diffraction theory for azimuthally structured fresnel zone

$: Diffraction theory for azimuthally structured Fresnel zone$
Diffraction theory for azimuthally structuredFresnel zone plate

Thordis Vierke and Jürgen Jahns*

FernUniversität in Hagen, Lehrgebiet Mikro- und Nanophotonik, Universitätsstr. 27/PRG, 58097 Hagen, Germany*Corresponding author: jahns@fernuni‑hagen.de

Received September 27, 2013; revised December 16, 2013; accepted December 18, 2013;posted December 20, 2013 (Doc. ID 198345); published January 23, 2014

A conventional Fresnel zone plate (FZP) consists of concentric rings with an alternating binary transmission ofzero and one. In an azimuthally structured Fresnel zone plate (aFZP), the light transmission of the transparentzones is modulated in the azimuthal direction, too. The resulting structure is of interest for extreme ultraviolet andx-ray imaging, in particular, because of its improved mechanical stability as compared to the simple ring structureof an FZP. Here, we present an analysis of the optical performance of the aFZP based on scalar diffraction theoryand show numerical results for the light distribution in the focal plane. These will be complemented by calcu-lations of the optical transfer function. © 2014 Optical Society of America

OCIS codes: (050.1965) Diffractive lenses; (110.4235) Nanolithography; (110.7440) X-ray imaging;(220.4000) Microstructure fabrication; (220.4241) Nanostructure fabrication; (350.1260) Astronomical optics.http://dx.doi.org/10.1364/JOSAA.31.000363

1. INTRODUCTIONFocusing and imaging of extreme ultraviolet (EUV) and x-rayradiation (i.e., at wavelengths from approximately 0.1 to100 nm) has many applications. In the technical sciences,for example, there exists a strong push toward developinglithographic systems at these wavelengths in order to reducestructural sizes further [1]. For life science applications, one isinterested in high-resolution x-ray microscopy and spectros-copy [2]. However, for these purposes, the use of conventionalrefractive lenses is not practical: first, at these wavelengths allmaterials are strongly absorbing, and, second, the values ofthe refractive index are very close to 1. Both phenomenaare expressed by the complex refractive index, which, at theseshort wavelengths, can be expressed as n ≈ 1 − δ� iβ with δ,β ≪ 1 [3]. Since δ assumes very small values of typically 10−5

and less, refraction is very weak for x rays. Nonetheless, it hasbeen suggested and demonstrated to use stacks of refractivelenses made of a suitable material, in order to achieve focus-ing of x-ray radiation [4].

As an alternative to using refraction, a diffractive imple-mentation is also of interest. Diffractive lenses based onthe classical Fresnel zone plate (FZP) have been consideredfor some time for x-ray applications [5], used, for example, in agrazing incidence configuration [6]. A conventional FZP con-sists of alternating opaque and transparent rings; see Fig. 1(a).When making such a structure with ring widths at the nano-scale, it is important to also consider its thermal and mechani-cal properties in order to avoid deformation during operationin a lithographic system, for example. Conventional FZPsare difficult to realize in this regard due to their ring structure.Problems may occur due to mechanical and thermal deforma-tions, which may lead to a degradation of the optical perfor-mance. Significant progress was made when the “photonsieve” was introduced several years ago [7]. The photon sieveconsists of a large number of pinholes in a thin membrane,

seemingly randomly distributed; see Fig. 1(b). This structureis very promising with respect to the problem mentioned,since the transparent areas are not contiguous. This has madepossible, for example, the implementation of large photonsieves for astronomical purposes [8].

An analytical description of the photon sieve is straightfor-ward by using the model of individual far-field diffraction [9].However, the photon sieve consists of thousands of pinholes,which makes an optimized design and fabrication potentially arather complex task. Hence, a third FZP-based structure is ofinterest and was introduced recently by Mitsuishi et al. [10].Here, again, the structure of a FZP is used. However, the trans-parent zones are modulated in a binary fashion in the azimu-thal direction with the transmission varying between 0 and 1,as shown in Fig. 1(c). In the following, we will denote thiselement as “azimuthally structured FZP” (aFZP). It should beadded that in the work of both Andersen [8] andMitsuishi et al.[10], the aspect of a lightweight implementation played an im-portant role. The interested reader will find interesting argu-ments regarding fabrication and experimental performance inthe references given above.

