More are better, but the details matter: combinations of multiple Fresnel zone plates for improved resolution and efficiency in X-ray microscopy

The properties of stacking multiple Fresnel zone plates together at intermediate distances are considered. One can enhance the nanofocusing efficiency, and other characteristics, by careful choice of design parameters.


Introduction
Fresnel zone plates are widely used as the nanofocusing optic in X-ray microscopes (Sakdinawat & Attwood, 2010;Attwood & Sakdinawat, 2017). They focus beams or image specimens with a Rayleigh resolution of r = 1.22dr N /m where dr N is the width of the finest outermost zone and m is the diffractive order used for focusing. Their first-order focusing efficiency depends on the refractive index n = 1 À À i of the zone material (Henke et al., 1993), with a scalar diffraction efficiency " m for a zone plate thickness t given by (Kirz, 1974) where ' = 2t= and =: The efficiency reaches a maximum at a thickness t opt satisfying the condition 2t opt ' À 2 ð3Þ or t opt ' 2 1 À 2 ; which for gold gives " 1 = 8.7% at t opt = 98 nm when using 0.5 keV soft X-rays, and " 1 = 32.7% at t opt = 2.0 mm when using 10 keV hard X-rays. Thus one can see that efficient highresolution multi-keV X-ray focusing requires the fabrication of high-aspect-ratio structures with narrow zone width dr N but large thickness t [see Fig. 1(a)]. There is in fact much progress in fabricating conventional Fresnel zone plates with high aspect ratios (Schneider et al., 1995;Spector et al., 1997;Jefimovs et al., 2007;Werner et al., 2014;Chang & Sakdinawat, 2014;Mohacsi et al., 2017;Li et al., 2017b). However, even if technological limits on high-aspect-ratio nanofabrication were removed, there remain fundamental limits: as zones increase in thickness, they begin to need to be tilted to match the Bragg condition (Maser & Schmahl, 1992) or even curved to match the converging wave as in a volume hologram (Pennington & Lin, 1965;Kogelnik, 1969;Goodman, 2005), as shown in Fig. 1(d).
An alternative approach to a monolithic three-dimensional optical structure is to stack multiple separate zone plates together. When doing so, one must pay attention to how the separation distances Áz compare with the depth of focus. The depth of focus ðDoFÞ of a Fresnel zone plate can be expressed as twice the depth resolution z or (Wang et al., 2000) DoF When multiple zone plates are located well within a depth of focus along the wavefield propagation direction, one can treat the net effect as being due to a zone plate with the combined thickness t as has been demonstrated (Shastri et al., 2001;Maser et al., 2002;Snigireva et al., 2007;Kagoshima et al., 2011;Feng et al., 2007;Aristov et al., 2007;Mohacsi et al., 2014Mohacsi et al., , 2017Rehbein et al., 2015). However, this close proximity is often challenging to achieve in practice, leading to an alternative approach of stacking multiple zone plates with separations larger than 2 z provided each individual zone plate is designed to focus to a common location (Vila-Comamala et al., 2013;Gleber et al., 2014). This has been studied within the approximation that each zone plate is optically thin, so that one can apply each zone plate's optical modulation to a wavefield, and then propagate it by a distance of Áz to the plane of the next zone plate to model the net focusing effect.
In this paper, we look in further detail at the optical properties of multiple stacked Fresnel zone plates. We make use (when necessary) of the multislice propagation technique (Cowley & Moodie, 1957) which can replicate (Li et al., 2017a) the results of coupled-wave theory which is otherwise required (Maser & Schmahl, 1992;Schneider, 1997;Schneider et al., 2008) when studying very high aspect ratio zone plates. We consider for the first time the following features of stacked zone plates: (i) While previous calculations (Vila-Comamala et al., 2013) and experiments (Gleber et al., 2014) considered intermediate distance stacking with zone plates with fixed outermost zone width dr N and adjustable diameter d, we consider the three cases of fixed dr N , fixed d and fixed zone number N in x2.
(ii) We examine in detail the exit wave from the last of a stacked set of Fresnel zone plates, and show in x3 that with multiple zone plates one begins to produce an effect like that of blazed zone plates but without the tradeoff of minimum linewidth otherwise required.
