of the oZP of ZP-A is larger than that of ZP-B with of 1.00 and 0.85, respectively. The outermost zone width of the two oZPs, Ár N_ou , is larger than twice that of the iZP, Ár N_in , of 84 nm. The effective total number of zones, N tot_eff , is sufficiently smaller than several thousands, which means that the chromatic aberration is negligible in usual synchrotron radiation beamlines equipping standard silicon double-crystal monochromators. Fig. 4 shows the calculated two-dimensional intensity distribution, I(r, Áz), of (a) ZP-A, (b) ZP-B and (c) iZP-only. The intensity is a normalized value by the peak intensity of iZP-only. The black dashed lines denote contours of 80% intensity relative to the self-peak intensity to make regions of each DoF clear. The ZP-A exhibits features of a pit-intensity focus and ZP-B exhibits features of a flat-intensity focus along the Áz axis. Fig. 5 shows intensity distributions (a) in the radial-direction, r, at Áz = 0 and (b) along the optical-axis, Áz, at r = 0 of ZP-A and ZP-B accompanied by those of iZP-only. The former corresponds to the point spread functions (PSFs) and the latter determines DoF. The PSF of iZP-only [black in Fig. 5(a)] is in accord with the theoretical values of ½2 J 1 x ð Þ=x 2 , where x = 2= ð Þr N in r=z (Born & Wolf, 1986). The intensity distribution along the optical axis of iZP-only [black in Fig. 5(b)] is also in accord with the theoretical values of sinðu=4Þ=ðu=4Þ ½ 2 , where u=4 = r 2 N in Áz=2z 2 (Born & Wolf, 1986).
In this paper, Á res is defined as a radius initially yielding effective null in the intensity distribution. Each Á res is shown in Fig. 5(a). According to equation (4), the increase of DoF can be assessed. The calculation results relating to Á res and DoF are summarized in Table 2. The calculation results relating to intensity are summarized in Table 3. Á res is 110 nm and 109 nm, and DoF is 461 mm and 412 mm for ZP-A and ZP-B, respectively. Because Á res determined by Ár fab is 102 nm research papers Table 1 Parameters of two promising IP-CZPs. ZP-A has a pit-intensity focus with the deepest DoF and ZP-B has a flat-intensity focus with deeper DoF than usual ZPs.

Figure 3
Optical system for diffraction integration with a monochromatic planewave illumination. (, ') and (x 0 , y 0 ) are coordinates in the IP-CZP plane, and (r, ) and (x, y) are those in the observation plane at a distance z (= f + Áz) from the IP-CZP. l is the distance from a point Q in the IP-CZP plane to a point P in the observation plane.
(= 1.22Ár fab ), the reduction in the spatial resolution is as small as 8% for both ZPs, from the point of view of fabrication. On the other hand, DoF can be 2.0 and 1.8 times deeper than that of iZP-only. The maximum intensity is reduced to 77% and 73% for ZP-A and ZP-B, respectively, in comparison with that of iZP-only. The integrated intensity, I int , within which the sum of intensities decreases by 20% from the self-peak intensity (inside the black dotted lines in Fig. 4), is reduced to 77% and 75% for ZP-A and ZP-B, respectively, in comparison with that of iZP-only. We found that a DoF of twice the depth could be expected with little reduction in the spatial resolution and with $ 1/4 intensity reduction. It should be mentioned here that the focused intensity of ZP-A and ZP-B normalized by the total incident X-ray intensity, namely efficiency, is low due to both intentionally adapting the inverse phase of oZP and to larger diameters than that of iZP-only. The normalized integrated intensity by the area of the ZPs, which corresponds to efficiency, is also shown in Table 3. The efficiency of ZP-A and ZP-B is about half that of iZP-only. In synchrotron radiation beamlines, the incident beam size at the experimental station is usually larger than the size of practical ZPs. The larger incident beam size is also required to allow for the beam drift. Thus, the actual focused intensity may be more practically important than the efficiency.