It is the purpose of this article to present a theory for thefocusing behavior of the aFZP. Although the mechanical as-pects of the elements are an essential part of the motivation,we will not consider them here since a thorough analysiswould be well outside the scope of this article. Rather, we willfocus on the optical properties of the elements and analyze theinfluence of the various optical design parameters. These arethe number M of rings (or zones) used and the number K ofopenings in each ring. We show at the beginning that for smallvalues ofM and K the light distribution in the focal plane mayexhibit significant blur. From basic optical considerations, it isclear that these occur due to the discrete phase contributionsfrom the different openings in each ring. In order to obtain avery sharp focus, the phases should be as evenly distributed as

T. Vierke and J. Jahns Vol. 31, No. 2 / February 2014 / J. Opt. Soc. Am. A 363

1084-7529/14/020363-10$15.00/0 © 2014 Optical Society of America

http://dx.doi.org/10.1364/JOSAA.31.000363

possible. This will tend to be the case for large values ofM andK , in particular, if K increases toward the outer rings. As wewill show, a statistical phase may be introduced in the designby adding a random shift in the azimuthal direction for each

ring pattern. The random azimuthal phase is the third designparameter.

This article is organized as follows: in Section 2, we presenta mathematical model for describing the structure of theelement and thereby introduce our notation. In Section 3,we apply well-known results from near-field diffraction to de-rive an analytical description for the focusing properties of theaFZP. In order to keep the equations manageable for numeri-cal calculations, two simplifications are introduced, and theirimpact on the results is discussed. In Section 4, several exam-ples for the light distribution obtained in the focal plane ofthe aFZP are given, showing the influence of the designparameters. In addition, as will be pointed out, the theory ofthe optical transfer function (OTF) is an appropriate tool toevaluate the performance of the aFZP. OTF calculations willcomplement the results for the focal spot distribution. Finally,Section 5 contains some concluding remarks.

2. MATHEMATICAL DESCRIPTION OF THEOBJECT FUNCTIONWe start with the conventional FZP. We denote its amplitudetransmission by a 1D function g�r0�, where r0 is the radial co-ordinate in the object plane. Here, we assume the validity ofthe paraxial approximation. In this case, a particularly simplemathematical description is obtained since an FZP is periodicin r20 [see Fig. 2(a)] and can be expressed by

g�r20� �XMm�1

vm�r20� (1)

with

vm�r20� � rect�r20 − �m − �1∕2��r21

r21∕2

�: (2)

g�r20� is the amplitude transmission function of the FZP.vm�r20� describes the transmittance of the mth individual ring,M denotes the total number of rings, and r21 is the period in r20.The periodicity in r20 finds its expression also in the well-known design rule for an FZP according to which in the para-xial case the radius of the mth zone is given as rm � ��

mp

r1.In order to describe the azimuthal modulation of the aFZP,

we first consider a single ring. It is shown in a Cartesian co-ordinate system in Fig. 2(b) and as a function of the azimuthalcoordinate ϕ in Fig. 2(c). We assume that the openings aredistributed regularly over the interval �0; 2π�. For themth ring,we describe the ϕ-dependent modulation by the term

wm�ϕ� �XKm

k�1

rect�ϕ − k�2π∕Km� − ϕs

π∕Km

�: (3)

Km denotes the number of openings in the mth ring (in thefigure: Km � 3). ϕs is the offset of the periodic pattern in theazimuthal coordinate. The complete aFZP consisting of Mrings can now be described by the summation over M struc-tured rings:

Fig. 1. Various diffractive elements used for focusing: (a) conven-tional FZP, (b) photon sieve, and (c) azimuthally structured FZP.

364 J. Opt. Soc. Am. A / Vol. 31, No. 2 / February 2014 T. Vierke and J. Jahns

ga�r20;ϕ� �XMm�1

vm�r20�wm�ϕ�: (4)

Comparison with Eq. (1) shows that for the azimuthally un-structured FZP, wm�ϕ� � 1. We note further that, in general,the aFZP is not separable in r0 and ϕ unless every ring consistsof the same number of openings positioned at the same azi-muthal positions, i.e., when K and ϕs are independent of m.As we will see during the following analysis, one will aim atincreasing the number of transparent openings with increas-ing ring index m in order to obtain a sharp focus. As men-tioned above and as will be investigated later, the phaseoffset ϕs may be used as a design parameter in order to opti-mize the performance of the element.