(iii) We show in x4 that the Talbot effect can be used to understand that there are optimum separation distances between stacked zone plates.
(iv) In x5 we consider 'complementary' zone plates where the positions of 'open' and 'filled' zones are reversed in some elements of a stacked combination, and show that this can reduce the effect of sidelobes off of the focus which can be advantageous for applications such as trace-element mapping using X-ray fluorescence.
(v) We consider 'refocusing' zone plates in x6, where some of the negative or diverging focal-order light from an upstream zone plate is captured and brought back into the positive first-order focus.
In addition to the above, in the supporting information we consider alignment tolerances of stacked zone plates as already discussed in the references cited, but provide more detail on misalignment effects.

Selecting multiple zone plate parameters
For larger separation distances between individual zone plates, the relationship between diameter d, outermost zone width dr N and zone number N of and the focal length of indicate that one has a choice in fixing any one of three parameters in order to adjust the ith zone plate to have a focal length f i so as to focus at the common location. These strategies are as follows: Zone plates modulate the phase and magnitude of a transmitted wave so as to direct it to a focus. This can be done in several ways, using magnitude and/or phase modulations on alternating zones of half-wavelength optical path difference to the focus. A simple Fresnel zone plate (a) has an outermost zone width of dr N and a thickness t ; it applies a constant magnitude reduction or phase shift across each zone, while a blazed zone plate (b) puts a staircase approximation of a phase ramp across the zone. One can stack several thin Fresnel zone plates (c) to manipulate the wave at several locations leading to the focus, or use thicker zones (d) which are individually tilted to meet the Bragg grating condition or even curved in a volume hologram approach to produce the converging wavefield.
(i) Fixed outermost zone width dr N : in this case, the diameter is adjusted according to d i = f i =dr N and the number of zones is adjusted to N i = d i =ð4dr N Þ.
(ii) Fixed diameter d: in this case, the outermost zone width is adjusted according to dr N;i = f i =d and the number of zones is adjusted to N i = d=ð4dr N;i Þ.
(iii) Fixed zone number N: in this case, the diameter is adjusted according to d i = 2ðNf i Þ 1=2 and the outermost zone width is adjusted to dr N;i = f i =d i .
While the fixed dr N approach has been studied previously (Vila-Comamala et al., 2013;Gleber et al., 2014), the fixed d and fixed N approaches have not.
Because many nanofabrication processes have limits on the achievable aspect ratio t=dr N , when considering approaches that vary the outermost zone width dr N;i we have also chosen to adjust the thickness of the ith zone plate according to for the fixed diameter d and fixed zone number N choices. In order to make clear the differences between these strategies, we show in Table 1 the parameters that result when four zone plates are used at 10 keV with a very large separation of Áz = 1 mm between each zone plate.
To understand the consequences of these different strategies, in Fig. 2 we show the efficiency and spatial resolution values that result. The case of fixed diameter d yields the highest efficiency and resolution, even though the downstream zone plates are assumed to have lower thickness t i and thus lower individual diffraction efficiency. In the case of fixed diameter d, the downstream zone plates capture a larger fraction of the zero-order undiffracted light passing through upstream zone plates, and the final zone plate has a smaller outermost zone width dr N;i which contributes to a higher spatial resolution. This increase in area more than compensates for the decrease in thickness t i . In contrast to this case, the fixed outermost zone width dr N case has the smallest diameter d i for the later zone plates. A factor not considered in the efficiency or energy fraction shown in Fig. 2 is the spectral bandwidth, which should be limited (Thieme, 1988) to In the case of fixed d, the change from N 1 = 450 to N 4 = 672 zones would reduce the acceptable spectral bandwidth by  The stacking of multiple thinner zone plates can lead to higher optical performance than can be obtained with a thicker single zone plate. On the left (a) is show the focusing efficiency (in terms of integrated energy fraction as a function of radius from the focal spot center) and, as an inset, the focused intensity profile for the three zone-plate design schemes and parameters outlined in Table 1. This is for the case of zone plates separated by a very large distance of Áz = 1000 mm relative to the focal length of f = 9074 mm for the first, upstream zone plate. The inset shows that the strategy of fixed diameter d gives a higher Rayleigh resolution (smaller radius for the first minimum of the intensity distribution) than the strategies of fixed outermost zone width dr N or fixed zone number N. On the right (b) is shown the integrated energy fraction for a single t = 2000 nm-thick zone plate versus four t = 500 nm-thick zone plates separated by Áz = 50 mm, and designed according to the fixed diameter d strategy. If multiple thinner zone plates can be aligned with sufficient accuracy, they can offer higher overall focusing efficiency (though in this case the separation between the zone plates is so small that they all have essentially the same numerical aperture, so there is no spatial resolution gain). Table 1 Parameters for n zp = 4 zone plates stacked at a separation distance of Áz = 1 mm, using the strategies of fixing the diameter d in mm, or the number of zones N, or the outermost zone width dr N in nm for the ith zone plate.
