Diffraction theory for azimuthally structured Fresnel zone plate

Transcription

1 T. Vierke and J. Jahns Vol. 31, No. 2 / February 2014 / J. Opt. Soc. A. A 363 Diffraction theory for aziuthally structured Fresnel zone plate Thordis Vierke and Jürgen Jahns* FernUniversität in Hagen, Lehrgebiet Mikro- und Nanophotonik, Universitätsstr. 27/PRG, Hagen, Gerany *Corresponding author: jahns@fernuni hagen.de Received Septeber 27, 2013; revised Deceber 16, 2013; accepted Deceber 18, 2013; posted Deceber 20, 2013 (Doc. ID ); published January 23, 2014 A conventional Fresnel zone plate (FZP) consists of concentric rings with an alternating binary transission of zero and one. In an aziuthally structured Fresnel zone plate (afzp), the light transission of the transparent zones is odulated in the aziuthal direction, too. The resulting structure is of interest for extree ultraviolet and x-ray iaging, in particular, because of its iproved echanical stability as copared to the siple ring structure of an FZP. Here, we present an analysis of the optical perforance of the afzp based on scalar diffraction theory and show nuerical results for the light distribution in the focal plane. These will be copleented by calculations of the optical transfer function Optical Society of Aerica OCIS codes: ( ) Diffractive lenses; ( ) Nanolithography; ( ) X-ray iaging; ( ) Microstructure fabrication; ( ) Nanostructure fabrication; ( ) Astronoical optics INTRODUCTION Focusing and iaging of extree ultraviolet (EUV) and x-ray radiation (i.e., at wavelengths fro approxiately 0.1 to 100 n) has any applications. In the technical sciences, for exaple, there exists a strong push toward developing lithographic systes at these wavelengths in order to reduce structural sizes further [1]. For life science applications, one is interested in high-resolution x-ray icroscopy and spectroscopy [2]. However, for these purposes, the use of conventional refractive lenses is not practical: first, at these wavelengths all aterials are strongly absorbing, and, second, the values of the refractive index are very close to 1. Both phenoena are expressed by the coplex refractive index, which, at these short wavelengths, can be expressed as n 1 δ iβ with δ, β 1 [3]. Since δ assues very sall values of typically 10 5 and less, refraction is very weak for x rays. Nonetheless, it has been suggested and deonstrated to use stacks of refractive lenses ade of a suitable aterial, in order to achieve focusing of x-ray radiation [4]. As an alternative to using refraction, a diffractive ipleentation is also of interest. Diffractive lenses based on the classical Fresnel zone plate (FZP) have been considered for soe tie for x-ray applications [5], used, for exaple, in a grazing incidence configuration [6]. A conventional FZP consists of alternating opaque and transparent rings; see Fig. 1(a). When aking such a structure with ring widths at the nanoscale, it is iportant to also consider its theral and echanical properties in order to avoid deforation during operation in a lithographic syste, for exaple. Conventional FZPs are difficult to realize in this regard due to their ring structure. Probles ay occur due to echanical and theral deforations, which ay lead to a degradation of the optical perforance. Significant progress was ade when the photon sieve was introduced several years ago [7]. The photon sieve consists of a large nuber of pinholes in a thin ebrane, seeingly randoly distributed; see Fig. 1(b). This structure is very proising with respect to the proble entioned, since the transparent areas are not contiguous. This has ade possible, for exaple, the ipleentation of large photon sieves for astronoical purposes [8]. An analytical description of the photon sieve is straightforward by using the odel of individual far-field diffraction [9]. However, the photon sieve consists of thousands of pinholes, which akes an optiized design and