3. SCALAR DIFFRACTION THEORY FORTHE aFZPFor the theoretical analysis of the focusing behavior of anFZP, we use scalar diffraction theory, in particular, the well-known results of near-field diffraction in the approximation ofthe Kirchhoff–Fresnel diffraction integral [11]:

u�x; y; z� ∝ eikz

λz

ZZu0�x0; y0�eik

2z��x�x0�2��y�y0�2 �dx0dy0: (5)

Here, u0�x0; y0� is the field in plane z � 0; in our analysis itis identical to the transmission function of the object.u�x; y; z� is the resulting near-field distribution in plane z > 0.In the following, we shall drop the term 1∕�λz� since we willassume both z and λ to be fixed, so that their product yields aconstant. For the given structure, it is convenient to use aformulation in polar coordinates [Fig. 3(a)]. The followingcoordinate transformations apply:

x0 � r0 cos ϕ und y0 � r0 sin ϕ (6)

and

x � r cos θ und y � r sin θ: (7)

For the 2D differential the coordinate transformation yieldsdx0dy0 � r0dr0dϕ. With r0dr0 � �1∕2�dr20, Eq. (5) can be re-written as

u�r;θ; z� � eikzZ

∞

r20�0

Z2π

ϕ�0ga�r20;ϕ�ei

πλz�r20�r2�e−i2π

rr0λz cos�ϕ−θ�dϕdr20:

(8)

By pulling the r2-dependent phase factor out of the integraland by disregarding nonrelevant terms, we obtain

u�r; θ; z� � eikzXMm�1

Z∞

r20�0vm�r20�Wm�r; θ�ei2π

r20

2λzdr20: (9)

Here, the order of summation and integration was inter-changed. The term denoted as Wm�r; θ� contains the integralover ϕ:

Wm�r; θ; z� �Z

2π

ϕ�0wm�ϕ�e−i2π

rr0λz cos�ϕ−θ�dϕ: (10)

As is known, for a paraxial FZP the periodicity in r20 leads toa discrete set of foci along the z axis. Mathematically, this maybe seen a result of McCutchen’s theorem [12], which statesthat the amplitude along the optical axis (i.e., for r � 0) isgiven as the Fourier transform of the transverse structure.The first focus occurs in plane z � f � r21∕2λ, which is theplane for which we calculate the field distribution [Fig. 3(b)].

4. INTERMEZZO: CONVENTIONAL FZPTo refer to the results later, we apply our theory to the case ofa conventional FZP. In that case, wm�ϕ� � 1. The structure ofthe FZP and the normalized amplitude distribution in the focalplane are shown in Fig. 4(a). One may use the r20 periodicity ofthe element for a relatively easy calculation of the field in thefocal plane [13]. In this case, the result is, of course, indepen-dent of θ, and one obtains

Wm�r; z� � 2πJ0

�2π

rr0λz

�: (11)

r02

r12 2r

12 3r

12

g(r0

2)

r12/2

1

(a)

φs

x0

y0

2π/K

(b)

φ

w(φ)

1

2π

∆φ=π/K2π/K

φs φ

lφ

u

(c)Fig. 2. (a) Transmission function of conventional FZP in radial direc-tion shown as a function of r20. Azimuthal modulation of a single ringis shown (b) for Cartesian coordinates and (c) along the azimuthalcoordinate ϕ for r0 � const: K , number of openings of the ring (laterreferred to as Km for the mth ring); ϕl and ϕu, lower and upper co-ordinates for single azimuthal opening; Δϕ, width of the opening; ϕs,random azimuthal offset of a ring.


For an FZP with radius R ��M

pr1, integration over r20 ac-

cording to Eq. (9) yields the well-known result for the focalplane z � f � r21∕2λ:

u�r; z � f � � �2πR2�ei2πr2λfJ1�2πRr∕λf �2πRr∕λf

: (12)

In Eqs. (11) and (12), J0 and J1 are the zeroth and firstBessel functions, respectively, which describe the focus gen-erated by a conventional FZP.