Under the assumption that a zone plate fabrication process has a limit to achievable aspect ratios t=dr N , zone plate thicknesses are also adjusted according to t i = t 1 dr N;i =dr N;1 where t 1 = 500 nm in this case.

Multiple zone plate exit waves
In near-field zone plate stacking, the optical effect of each zone plate is simply superimposed on the incident wavefield with no interceding propagation-based wavefield evolution. In non-near-field stacking, simulations (Vila-Comamala et al., 2013) and experiments (Gleber et al., 2014) have calculated the intensity profile near the focal region. In Fig. 3, we show for the first time the nature of the exit wave from the last of a set of stacked zone plates. (This exit wave then converges to produce the focal spot.) When multiple stacked zone plates are used, the exit wave from multiple stacked zone plates evolves towards that of a blazed zone plate (Fig. 1b). Because of this, one can achieve a higher diffraction efficiency with a set of separated stacked zone plates than would be expected simply from the sum of the zone plate thicknesses and equation (1). A staircase approximation to single-optic blazed zone plates has been realized by using multiple overlaid lithography steps to produce a single optical structure with improved focusing efficiency (Krasnoperova et al., 1993;Di Fabrizio et al., 1994Yun et al., 1999). However, if k lithographic overlays are used to produce a zone profile with k stair steps, the finest transverse feature size in the last stair step must be dr N =k to achieve a net width dr N of the finest outermost half of the blazed structure. In other words, if the finest transverse feature size that a given lithographic process can produce is a, then the finest zone half period is ka rather than having dr N = a. In this case the increase in efficiency of using k steps to produce a staircase approximation of a blazed zone plate comes at a cost in achievable spatial resolution.
Separated stacked zone plates can avoid this undesirable tradeoff. In Fig. 3, we show the phase of the wavefield exiting the final downstream zone plate for the case of a single t opt = 2000 nm-thick [equation (4)] zone plate of gold for use at 10 keV for which equation (1) gives a diffraction efficiency of " 1 = 32.7%, as well as the exit waves for four zone plates each with 500 nm thickness, and ten zone plates each with 200 nm thickness (in all cases the separation distance was Áz = 10 mm). The wavefield propagation through each individual zone plate was calculated using a multislice approach (Cowley & Moodie, 1957;Li et al., 2017a). As was noted, the ten zone plate case produces a last-zone-plate exit wave strongly resembling what one would have from a blazed zone plate.
In order to explore this further, we first consider the case of the number of zone plates to be used and the resulting diffraction efficiency. While the optimum thickness for a single zone plate of gold at 10 keV is t opt = 2.0 mm, in Fig. 4 we consider cumulative zone plate thicknesses as high as t = 4.0 mm while in fact using multiple zone plates with individual thicknesses t i ranging from t i = 0.1 mm (so that n zp = 40 such zone plates would have an accumulated thickness of t = 4.0 mm if they had the same design parameters) to t i = 2.0 mm (so that only n zp = 2 zone plates would be stacked). In all cases, a separation distance of Áz = 10 mm was used. As this figure shows, one can obtain a diffraction efficiency of " = 66% if one uses n zp = 30 zone plates that are each only t i = 0.1 mm thick at Áz = 10 mm spacing, where each is designed according to the fixed diameter d strategy described in x2. Even if fewer zone plates are used with a cumulative thickness limited to t opt = 2.0 mm, one still sees gains over single zone plates: for example, n zp = 4 zone plates each with t i = 0.5 mm give an efficiency of " = 41.5% while if one uses n zp = 10 zone plates with t i = 0.2 mm the efficiency is " = 48.9%. Fig. 4(a) used a constant separation distance of Áz = 10 mm between zone plates, which is in practice a very small separation distance. Fig. 4(b) shows how the performance of multiple zone plates (each with t i = 500 nm thickness) changes as one increases the separation distance Áz. As can be seen, smaller separation distances are preferred, but even with Áz = 100 mm one can obtain a first-order diffraction efficiency of " 1 = 44.8% by using n zp = 5 zone plates.