fabrication potentially a rather coplex task. Hence, a third FZP-based structure is of interest and was introduced recently by Mitsuishi et al. [10]. Here, again, the structure of a FZP is used. However, the transparent zones are odulated in a binary fashion in the aziuthal direction with the transission varying between 0 and 1, as shown in Fig. 1(c). In the following, we will denote this eleent as aziuthally structured FZP (afzp). It should be added that in the work of both Andersen [8] and Mitsuishi et al. [10], the aspect of a lightweight ipleentation played an iportant role. The interested reader will find interesting arguents regarding fabrication and experiental perforance in the references given above. It is the purpose of this article to present a theory for the focusing behavior of the afzp. Although the echanical aspects of the eleents are an essential part of the otivation, we will not consider the here since a thorough analysis would be well outside the scope of this article. Rather, we will focus on the optical properties of the eleents and analyze the influence of the various optical design paraeters. These are the nuber M of rings (or zones) used and the nuber K of openings in each ring. We show at the beginning that for sall values of M and K the light distribution in the focal plane ay exhibit significant blur. Fro basic optical considerations, it is clear that these occur due to the discrete phase contributions fro the different openings in each ring. In order to obtain a very sharp focus, the phases should be as evenly distributed as /14/ $15.00/ Optical Society of Aerica

2 364 J. Opt. Soc. A. A / Vol. 31, No. 2 / February 2014 T. Vierke and J. Jahns ring pattern. The rando aziuthal phase is the third design paraeter. This article is organized as follows: in Section 2, we present a atheatical odel for describing the structure of the eleent and thereby introduce our notation. In Section 3, we apply well-known results fro near-field diffraction to derive an analytical description for the focusing properties of the afzp. In order to keep the equations anageable for nuerical calculations, two siplifications are introduced, and their ipact on the results is discussed. In Section 4, several exaples for the light distribution obtained in the focal plane of the afzp are given, showing the influence of the design paraeters. In addition, as will be pointed out, the theory of the optical transfer function (OTF) is an appropriate tool to evaluate the perforance of the afzp. OTF calculations will copleent the results for the focal spot distribution. Finally, Section 5 contains soe concluding rearks. 2. MATHEMATICAL DESCRIPTION OF THE OBJECT FUNCTION We start with the conventional FZP. We denote its aplitude transission by a 1D function gr 0, where r 0 is the radial coordinate in the object plane. Here, we assue the validity of the paraxial approxiation. In this case, a particularly siple atheatical description is obtained since an FZP is periodic in r 2 0 [see Fig. 2(a)] and can be expressed by with gr 2 0 XM 1 v r 2 0 (1) r v r rect 0 1 2r 2 1 r : (2) gr 2 0 is the aplitude transission function of the FZP. v r 2 0 describes the transittance of the th individual ring, M denotes the total nuber of rings, and r 2 1 is the period in r2 0. The periodicity in r 2 0 finds its expression also in the wellknown design rule for an FZP according to which in the paraxial case the radius of the th zone is given as r r1. p In order to describe the aziuthal odulation of the afzp, we first consider a single ring. It is shown in a Cartesian coordinate syste in Fig. 2(b) and as a function of the aziuthal coordinate ϕ in Fig. 2(c). We assue that the openings are distributed regularly over the interval 0; 2π. For the th ring, we describe the ϕ-dependent odulation by the ter Fig. 1. Various diffractive eleents used for focusing: (a) conventional FZP, (b) photon sieve, and (c) aziuthally structured FZP. w ϕ XK ϕ k2π K rect ϕ s : (3) π K k1 possible. This will tend to be the case for large values of M and K, in particular, if K increases toward the outer rings. As we will show, a statistical phase ay be introduced in the design by adding a rando shift in the aziuthal direction for each K denotes the nuber of openings in the th ring (in the figure: K 3). ϕ s is the offset of the periodic pattern in the aziuthal coordinate. The coplete afzp consisting of M rings can now be described by the suation over M structured rings:

3 T. Vierke and J. Jahns Vol. 31, No. 2 / February 2014 / J. Opt. Soc. A. A 365 g(r 02 ) 1 r 12 /2 r 1 2 2r 1 2 3r 1 2 y 0 (a) 2π/K r 0 2 ZZ ux; y; z eikz u 0 x 0 ;y 0 e ik 2z xx 0 2 yy 0 2 dx 0 dy 0 : (5) Here, u 0 x 0 ;y 0 is the field in plane z 0; in our analysis it is identical to the transission function of the object. ux; y; z is the resulting near-field distribution in plane z>0. In the following, we shall drop the ter 1 since we will assue both z and λ to be fixed, so that their product yields a constant. For the given structure, it is convenient to use a forulation in polar coordinates [Fig. 3(a)]. The following coordinate transforations apply: φ s x 0 and x 0 r 0 cos ϕ und y 0 r 0 sin ϕ (6) x r cos θ und y r sin θ: (7) 1 w(φ) 2π/K φ=π/k (b) φ s φ l φ u (c) Fig. 2. (a) Transission function of conventional FZP in radial direction shown as a function of r 2 0. Aziuthal odulation of a single ring is shown (b) for Cartesian coordinates and (c) along the aziuthal coordinate ϕ for r 0 const:k, nuber of openings of the ring (later referred to as K for the th ring); ϕ l and ϕ u, lower and upper coordinates for single aziuthal opening; Δϕ, width of the opening; ϕ s, rando aziuthal offset of a ring. g a r 2 0 ; ϕ XM v r 2 0 w ϕ: (4) 1 Coparison with Eq. (1) shows that for the aziuthally unstructured FZP, w ϕ 1. We note further that, in general, the afzp is not separable in r 0 and ϕ unless every ring consists of the sae nuber of openings positioned at the sae aziuthal positions, i.e., when K and ϕ s are independent of. As we will see during the following analysis, one will ai at increasing the nuber of transparent openings with increasing ring index in order to obtain a sharp focus. As entioned above and as will be investigated later, the phase offset ϕ s ay be used as a design paraeter in order to optiize the perforance of the eleent. 3. SCALAR DIFFRACTION THEORY FOR THE afzp For the theoretical analysis of the focusing behavior of an FZP, we use scalar diffraction theory, in particular, the wellknown results of near-field diffraction in the approxiation of the Kirchhoff Fresnel diffraction integral [11]: 2π φ For the 2D differential the coordinate transforation yields dx 0 dy 0 r 0 dr 0 dϕ. With r 0 dr 0 1 2dr 2 0, Eq. (5) can be rewritten as ur;θ;ze ikz Z r Z 2π ϕ0 g a r 2 0 ; ϕei π r2 0 r2 e i2πrr 0 cosϕ θ dϕdr 2 0 : By pulling the r 2 -dependent phase factor out of the integral and by disregarding nonrelevant ters, we obtain (8) Z XM ikz ur;θ;ze v r 2 1 r W r; θe i2π r 2 0 2dr 2 0 : (9) Here, the order of suation and integration was interchanged. The ter denoted as W r;θ contains the integral over ϕ: Z 2π W r;θ;z w ϕe i2πrr 0 cosϕ θ dϕ: (10) ϕ0 As is known, for a paraxial FZP the periodicity in r 2 0 leads to a discrete set of foci along the z axis. Matheatically, this ay be seen a result of McCutchen s theore [12], which states that the aplitude along the optical axis (i.e., for r 0) is given as the Fourier transfor of the transverse structure. The first focus occurs in plane z f r 2 1 2λ, which is the plane for which we calculate the field distribution [Fig. 3(b)]. 4. INTERMEZZO: CONVENTIONAL FZP To refer to the results later, we apply our theory to the case of a conventional FZP. In that case, w ϕ 1. The structure of the FZP and the noralized aplitude distribution in the focal plane are shown in Fig. 4(a). One ay use the r 2 0 periodicity of the eleent for a relatively easy calculation of the field in the focal plane [13]. In this case, the result is, of course, independent of θ, and one obtains W r; z 2πJ 0 2π rr 0 : (11)