The performance of a lens (or a complete imaging system,respectively) may be suitably analyzed by using the concept ofthe OTF that describes the transmission characteristics of thelens or system [11]. It is given as the normalized autocorrela-tion function of the lens pupil. In our case with the real-valued,binary transmission function ga�r20;ϕ�, we can write

OTF�r20;ϕ� �Rr20

Rϕ ga�r020 ;ϕ0�ga�r020 − r20;ϕ

0− ϕ�dr020 dϕ0

Rr20

Rϕ g

2a�r020 ;ϕ0�dr020 dϕ0 : (13)

Since a Fourier relationship exists between the OTF andthe intensity in the focal plane, the OTF can also be calculatedby an inverse Fourier transformation [11]. Here, however, weuse Eq. (13), which is particularly convenient to calculatewhen we represent the transmission function of an FZP inan �r20;ϕ�-coordinate system [Fig. 4(b)]. Figures 4(c) and 4(d),respectively, show the normalized amplitude distribution inthe focal plane and the corresponding OTF. Note that theOTF exhibits a decrease in the spatial coordinate due to thefinite extension of the FZP in r20. However, it remains constantin ϕ because of the cyclic nature of the azimuthal coordinate.

Before we return to further discussion of the aFZP, wewould like to add a few remarks on the scalar approach andthe choice of parameters for the subsequent simulations. In allthe examples to follow, we use λ � 1 μm, f � 10; 000 μm. Forthese values, r1 ≈ 141 μm. It is easy to show that for a binaryFZP, the minimum feature size wmin, i.e., the width of the

outermost ring, is wmin ≈ r1∕�4��M

p� as long as M ≫ 1 [14].

The scalar approach is known to be justified as long aswmin ≫ λ. The largest value ofM that is used in the subsequentcalculations is 50, which corresponds to a minimum ring

Fig. 3. (a) Notation used for polar coordinates in object and obser-vation plane. (b) Setup considered consisting of aFZP illuminated by aplane wave of wavelength λ. The focus is generated at a distancez � r21∕2λ from the aFZP.

Fig. 4. Conventional FZP: (a) transmission function in �x0; y0� forM � 5. (b) FZP shown in �r20;ϕ� diagram. (c) 2D amplitude distribu-tion in the focal plane (f , focal length; D, diameter of the FZP:D � 2R). (d) Optical transfer function as autocorrelation functionin �r20;ϕ�-coordinates.


width of wmin ≈ 5 μm. For the sake of completeness, wewould like to add that the f-number of the FZP may also beexpressed directly by the number of periods (or zones): sincethe diameter is 2R � 2rM � 2

��M

pr1, the f -number is f ∕no: �

r1∕�4��M

pλ� � wmin∕λ.

Note that by suitable scaling, the calculation can also beapplied to other (in particular, x ray) wavelengths and devicedimensions. Obviously, linear scaling of wavelength andperiod by the parameter s (i.e., λ → sλ, r1 → sr1) also scalesthe focal length of the FZP linearly: f → �sr1�2∕�2sλ� � sf . Forexample, if we reduce the wavelength to λ � 50 nm (i.e.,s � 1∕20) and scale the device features accordingly, a focallength f � 500 μm would result. This value would actuallybe quite small for x-ray lithography. Practical values are con-siderably larger, which implies that for these applications, thescalar approach is well justified.

A. r20 IntegralThe first simplification is obtained by performing the r0 inte-gration over infinitesimal ring widths rather than rings withfinite widths. This is possible since the shape of the focal spotis determined by the diameter of the aFZP, not by the width ofan individual ring. Mathematically, we treat this situation byexpressing the case of infinitesimal rings by reducing the r20-related part of the object transmittance [compare Eq. (1)] to

vm�r20� � δ�r20 − r2m� (14)

with r2m � mr21. As a side remark, we note that no m-depen-dent weighting factor occurs here. This reflects the fact thatthe all rings in an FZP have the same area, a feature that wemaintain in the case of infinitesimal rings as well. Equation (9)can now be rewritten as

u�r; θ; z� ≈XMm�1

Z∞

r0�0δ�r20 − r2m�Wm�r; θ; z�ei2π

r20

2λzdr20: (15)

By using the sifting property of the delta function, oneimmediately obtains


eiπλzmr21Wm�rm; θ; z�: (16)

If we consider the focal plane, where z � f � r21∕2λ, allexponential terms are equal to 1 and hence

u�r; θ; z � f � ≈XMm�1

Wm�rm; θ; f �: (17)

In order to estimate the error that is made by using infini-tesimal as compared to finite ring widths, we first consider thecase of a conventional, i.e., azimuthally unstructured, FZPwith wm�ϕ� � 1. In this case, one can use Eq. (11):

u�r; θ; z � f � ≈ 2πXMm�1

J0

�2π

rmrλz

�: (18)

We use Eqs. (12) and (18) to compare the results of the cal-culations using finite and infinitesimal ring widths (Figs. 5–7).