The Talbot effect and zone plate spacing
The Talbot effect (Talbot, 1836;Lord, 1881) involves the replication of a transmission grating pattern of period a at a Talbot distance Phase profile of the exit wave from (top) a single t opt = 2.0 mm-thick zone plate of gold for 10 keV X-ray focusing, (middle) four 0.5 mm-thick zone plates, and (bottom) ten 0.2 mm-thick zone plates. In each case the phase of the exit wave from the final downstream zone plate is shown. By using multiple thin zone plates, one can better approximate the phase profile of a blazed zone plate (Fig. 1b).
when zero and first diffraction orders constructively interfere (Fig. 5). This suggests that there might be an optimum separation distance Áz for stacked zone plate. Because the local grating period in a Fresnel zone plate varies constantly with radius, it is less clear whether the Talbot effect should apply based on the period 2dr N of the finest zone width, or on the period 4dr N of the zones at half the radius, or whether it is 'washed out' by the variation in zone width. While Fig. 4(b) showed the effect of only a few different separation distances Áz, in Fig. 6 we show the effect of a much larger number of more finely varied separation distances. In this figure, for each value of individual zone plate thickness t i and separation distance Áz, the optimum number n zp of individual zone plates was chosen as is shown in Fig. 4(a) and the average value of n zp is shown for each thickness t i . Clearly, we see maxima and minima of efficiencies with respect to stacking separations, and a dependence on outermost zone width dr N .
In order to better compare the separation distances leading to the first two maxima (z max1 ; z max2 ) and minima (z min1 ; z min2 ) in Fig. 6, in Fig. 7 we show these extrema both as a function of varying the outermost zone width dr N in Fig. 7(a), and the zone plate diameter d in Fig. 7(b). Obviously the efficiency maxima and minima depend on outermost zone width dr N rather than diameter d. In Fig. 7(a), we also indicate the Talbot distance z Talbot of a grating with a period 2dr N corresponding to the outermost zone width. While the positions of the maxima and minima show the expected scaling with ðdr N Þ 2 , the separation distance z max1 corresponding to the first efficiency maximum is about one quarter of the value 2ð2 dr N Þ 2 = that one would predict from equation (10) based on the outermost zone width. As shown in Fig. 5, at this distance a constantperiod grating shows an interference pattern at half the period of the grating, and with a contrast inversion.

High diffraction orders and complementary zone plates
In nanofabrication using electron beam lithography, transverse spreading of the electron beam in the photoresist (the  Illustration of the Talbot effect. When a periodic diffraction grating is illuminated by a plane wave, the image of the grating is repeated at a Talbot distance of z Talbot = 2a 2 = [equation (10)] where a is the period of the diffraction grating. For a grating period of a = 100 nm, and = 0.124 nm (corresponding to 10 keV X-rays), the Talbot distance is z Talbot = 161 mm. The grating here was assumed to be fully absorptive.