4 366 J. Opt. Soc. A. A / Vol. 31, No. 2 / February 2014 T. Vierke and J. Jahns Fig. 3. (a) Notation used for polar coordinates in object and observation plane. (b) Setup considered consisting of afzp illuinated by a plane wave of wavelength λ. The focus is generated at a distance z r 2 1 2λ fro the afzp. p For an FZP with radius R M r1, integration over r 2 0 according to Eq. (9) yields the well-known result for the focal plane z f r 2 1 2λ: ur;z f 2πR 2 e i2πr2 λf J 1 2πRr λf : (12) 2πRr λf In Eqs. (11) and (12), J 0 and J 1 are the zeroth and first Bessel functions, respectively, which describe the focus generated by a conventional FZP. The perforance of a lens (or a coplete iaging syste, respectively) ay be suitably analyzed by using the concept of the OTF that describes the transission characteristics of the lens or syste [11]. It is given as the noralized autocorrelation function of the lens pupil. In our case with the real-valued, binary transission function g a r 2 0 ; ϕ, we can write R Rr OTFr 2 0 ; ϕ 2 ϕ g ar 02 0 ; ϕ0 g a r 02 0 r2 0 ; ϕ0 ϕdr 02 0 dϕ0 0 R Rϕ g2 ar 02 0 ; ϕ0 dr 02 : (13) 0 dϕ0 r 2 0 Since a Fourier relationship exists between the OTF and the intensity in the focal plane, the OTF can also be calculated by an inverse Fourier transforation [11]. Here, however, we use Eq. (13), which is particularly convenient to calculate when we represent the transission function of an FZP in an r 2 0 ; ϕ-coordinate syste [Fig. 4(b)]. Figures 4(c) and 4(d), respectively, show the noralized aplitude distribution in the focal plane and the corresponding OTF. Note that the OTF exhibits a decrease in the spatial coordinate due to the finite extension of the FZP in r 2 0. However, it reains constant Fig. 4. Conventional FZP: (a) transission function in x 0 ;y 0 for in ϕ because of the cyclic nature of the aziuthal coordinate. M 5. (b) FZP shown in r 2 0 ; ϕ diagra. (c) 2D aplitude distribution in the focal plane (f, focal length; D, diaeter of the FZP: Before we return to further discussion of the afzp, we would like to add a few rearks on the scalar approach and D 2R). (d) Optical transfer function as autocorrelation function the choice of paraeters for the subsequent siulations. In all in r 2 0 ; ϕ-coordinates. the exaples to follow, we use λ 1 μ, f 10; 000 μ. For these values, r μ. It is easy to show that for a binary The scalar approach is known to be justified as long as FZP, the iniu feature size w in, i.e., the width of the p outerost ring, is w in r 1 4 w in λ. The largest value of M that is used in the subsequent M as long as M 1 [14]. calculations is 50, which corresponds to a iniu ring

5 T. Vierke and J. Jahns Vol. 31, No. 2 / February 2014 / J. Opt. Soc. A. A 367 width of w in 5 μ. For the sake of copleteness, we would like to add that the f-nuber of the FZP ay also be expressed directly by the nuber of periods (or zones): since p the diaeter is 2R 2r M 2 M r1, the f -nuber is f no: p r 1 4 M λwin λ. Note that by suitable scaling, the calculation can also be applied to other (in particular, x ray) wavelengths and device diensions. Obviously, linear scaling of wavelength and period by the paraeter s (i.e., λ sλ, r 1 sr 1 ) also scales the focal length of the FZP linearly: f sr 1 2 2sλ sf. For exaple, if we reduce the wavelength to λ 50 n (i.e., s 1 20) and scale the device features accordingly, a focal length f 500 μ would result. This value would actually be quite sall for x-ray lithography. Practical values are considerably larger, which iplies that for these applications, the scalar approach is well justified. A. r 2 0 Integral The first siplification is obtained by perforing the r 0 integration over infinitesial ring widths