Figure 5 shows the intensities I�r� � ju�r; z � f �j2 in the focalplane calculated by both equations. It also shows the differ-ence ΔI�r� � Ieq: 12�r� − Ieq: 1218�r� obtained for M � 5. Themaximum deviation is about 8% (relative to the maximumvalue). Figure 6 shows the same comparison for M � 50.Notice that the maximum error has decreased significantly;it is now approximately 0.8% The maximum error ΔImax isplotted as a function of M in Fig. 7. We observe a continuousdecrease of ΔImax as a function of M . Therefore, we concludethat for large enough values of M it is justified to use the

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

I(r)

r

infinitesimal

finite (solid) M=5

(a)

10 20 30 40 50

-0.08

-0.06

-0.04

-0.02

0.02

0.04

∆I(r)

r

∆Imax

(b)

Fig. 5. Conventional FZP: (a) normalized intensities in the focalplane, calculated for finite ring widths (solid line) and infinitesimalring widths (dashed line). (b) Difference of intensities. Here M � 5.

5 10 15 20 25

0.2

0.4

0.6

0.8

1.0

I(r)

r

infinitesimal (dashed)

finite (solid)M=50

(a)

5 10 15 20 25

-0.008

-0.006

-0.004

-0.002

0.002

∆Imax

∆I(r)

r

(b)

Fig. 6. Conventional FZP: (a) normalized intensities in the focalplane, calculated for finite ring widths (solid line) and infinitesimalring widths (dashed line). (b) Difference of intensities. Here M � 50.


approximation given by Eq. (17) to calculate the focal spotamplitude, since eventually the error becomes negligible.

B. ϕ IntegralWe now consider the integral over the azimuthal variable ϕ inEqs. (10) and (17), respectively. The modulation along the ϕcoordinate is assumed to be binary with transmission valuesof either 0 or 1. Hence, integration along the azimuthal coor-dinate is performed over a finite interval Δϕ�k� � ϕu�k� −ϕl�k� � π∕Km for each individual opening, i.e., from ϕ �ϕl�k� to ϕ � ϕu�k� [see Figs. 2(b) and 2(c)]. Equation (10)can thus be expressed as

Wm �XKm

k�1

Zϕu�k�

ϕl�k�ei2π

rrmλz cos�ϕ−θ�dϕ: (19)

The numerical integration is not directly possible, in gen-eral, and can be quite tedious for large values of m andKm. However, the integration can be simplified and madesuitable for numerical evaluation by using the Jacobi–Angerexpansion [15]. With the substitution φ � ϕ − θ, it reads

ei2πrrmλz cos φ � J0

�2π

rrmλz

�� 2

X∞n�1

inJn

�2π

rrmλz

�cos φ: (20)

With this and the notation Δφ � φu − φl we can write

Zφu

φl

ei2πrrmλz cos φdφ � J0

�2π

rrmλz

�Δφ

�Z

φu

φl

2X∞n�1

inJn

�2π

rrmλz

�cos φdφ

� J0

�2π

rrmλz

�Δφ

� 2X∞n�1

−in

nJn

�2π

rrmλz

��sin�nφu�

− sin�nφl��: (21)

The azimuthal coordinates of the kth opening are φu�k� �k�2π∕Km� − θ and φl�k� � k�2π∕Km� − �π∕Km� − θ, and thewidth of the azimuthal interval is Δφ � �π∕Km�. Summationover all Km openings yields

Wm �XKm

k�1

�J0

�2π

rrmλz

�π

Km� 2

X∞n�1

−in

nJn

�2π

rrmλz

�

× fsin�nφu�k�� − sin�n�φl�k��g�: (22)

After back substitution to ϕ we get

Wm � πJ0

�2π

rrmλz

�� 2

XKm

k�1

X∞n�1

−in

nJn

�2π

rrmλz

�

× fsin�n�ϕu�k� − θ�� − sin�n�ϕl�k� − θ��g: (23)

From a computational perspective, the final problem thatneeds to be solved is the infinite number of terms in thesum over index n. As it turns out, the computation can be sim-plified by limiting the number of terms in the Jacobi–Angerexpansion to a finite number. Simulation results not presentedhere showed that one can approximate the infinite sum over nby a finite sum over nmax � Km terms:

Wm ≈ πJ0

�2π

rrmλz

�� 2

XKm

k�1

XKm

n�1

−in

nJn

�2π

rrmλz

�

× fsin�n�ϕu�k� − θ�� − sin�n�ϕl�k� − θ��g: (24)

Remark: In our investigations we calculated the sum fornmax ≫ Km and for nmax � Km. By comparison, we find thatthis approximation appears to be very precise particularly forsmall values of r, typically for the coordinates of the centralpeak and the first sidelobes. For larger values of r, some de-viations occur. However, these potentially cancel if Kmax

varies with the ring index m. Hence, although without formalproof, we feel confident that Eq. (24) is justified. It can now beinserted in Eq. (16) to obtain


ei2πmr2

12λz Wm�r; θ; z�: (25)

As before, if we look at the focal plane z � f � r21∕2λ, thisexpression simply becomes

u�r; θ; z � f � ≈XMm�1

Wm�r; θ; f �; (26)

where Wm is calculated according to Eq. (24). This is the finalresult of the analysis: Eq. (26) in combination with Eq. (24)represents a mathematical formulation that lends itself toan efficient numerical evaluation of the aFZP.

5. SIMULATION RESULTSThe design parameters that can be used for an aFZP arethe number of rings M , the number of openings in the inner-most ring K1, the increase of openings ΔK , and the azimuthaloffset ϕs�m�. For given M and K1, the remaining parametersare ΔK and ϕs�m�, which we want to analyze first. Thisleads to four cases (Table 1), for which results will be pre-sented below.

5 10 15 20 25 30 35 40 45 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

|∆Imax

|

M

Fig. 7. Error ΔImax as a function of the number of rings M .


A. Influence of K and ϕs

We investigate the influence of the number of openings byevaluation of Eq. (25) for different values of K . Obviously,one may use the same number of openings in each ring(Km � const:) or increase it, for example, linearly, i.w.,Km � K1 � ΔK . In the examples below, the increase in K willalways be linear. In order to visualize the effects, we beginwith a small number of rings, i.e., for M � 5.

1. Case 1 (ΔK � 0 and ϕs � 0):Here, the number of openings Km is constant for each ring. Inthis example, it is M � 5 and K � 11 [Fig. 8(a)]. The ampli-tude distribution in the focal plane is shown in Fig. 8(b).One can tell that the symmetry of the element leads to rela-tively strong sidelobes with a corresponding symmetry. Theperiodicity of the element in the azimuthal direction withopenings all appearing at the same positions [see Fig. 8(c)]leads to a ϕ-periodic OTF for that element with a full modu-lation [Fig. 8(d)].

2. Case 2 (ΔK > 0 and ϕs � 0):Here, the number of openings increases with m; in the sim-plest case, we assume a linear increase. In our particular ex-ample, it is ΔK � 2, again for M � 5, K1 � 11 [Fig. 9(a)]. Theamplitude distribution in the focal plane in Fig. 9(b) shows areduced, yet still visible, structuring of the sidelobes. Since theelement has a less pronounced periodicity in ϕ [see Fig. 9(c)],the OTF smears out significantly, which is desirable. However,it also shows a strong decay in the azimuthal direction[Fig. 9(d)]. This is caused by the relatively large variationof Km from the innermost to the outermost ring.

3. Case 3 (ΔK � 0 and ϕs ≠ 0, Random):Figure 10(a) shows a structure with a constant number ofopenings in each ring as in Case 1 (ΔK � 0), however, witha variable random azimuthal phase shift. ϕs varies with thering index m. Obviously, the random phase leads to a signifi-cant reduction of the sidelobes [Fig. 10(b)]. The calculatedplot shows a virtually diffraction-limited spot, even for thesmall values of M and K used here. This is remarkable, sincethe element still exhibits a strong periodicity, as one can tellfrom the �r20;ϕ� representation [Fig. 10(c)] and the corre-sponding OTF shown in Fig. 10(d). Note that the OTF is verysimilar as for Case 1. Yet, due to the random azimuthal phaseoffset, there are slight differences in both the r20 and the ϕ

dependency. Notice, for example, that for the r20 � r2p thepeaks have broadened. Furthermore, for different values ofr20∕r

2p, an offset occurs in the azimuthal direction. This is

the reason the periodic sidelobes have disappeared [compareFig. 8(b)].