Figure 4
While there might be practical limitations to the number of zone plates n zp that can be stacked together, one can obtain gains in first-order diffraction efficiency " 1 that go well beyond the simple thin zone plate expression of equation (1). In (a) we show the efficiency as a function of individual zone plate thickness t i and cumulative thickness t, where n zp = t=t i zone plates are used with with a separation of Áz = 10 mm (for d = 45 mm, and dr N = 25 nm for the first zone plate at 10 keV). A single gold zone plate with the optimum thickness t opt = 2.0 mm would give " 1 = 32.7%, whereas much higher efficiencies can be obtained by using many more zone plates with slightly higher cumulative thickness. In (b) we show how the stacking of n zp = 8 zone plates, each with a thickness t i = 0.5 mm, leads to differences in diffraction efficiency as one changes the separation distance Áz between zone plates. Smaller separation distances Áz are preferable but might be impractical, but even with larger separation distances like Áz = 1000 mm one can still obtain an efficiency of " 1 = 36% if n zp = 5 zone plates are used. All calculations were for gold zone plates at 10 keV. proximity effect) can complicate the fabrication of dense narrow-linewidth structures such as the zones in a Fresnel zone plate. For this reason, a variety of interlacing approaches have been used in which every other zone is written in one operation, with a subsequent identical operation used to write the alternating zones either on the same side of a thin window (Chao et al., 2005) as shown in Fig. 8(b), or on the opposite side (Mohacsi et al., 2017) as shown in Fig. 8(c). Another way to reduce the limitations of the proximity effect is to write narrower zones in a low-density template, and use atomic layer deposition (ALD) to deposit high-density material on these templates in a process known as zone doubling (Jefimovs et al., 2007). Yet another approach is to consider the fabrication of zone plates with line : space ratios other than 1 : 1 where the separation distance between written structures can be increased (again reducing the proximity effect in electron beam lithography), while working in higher diffraction orders (Schneider, 1997).
Inspired by these approaches, in Fig. 8 we consider several options for zone plate stacking: Diffraction efficiency " 1 versus separation distance Áz for different thicknesses t i of individual zone plates, and for two different values of outermost zone width: dr N = 20 nm (a) and dr N = 30 nm (b). For each individual zone thickness t i and separation distance Áz, the number of stacked zone plates n zp was chosen to give maximum efficiency, as shown in Fig. 4(b), and the average value of n zp corresponding to a particular value of t i is indicated. As can be seen, there is a pattern of maxima and minima in the efficiences as a function of separation distance Áz, with the first two maxima denoted by z max1 and z max2 , and the first two minima denoted by z min1 and z min2 . These maxima and minima are compared with the Talbot distance [equation (10)] for the outermost zones in Fig. 7.

Figure 7
Positions of the efficiency maxima ðz max1 ; z max2 Þ and minima ðz min1 ; z min2 Þ shown in Fig. 6 as a function of varying outermost zone width dr N (a) or diameter d (b). This was done for an individual zone plate thickness of t i = 0.9 mm. Clearly the separation distances Áz show maxima and minima that scale with outermost zone width dr N rather than diameter d. Also shown on the left is the Talbot distance z Talbot based on equation (10) for a grating with a period a = 2dr N determined by the outermost zone width. The efficiency maxima and minima scale with dr N , but the ideal separation distance Áz corresponding to z max1 is at about one-quarter of the distance 2ð2 dr N Þ 2 =.
(i) The first of these is the use of line : space ratios with values such as 1 : 3, giving more space between lithographically patterned zones and thereby reducing the limitations set by the proximity effect. With a 1 : 1 zone plate, the outermost zone period is p N = 2dr N and the Rayleigh resolution is r = 1:22 dr N =m. With other values of the line:space ratio, it is better to base the Rayleigh resolution expression using the outermost zone period p N , giving When the line : space ratio in a grating is changed from 1 : 1 to other values, the energy distribution into various diffractive orders is modified. For example, a 1 : 3 zone plate operated in the m = 2 diffraction order can in principle have twice the spatial resolution for a given period p N as shown in equation (11), and there can be specific thicknesses and line : space ratios which deliver high focusing efficiency into the m = 2 order as calculated using coupled wave theory (Schneider, 1997). By working in the second diffraction order, one obtains a factor of two improvement in spatial resolution compared with the minimum zone width.