rather than rings with finite widths. This is possible since the shape of the focal spot is deterined by the diaeter of the afzp, not by the width of an individual ring. Matheatically, we treat this situation by expressing the case of infinitesial rings by reducing the r related part of the object transittance [copare Eq. (1)] to v r 2 0 δr2 0 r2 (14) with r 2 r 2 1. As a side reark, we note that no -dependent weighting factor occurs here. This reflects the fact that the all rings in an FZP have the sae area, a feature that we aintain in the case of infinitesial rings as well. Equation (9) can now be rewritten as ur; θ;z XM 1 Z r 0 0 δr 2 0 r2 W r; θ;ze r2 i2π 0 2dr 2 0 : (15) I(r) I(r) finite (solid) M=5 infinitesial (a) I ax (b) Fig. 5. Conventional FZP: (a) noralized intensities in the focal plane, calculated for finite ring widths (solid line) and infinitesial ring widths (dashed line). (b) Difference of intensities. Here M 5. Figure 5 shows the intensities Ir jur; z f j 2 in the focal plane calculated by both equations. It also shows the difference ΔIr I eq: 12 r I eq: 1218 r obtained for M 5. The axiu deviation is about 8% (relative to the axiu value). Figure 6 shows the sae coparison for M 50. Notice that the axiu error has decreased significantly; it is now approxiately 0.8% The axiu error ΔI ax is plotted as a function of M in Fig. 7. We observe a continuous decrease of ΔI ax as a function of M. Therefore, we conclude that for large enough values of M it is justified to use the r r By using the sifting property of the delta function, one iediately obtains ur;θ;z XM 1 e i π r2 1W r ; θ;z: (16) If we consider the focal plane, where z f r 2 1 2λ, all exponential ters are equal to 1 and hence ur;θ;z f XM 1 W r ; θ;f: (17) I(r) I(r) M=50 finite (solid) infinitesial (dashed) (a) r In order to estiate the error that is ade by using infinitesial as copared to finite ring widths, we first consider the case of a conventional, i.e., aziuthally unstructured, FZP with w ϕ 1. In this case, one can use Eq. (11): ur;θ;z f 2π XM 1 J 0 2π r r : (18) We use Eqs. (12) and (18) to copare the results of the calculations using finite and infinitesial ring widths (Figs. 5 7) I ax (b) Fig. 6. Conventional FZP: (a) noralized intensities in the focal plane, calculated for finite ring widths (solid line) and infinitesial ring widths (dashed line). (b) Difference of intensities. Here M 50. r

6 368 J. Opt. Soc. A. A / Vol. 31, No. 2 / February 2014 T. Vierke and J. Jahns approxiation given by Eq. (17) to calculate the focal spot aplitude, since eventually the error becoes negligible. B. ϕ Integral We now consider the integral over the aziuthal variable ϕ in Eqs. (10) and (17), respectively. The odulation along the ϕ coordinate is assued to be binary with transission values of either 0 or 1. Hence, integration along the aziuthal coordinate is perfored over a finite interval Δϕk ϕ u k ϕ l k π K for each individual opening, i.e., fro ϕ ϕ l k to ϕ ϕ u k [see Figs. 2(b) and 2(c)]. Equation (10) can thus be expressed as W XK k1 Z ϕu k ϕ l k e i2πrr cosϕ θ dϕ: (19) The nuerical integration is not directly possible, in general, and can be quite tedious for large values of and K. However, the integration can be siplified and ade suitable for nuerical evaluation by using the Jacobi Anger expansion [15]. With the substitution φ ϕ θ, it reads e i2πrr cos φ J 0 2π rr 2 X i n J n 2π rr cos φ: (20) n1 Z φu φ l I ax Fig. 7. Error ΔI ax as a function of the nuber of rings M. With this and the notation Δφ φ u φ l we can write e i2π rr cos φ dφ J 0 2π rr Δφ Z φu 2 X i n J n 2π rr cos φdφ φ l n1 J 0 2π rr Δφ 2 X n1 i n n J n 2π rr sinnφ u sinnφ l : (21) The aziuthal coordinates of the kth opening are φ u k k2π K θ and φ l k k2π K π K θ, and the width of the aziuthal interval is Δφ π K. Suation over all K openings yields M W XK J 0 2π rr π 2 X i n K k1 n J n 2π rr n1 fsinnφ u k sinnφ l kg : (22) After