Dependence on photon energy
The focusing property dependence on photon energy has also been investigated. Fig. 6 shows PSFs of (a) ZP-A and (b) ZP-B for several photon energies from 8 to 12 keV, respectively. All PSFs self-normalized by the intensity at r = 0 have almost the same profile (not shown in the figure). Thus, the dependence of Á res on wavelength is negligible in the photon energy range. On the other hand, regarding intensity distributions along the optical axis, Áz, at r = 0, the situation is somewhat different as shown in Fig. 7, where the abscissa, Áz,       is a distance from a focus position of each photon energy. ZP-A almost conserves the line symmetry about the line of Áz = 0, while the line symmetry breaks for ZP-B in the lower photon energies. It is fortunate for ZP-A that the normalized intensity of the pits (Áz = 0) by the self-maximum intensity is constantly 0.8, and thus the deepest DoF can be maintained in the photon energy range. These respective properties may not be simply understood, and further explanation is outside the scope of this paper. Fig. 8 shows (a) intensity changes at Áz = 0 and r = 0, and (b) DoF and DoF/f changes. The diffraction efficiency of a phase zone plate depends on both the real part and the imaginary part of the refractive index of the zone material (Kirz, 1974). The same theory can be applied to the present IP-CZP as a linear combination of complex amplitude of the iZP and oZP. The L-absorption edges of the zone material (tantalum) are indicated as a reference by vertical black arrows. The absorption edges have no influence on DoF. DoF increases linearly, while DoF/f is almost constant because f is proportional to the photon energy.

Future prospects
Difficulties in the fabrication are not considered in this paper.
Actually, the aspect ratio of the IP-CZP presented in this paper is 31.5, which is too high to realize by the presently popular fabrication technique of electron beam lithography. Solutions for fabricating high-aspect-ratio ZPs were demonstrated by stacking two identical zone plates (Snigireva et al., 2007;Kagoshima et al., 2011), double-sided zone plates (Mohacsi et al., 2017) and zone plates of the highest aspect ratio fabricated by using metal-assisted chemical etching (Li et al., 2017). Progress on these advanced fabrication technologies will make the proposed IP-CZPs available.
Another surpassing way to produce a blurred focal spot with an extended DoF has been studied (Li & Jacobsen, 2018). It uses multiple zone plate stacking with misalignment along the optical axis. The merit of producing blurred or extended DoF by multiple zone plate stacking is higher efficiency, i.e. zone plate stacking brings more energy to the focus. On the other hand, our IP-CZP has the great advantage that it is a single lens on a single substrate, and thus it can be easily handled and operated as a thin lens though the efficiency is low. Further, the focused intensity can be compensated by using brighter sources. Since Á res and DoF are naturally determined by the research papers   Calculated intensity distributions along the optical axis, Áz, at r = 0 at several photon energies: (a) ZP-A and (b) ZP-B. The ordinate is a relative value to the peak intensity of iZP-only at the photon energy of 10 keV. The abscissa, Áz, is a distance from a focus position of each photon energy.   (left) and DoF/f (right) at several photon energies. The ordinate of (a) is a relative value to the peak intensity of iZP-only at the photon energy of 10 keV. In (b), the solid and dotted lines correspond to DoF and DoF/f, respectively. L-absorption edges of tantalum are indicated. numerical aperture of an optical device itself, they are unable to be gained by the source intensity. Therefore, the IP-CZP will be practically valuable for thicker samples especially with future brighter sources.

Conclusions
In order to defeat the diffraction limit restricting the relation between spatial resolution and depth of focus, an inversephase composite zone plate is proposed. The structure is a combination of an iZP functioning as a conventional phase zone plate and an oZP functioning with third-order diffraction with opposite phase to the iZP. The focusing properties of spatial resolution and DoF at a photon energy of 10 keV have been investigated by diffraction integration. Two characteristic promising cases were found of a pit-intensity focus with the deepest DoF and a flat-intensity focus with deeper DoF than usual ZPs. We found that a DoF of nearly twice the depth could be expected with little reduction in the spatial resolution, and that DoF and the spatial resolution were almost unchanged in the relatively wide photon energy range.

Related literature
The following references, not cited in the main body of the paper, have been cited in the supporting information: Kang et al. (2005Kang et al. ( , 2006; Maser & Schmahl (1992); Schneider (1997); Schnopper et al. (1977).