(ii) The second of these is the use of complementary zone plates. In the normal case for a Fresnel zone plate, the zeroth or central zone is filled with material up to a radius r 1 as given by after which one alternates between open and material-filled zones such that the next material-filled zone is bounded by r 2 and r 3 . However, in a complementary zone plate the central zone is open and the first material-filled zone is bounded by r 1 and r 2 . Complementary zone plates do not by themselves provide any reduction in the proximity effect, but they can be used with any line : space ratio including 1 : 1 or 1 : 3. Variation of the line : space ratio and the use of regular or complementary zone plates give us additional options in stacked zone plate design. In addition, when using separated stacked zone plates rather than the interlacing schemes of Figs. 8(b) or 8(c), one gains the ability to use processes that are difficult to interlace on one window. One of these is metalassisted chemical etching (MACE) to produce extremely high aspect ratio zone-doubling template structures for ALD (Chang & Sakdinawat, 2014;Li et al., 2017b).
We consider the second diffraction order (m = 2) focusing properties of zone plates with differing line : space ratio in Fig. 9 where we also consider the option of having either a regular or complementary second zone plate. These simulations were carried out with a thickness of t = 1 mm of Au and an incident X-ray energy of 10 keV. In Fig. 9(a), we show the radial focal profiles, and radially integrated intensity, as a function of radius from the optical axis in the case where the outermost zone period is p N = 100 nm and the separation is Áz = 50 mm. As can be seen, for second-order focusing a 1 : 3 line : space ratio is preferred over a 1 : 1 line : space ratio, as expected (for a thin zone plate, a 1 : 1 line : space ratio would give zero efficiency for the second-order focus). Having the  A normal Fresnel zone plate with 1 : 1 line : space ratio (a) can be constructed by interlacing two fabrication processes, either on the same side of a thin window (Chao et al., 2005) (b) or on opposite sides (Mohacsi et al., 2017) (c). Both of those interlaced processes reduce the limitations produced by the proximity effect in electron beam lithography in the separate fabrication processes. The proximity effect is reduced further if one fabricates zones with a line : space ratio of 1 : 3 and uses them in the second diffraction order. One can also use complementary zone plates with an opposite pattern of material-filled/open zones. That leads to several options in zone plate stacking: stacking with 1 : 3 line : space ratio with a regular (d) or complementary (e) second zone plate, or with 1 : 1 line : space ratio with a regular ( f ) or complementary second zone plate. These various combinations give different properties for focusing efficiency and focal spot sidelobes, as shown in Fig. 9. second zone plate being a regular zone plate gives higher focusing efficiency than using a complementary zone plate.
One very interesting feature of the complementary 1 : 3 line : space ratio case is that there is very little increase in light in 'sidelobes' outside the central focus spot (though the central focus spot is slightly widened). This can be of advantage in methods such as the mapping of trace elements by X-ray fluorescence, since a focal probe without sidelobes will allow for better quantitation of the elemental content within the central focus spot with little or no signal contributed from other nearby positions.
When working in the first diffraction order, downstream stacked zone plates will be of the same type as the first one (that is, regular rather than complementary zone plates) with their parameters modified with separation distance as described in x2. However, when working in the second diffraction order, the wavefield converging from the first zone plate will converge at twice the normal angle, or = 2=p N , which when multiplied by the separation distance Áz gives a reduction of radius for a given zone number of 2Áz=p N rather than 1Áz=p N . If one sets the extra radius reduction of 1Áz c =p N equal to one zone period p N , one finds That is, at a distance of Áz c the second zone plate will work with its zones shifted by one period, and at half that distance one will have a transition to requiring that the second zone plate be a complementary one rather than a regular one. For p N = 100 nm and = 0.124 nm corresponding to 10 keV, the 'matched' distance is Áz c = 81 mm and the transition to a complementary zone plate should occur at half that distance or about Áz c =2 = 40 mm. This effect is shown in Fig. 9(b), where when using stacked 1 :3 line : space ratio zone plates in the second diffraction order one finds that a complementary zone plate is preferred at distances smaller than Áz c =2, and at Áz c =2 and above there is a change to preferring a regular zone plate.