back substitution to ϕ we get W πj 0 2π rr 2 XK X k1 n1 i n n J n 2π rr fsinnϕ u k θ sinnϕ l k θg: (23) Fro a coputational perspective, the final proble that needs to be solved is the infinite nuber of ters in the su over index n. As it turns out, the coputation can be siplified by liiting the nuber of ters in the Jacobi Anger expansion to a finite nuber. Siulation results not presented here showed that one can approxiate the infinite su over n by a finite su over n ax K ters: W πj 0 2π rr 2 XK X K k1 n1 i n n J n 2π rr fsinnϕ u k θ sinnϕ l k θg: (24) Reark: In our investigations we calculated the su for n ax K and for n ax K. By coparison, we find that this approxiation appears to be very precise particularly for sall values of r, typically for the coordinates of the central peak and the first sidelobes. For larger values of r, soe deviations occur. However, these potentially cancel if K ax varies with the ring index. Hence, although without foral proof, we feel confident that Eq. (24) is justified. It can now be inserted in Eq. (16) to obtain ur;θ;z XM 1 e i2πr2 1 2 W r; θ;z: (25) As before, if we look at the focal plane z f r 2 1 2λ, this expression siply becoes ur;θ;z f XM 1 W r; θ;f; (26) where W is calculated according to Eq. (24). This is the final result of the analysis: Eq. (26) in cobination with Eq. (24) represents a atheatical forulation that lends itself to an efficient nuerical evaluation of the afzp. 5. SIMULATION RESULTS The design paraeters that can be used for an afzp are the nuber of rings M, the nuber of openings in the innerost ring K 1, the increase of openings ΔK, and the aziuthal offset ϕ s. For given M and K 1, the reaining paraeters are ΔK and ϕ s, which we want to analyze first. This leads to four cases (Table 1), for which results will be presented below.

7 T. Vierke and J. Jahns Vol. 31, No. 2 / February 2014 / J. Opt. Soc. A. A 369 Table 1. Four Design Variations ΔK 0 >0 ϕ s 0 case 1 case 2 rando case 3 case 4 A. Influence of K and ϕ s We investigate the influence of the nuber of openings by evaluation of Eq. (25) for different values of K. Obviously, one ay use the sae nuber of openings in each ring (K const:) or increase it, for exaple, linearly, i.w., K K 1 ΔK. In the exaples below, the increase in K will always be linear. In order to visualize the effects, we begin with a sall nuber of rings, i.e., for M Case 1 (ΔK 0 and ϕ s 0): Here, the nuber of openings K is constant for each ring. In this exaple, it is M 5 and K 11 [Fig. 8(a)]. The aplitude distribution in the focal plane is shown in Fig. 8(b). One can tell that the syetry of the eleent leads to relatively strong sidelobes with a corresponding syetry. The periodicity of the eleent in the aziuthal direction with openings all appearing at the sae positions [see Fig. 8(c)] leads to a ϕ-periodic OTF for that eleent with a full odulation [Fig. 8(d)]. 2. Case 2 (ΔK >0 and ϕ s 0): Here, the nuber of openings increases with ; in the siplest case, we assue a linear increase. In our particular exaple, it is ΔK 2, again for M 5, K 1 11 [Fig. 9(a)]. The aplitude distribution in the focal plane in Fig. 9(b) shows a reduced, yet still visible, structuring of the sidelobes. Since the eleent has a less pronounced periodicity in ϕ [see Fig. 9(c)], the OTF sears out significantly, which is desirable. However, it also shows a strong decay in the aziuthal direction [Fig. 9(d)]. This is caused by the relatively large variation of K fro the innerost to the outerost ring. 3. Case 3 (ΔK 0 and ϕ s 0, Rando): Figure 10(a) shows a structure with a constant nuber of openings in each ring as in Case 1 (ΔK 0), however, with a variable rando aziuthal phase shift. ϕ s varies with the ring index. Obviously, the rando phase leads to a significant reduction of the sidelobes [Fig. 10(b)]. The calculated plot shows a virtually diffraction-liited spot, even for the sall values of M and K used here. This is