For two zone plates stacked in close proximity (such as interlaced zone plates) operating in the second diffraction order, a line : space ratio of 1 : 3 should give near-optimum research papers Figure 9 Stacking of two zone plates with 100 nm outermost zone period, each with t i = 1 mm thickness, at 10 keV photon energy. Using the four different stacking schemes shown in Fig. 8, in (a) we show both the radial intensity profile and also the radially integrated energy for m = 2 second diffraction order focusing. The 1 : 3 line : space approach with the second zone plate being a complementary zone plate has a unique property of having very little energy in sidelobes around the central focus spot. In (b) we show the normalized focus intensity for 1 : 3 line : space stacking with regular and complementary second zone plate as a function of separation distance Áz, demonstrating a crossover between which approach is preferred at half of Áz c of equation (13). In (c) we show the focal spot intensity as the line : space ratio is adjusted in both the upstream and (regular) downstream zone plate; the optimum line : space ratio for a variety of separation distances Áz is shown in (d).
diffraction efficiency. However, the optimum line : space ratio can be different when two zone plates become more separated. We therefore undertook simulations where the line : space ratio of both the first and second zone plates was adjusted. The results for a fixed separation distance of Áz = 50 mm are shown in Fig. 9(c), which indicates that a line : space ratio of about 1 : 2.7 is preferred to 1 : 3 for both zone plates in regular stacking. However, given that complementary zone plates can be preferred at certain distances as shown in Fig. 9(b) and as discussed above, in Fig. 9(d) we show the focus intensity for the optimum line : space ratio, and also the value of that optimum line : space ratio, for two stacked zone plates as a function of separation distance Áz.
The calculations shown in Fig. 9 were for the case of a fixed zone thickness of t = 1 mm. In Fig. 10, we show how the optimal line : space ratio depends on both the thickness t i of the two individual zone plates, and the stacking separation distance Áz. At a zone plate thickness of t i = 0.5 mm, the differences between having the second zone plate being regular or complementary are not so significant, whereas they are quite noticeable with t i = 2 mm, which corresponds to a phase shift. This makes it clear that the differences between using a regular or a complementary zone plate for the second zone plate are volume diffraction effects.

Refocusing zone plates
In the simulations shown above, only the zeroth or positive focal orders are used from upstream zone plates. However, significant energy goes into negative diffraction orders, which diverge from a virtual focus located upstream. We now consider schemes to recapture a fraction of this energy using a stacked refocusing zone plate. The idea is shown schematically in Fig. 11(a). The refocusing zone plate can be designed to operate in m r = 1 or first diffraction order, but this requires a finer outermost zone width dr N in the refocusing zone plate than is used in the first or second (stacked) zone plate. Therefore another option to consider is the use of m r = 3 or third-order diffraction by the refocusing zone plate, so that the minimum zone width dr N in the refocusing zone plate is larger than the minimum zone width dr N in the conventional and stacked zone plate. For the refocusing zone plate, the usual expression for conventional zone plate zone radii r k of equation (12) is replaced with where ' indexes the zones in the refocusing zone plate, and M is the magnification of the source to its image. In order to understand the potential improvements that might be provided by using a refocusing zone plate, we first calculated three cases where we did not include a stacked zone plate: a conventional zone plate with central stop [case C in Fig. 11(b)], the conventional zone plate plus a refocusing zone plate operated with m r = 1 or first diffraction order [case CR1 in Fig. 11(b)], and a conventional zone plate plus a refocusing zone plate operated with m r = 3 or third diffraction order [case CR3 in Fig. 11(b)]. The parameters of the zone plates considered can be found in Table 2. This figure demonstrates that the refocusing zone plate can add substantially to the conventional zone plate's focusing efficiency.
Having seen the effect of a refocusing zone plate alone in Fig. 11(b), we add a stacked zone plate in the position indicated in Fig. 11(a) to give the results shown in Fig. 12. In this case, a comparison was made between a conventional zone plate (C) with and without a stacked zone plate (S), and with and without a third-order refocusing zone plate (R3), using the parameters shown in Table 3. As can be seen, the refocusing zone plate adds to the intensity in the first-order focus, though the stacked zone plate S plays a more important role than the refocusing zone plate R3. We note that the stacked S and refocusing R3 zone plates can be fabricated on the same window, so the refocusing zone plate adds little additional complexity. Also, as the separation distance Áz between the conventional zone plate (C) and stacked/refo- Optimum line : space ratio (a) and resulting relative focusing intensity (b) for m = 2 second-order focusing as a function of both zone thickness t i and separation distance Áz. The differences between using a regular or a complementary zone plate for the second optic become larger when thickness t i approaches 2 mm which corresponds to a phase shift.