rearkable, since the eleent still exhibits a strong periodicity, as one can tell fro the r 2 0 ; ϕ representation [Fig. 10(c)] and the corresponding OTF shown in Fig. 10(d). Note that the OTF is very siilar as for Case 1. Yet, due to the rando aziuthal phase offset, there are slight differences in both the r 2 0 and the ϕ dependency. Notice, for exaple, that for the r 2 0 r2 p the peaks have broadened. Furtherore, for different values of r 2 0 r2 p, an offset occurs in the aziuthal direction. This is the reason the periodic sidelobes have disappeared [copare Fig. 8(b)]. Fig. 8. Case 1: afzp in x 0 ;y 0 for M 5 rings, K 1 11 openings, and ΔK 0. (b) afzp shown in r 2 0 ; ϕ diagra. (c) 2D aplitude distribution in focal plane. (d) Autocorrelation in r 2 0 ; ϕ. 4. Case 4 (ΔK >0 and ϕ s 0, Rando): Of course, it is also possible to cobine a variable K and a variable ϕ s. The results are shown in Fig. 11. Here, however, the rando phase was chosen to be saller than in Case 3 to show the effect [Fig. 11(a)]. The aplitude is of siilar quality

8 370 J. Opt. Soc. A. A / Vol. 31, No. 2 / February 2014 T. Vierke and J. Jahns Fig. 9. Case 2: afzp in x 0 ;y 0 for M 5 rings, K 1 11 openings, and ΔK 2. (b) afzp shown in r 2 0 ; ϕ diagra. (c) 2D aplitude distribution in focal plane. (d) Autocorrelation in r 2 0 ; ϕ. Fig. 10. Case 3: afzp in x 0 ;y 0 for M 5 rings, K 1 11 openings, ΔK 0, and a rando aziuthal phase added. (b) afzp shown in r 2 0 ; ϕ diagra. (c) 2D aplitude distribution in focal plane. (d) Autocorrelation in r 2 0 ; ϕ. as in Case 3; however, a weakly odulated sidelobe can be seen in Fig. 11(b). Due to the increasing nuber of openings in each ring [see also Fig. 11(c)], the OTF shows nearly constant side peaks in Fig. 11(d). Note that in coparison to Fig. 9(d), the OTF is soother and does not exhibit a significant decay in the aziuthal direction. In conclusion, we ay say that undesired sidelobes that occur for Cases 1 and 2 are reduced by increasing the nuber of

9 T. Vierke and J. Jahns Vol. 31, No. 2 / February 2014 / J. Opt. Soc. A. A M= r in µ (a) M= r in µ (b) Fig. 12. Large afzp: intensity plots for (a) M 10 rings, K 1 20 openings, and ΔK 2; and (b) M 50 rings, K 1 10 openings, and ΔK 1. In both cases, ϕ s 0. Gray curves show the intensity of a focal plot generated by a conventional FZP. values of K and for ΔK >0, the sidelobes get less pronounced since the distribution of the aziuthal positions is ore evenly distributed in the interval 0; 2π and thus the associated phase contributions will tend to cancel out. This becoes very obvious for large values of K and ΔK >0. This effect can be further enhanced by adding a statistical phase ϕ s. Fig. 11. Case 4: afzp in x 0 ;y 0 for M 5 rings, K 1 11 openings, ΔK 2, and a rando aziuthal phase added. (b) afzp shown in r 2 0 ; ϕ diagra. (c) 2D aplitude distribution in focal plane. (d) Autocorrelation in r 2 0 ; ϕ. openings K. The influence of K is easily understood: for sall values of K as well as for ΔK 0, undesired sidelobes occur around the focus. In particular, if ΔK 0, even for large values of K 1, these sidelobes are significant. By increasing the B. Influence of M: Large afzp The exaples shown above were shown for sall eleents with only five rings in order to ake the influence of the various paraeters visible. It is obvious that for a large nuber of rings and a varying nuber of openings, the phase values generated by the nuerous openings in the afzp will lead to a strong reduction of undesired sidelobes. In Fig. 12, we show the intensity plots for two afzps with relatively large nubers of rings. In both cases, no rando phase shift was used in the design, i.e., ϕ s 0. First, in Fig. 12(a), the plot is shown for the afzp shown in Fig. 1(c). This eleent has M 10 rings, K 1 20 and ΔK 2. For coparison, Fig. 12(a) also shows the noralized intensity plot for a conventional FZP with the sae nuber of rings (gray curve). One can notice a sall, yet significant, difference in the two intensity plots. Finally, in Fig. 12(b) we do the sae coparison, however, here for M 50 rings, K 1 10, and ΔK 2. Obviously, here the difference is nearly zero between the focal spot profiles generated by afzp and conventional FZP. 6. CONCLUSION A odel to analyze the focusing properties of afzps has been presented. The calculation offers two significant siplifications that are useful in order to keep the coputational effort at a iniu. The exaples presented show that good focal properties can be obtained with an afzp that, for large values

10 372 J. Opt. Soc. A. A / Vol. 31, No. 2 / February 2014 T. Vierke and J. Jahns of rings and openings, does not differ significantly fro the perforance of a conventional FZP. The ost significant influence coes fro a rando aziuthal phase that helps to avoid deterinistic interference patterns around the focus. Here, we also pointed out the usefulness of the well-known OTF theory for the analysis of FZPs. The virtue of the OTF concept lies in the possibility to perfor the analysis based on the calculation of an autocorrelation, which is relatively easy to carry out for binary structures. We conclude with two further rearks: as an additional degree of freedo, one ay use the widths of the zones in order to reduce sidelobes and even obtain a Gaussian focal spot [16]. Finally, as entioned at the beginning, analysis of the echanical perforance of the different types of zone plate structures is an interesting goal for further investigation. ACKNOWLEDGMENTS The authors thank Qing Cao (Shanghai University), Ji Fienup (University of Rochester), and Jürgen Mohr (Karlsruhe Insitute of Technology, Gerany) for interesting discussions. Furtherore, useful feedback fro the reviewers is highly appreciated. REFERENCES 1. A. Heuberger, X-ray lithography, J. Vac. Sci. Technol. B 6, (1985). 2. W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, Soft x-ray icroscopy at a spatial resolution better than 15 n, Nature 435, (2005). 3. D. T. Attwood, Soft X-Rays and Extree Ultraviolet Radiation: Principles and Applications (Cabridge University, 1999). 4. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, A copound refractive lens for focusing high-energy x-rays, Nature 384, (1996). 5. G. Schahl and D. Rudolph, High-power zone plates as iage foring systes for soft x-rays, Optik 29, (1969). 6. J. Als-Nielsen, D. Jacqueain, K. Kjaer, F. Leveiller, M. Lahav, and L. Leiserowitz, Principles and applications of grazing incidence x-ray and neutron scattering fro ordered olecular onolayers at the air-water interface, Phys. Rep. 246, (1994). 7. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Har, and R. Seeann, Sharper iages by focusing soft x-rays with photon sieves, Nature 414, (2001). 8. G. Andersen, Large optical photon sieves, Opt. Lett. 30, (2005). 9. Q. Cao and J. Jahns, Focusing analysis of the pinhole photon sieve: individual far-field odel, J. Opt. Soc. A. A 19, (2002). 10. I. Mitsuishi, Y. Ezoe, M. Koshiishi, M. Mita, Y. Maeda, N. Y. Yaasaki, K. Mitsuda, T. Shirata, T. Hayashi, T. Takano, and R. Maeda, Evaluation of the soft x-ray reflectivity of icropore optics using anisotropic wet etching of silicon wafers, Appl. Opt. 49, (2010). 11. A. W. Lohann, Optical Inforation Processing (TU Ilenau University, 2006). 12. C. W. McCutchen, Generalized aperture and the threediensional diffraction iage, J. Opt. Soc. A. 54, (1964). 13. J. Jahns and S. Helfert, Introduction to Micro- and Nanooptics (VCH-Wiley, 2012). 14. J. Jahns and S. J. Walker, Two-diensional array of diffractive icrolenses fabricated by thin fil deposition, Appl. Opt. 29, (1990). 15. M. Abraowitz and I. A. Stegun, Handbook of Matheatical Functions with Forulas, Graphs, and Matheatical Tables (Dover, 1964). 16. Q. Cao and J. Jahns, Modified Fresnel zone plates that produce sharp Gaussian focal spots, J. Opt. Soc. A. A 20, (2003).