cusing zone plate (SR3) is increased, the area of the stacked zone plate will decrease while the area of the refocusing zone plate will increase and the improvement from using a refocusing zone plate should also increase. Finally, the use of m r = 3 third-order diffraction in the refocusing zone plate R3 leads to a 1=m 2 r sharpening of the axial intensity profile as shown in Fig. 12(b).

Conclusion
Following the introduction of the concept of stacking multiple zone plates at beyond-proximity distances (Vila-Comamala et al., 2013), we have considered here a variety of design options. Using multislice propagation (Cowley & Moodie, 1957;Li et al., 2017a) to handle the case of wavefield propagation within thicker zone plates, we studied the effects of stacked zone plate parameter design schemes and choice of separation distances. In the supporting information, we also follow prior work on understanding the effects of alignment errors (Vila-Comamala et al., 2013;Gleber et al., 2014) by providing a more detailed look at the effects on the focal spot.
Combining multiple zone plates can lead to higher focusing efficiency and focal spots with different characteristics (such as reduced sidelobes), but the design details matter. One can also understand the potential improvements by realizing that the stacking of multiple planar zone plates leads one toward the situation of a volume grating. Separate mounting and aligning of up to five stacked zone plates has already been demonstrated (Gleber et al., 2014), and this approach with mechanical adjustment has the advantage of allowing one to tune the incident photon energy and then adjust the separation distance between zone plates as required. For single-wavelength operation, one can also use monolithic mounting approaches (Feng et al., 2007) where multiple zone plates are pre-aligned and fixed in place prior to use. In general, more zone plates are better, but the details matter!

Related literature
The following references, not cited in the main body of the paper, have been cited in the supporting information: Pratsch et al. (2014); Simpson & Michette (1983).

Figure 11
Refocusing zone plates can recapture some of the m = À1-order focus light from a conventional zone plate. This is shown schematically in (a), where one can produce a combined optic with a stacked zone plate in the inner diameter and a refocusing zone plate operating in either m r = 1 or first diffraction order, or m r = 3 or third diffraction order. The effect on the focus profile and integrated energy as a function of radius is shown in (b), for the cases of a conventional zone plate alone (case C), or with the addition of a refocusing zone plate operating in first diffraction order (case CR1), or with a refocusing zone plate operating in third diffraction order (case CR3). Both refocusing zone plates offer an increase in focal efficiency, with the CR1 case also offering an improvement in spatial resolution. However, in the CR1 case the refocusing zone plate must have finer zone width dr N than the conventional zone plate, and usually the conventional zone plate is fabricated out to the limits of what can be achieved in nanolithography; in the CR3 case, the finest zone width dr N is the same for the conventional (C) and third-order refocusing (R3) zone plates. The parameters for the zone plates used for the calculation of focal intensities (b) are given in Table 2. Table 2 Zone plate parameters used for the calculation shown in Fig. 11(b).
In the case of a refocusing zone plate operating in first diffraction order, one can see that a much smaller outermost zone width dr N is required than in the conventional zone plate. Because nanolithography processes are often limited in their achievable aspect ratio t i =dr N , the thickness of R1 was decreased to maintain a limiting aspect ratio of 20.

Figure 12
Effect of using both stacked and refocusing zone plates. Four combinations were compared: a conventional zone plate alone (C), or optionally with a stacked zone plate (S), or optionally with a third diffraction order refocusing zone plate (R3). This gives combinations C, CS, CR3 and CSR3 as indicated. On the left (a) is shown the focused intensity profile and radial integral of energy, while on the right (b) is shown the set of axial intensity profiles. Adding a refocusing zone plate provides some increase in focused intensity, but only over a narrow depth of focus range reduced by a factor of 1=m 2 due to the presence of the third diffraction order refocusing zone plate (R3). The parameters for the individual zone plates are shown in Table 3.