4. Case 4 (ΔK > 0 and ϕs ≠ 0, Random):Of course, it is also possible to combine a variable Km and avariable ϕs. The results are shown in Fig. 11. Here, however,the random phase was chosen to be smaller than in Case 3 toshow the effect [Fig. 11(a)]. The amplitude is of similar quality

Table 1. Four Design Variations

ΔK

0 >0

ϕs 0 case 1 case 2random case 3 case 4

Fig. 8. Case 1: aFZP in �x0; y0� for M � 5 rings, K1 � 11 openings,and ΔK � 0. (b) aFZP shown in �r20;ϕ� diagram. (c) 2D amplitudedistribution in focal plane. (d) Autocorrelation in �r20;ϕ�.


as in Case 3; however, a weakly modulated sidelobe can beseen in Fig. 11(b). Due to the increasing number of openingsin each ring [see also Fig. 11(c)], the OTF shows nearly con-stant side peaks in Fig. 11(d). Note that in comparison to

Fig. 9(d), the OTF is “smoother” and does not exhibit a signifi-cant decay in the azimuthal direction.

In conclusion, we may say that undesired sidelobes that oc-cur for Cases 1 and 2 are reduced by increasing the number of

Fig. 9. Case 2: aFZP in �x0; y0� for M � 5 rings, K1 � 11 openings,and ΔK � 2. (b) aFZP shown in �r20;ϕ� diagram. (c) 2D amplitudedistribution in focal plane. (d) Autocorrelation in �r20;ϕ�.

Fig. 10. Case 3: aFZP in �x0; y0� for M � 5 rings, K1 � 11 openings,ΔK � 0, and a random azimuthal phase added. (b) aFZP shown in�r20;ϕ� diagram. (c) 2D amplitude distribution in focal plane. (d) Auto-correlation in �r20;ϕ�.


openings K . The influence of K is easily understood: for smallvalues of K as well as for ΔK � 0, undesired sidelobes occuraround the focus. In particular, if ΔK � 0, even for large val-ues of K1, these sidelobes are significant. By increasing the

values ofK and forΔK > 0, the sidelobes get less pronouncedsince the distribution of the azimuthal positions is moreevenly distributed in the interval �0; 2π� and thus the associ-ated phase contributions will tend to cancel out. This becomesvery obvious for large values of K and ΔK > 0. This effect canbe further enhanced by adding a statistical phase ϕs�m�.

B. Influence of M: Large aFZPThe examples shown above were shown for small elementswith only five rings in order to make the influence of the vari-ous parameters visible. It is obvious that for a large number ofrings and a varying number of openings, the phase values gen-erated by the numerous openings in the aFZP will lead to astrong reduction of undesired sidelobes. In Fig. 12, we showthe intensity plots for two aFZPs with relatively large numbersof rings. In both cases, no random phase shift was used in thedesign, i.e., ϕs � 0. First, in Fig. 12(a), the plot is shown for theaFZP shown in Fig. 1(c). This element has M � 10 rings,K1 � 20 and ΔK � 2. For comparison, Fig. 12(a) also showsthe normalized intensity plot for a conventional FZP with thesame number of rings (gray curve). One can notice a small, yetsignificant, difference in the two intensity plots. Finally, inFig. 12(b) we do the same comparison, however, here forM � 50 rings, K1 � 10, and ΔK � 2. Obviously, here the dif-ference is nearly zero between the focal spot profiles gener-ated by aFZP and conventional FZP.

6. CONCLUSIONA model to analyze the focusing properties of aFZPs has beenpresented. The calculation offers two significant simplifica-tions that are useful in order to keep the computational effortat a minimum. The examples presented show that good focalproperties can be obtained with an aFZP that, for large values

Fig. 11. Case 4: aFZP in �x0; y0� for M � 5 rings, K1 � 11 openings,ΔK � 2, and a random azimuthal phase added. (b) aFZP shown in�r20;ϕ� diagram. (c) 2D amplitude distribution in focal plane. (d) Auto-correlation in �r20;ϕ�.

5 10 15r in µm

r in µm

20 25

0.2

0.4

0.6

0.8

1.0

M=10

(a)

5 10 15 20 25

0.2

0.4

0.6

0.8

1.0

M=50

(b)Fig. 12. Large aFZP: intensity plots for (a) M � 10 rings, K1 � 20openings, and ΔK � 2; and (b) M � 50 rings, K1 � 10 openings,and ΔK � 1. In both cases, ϕs � 0. Gray curves show the intensityof a focal plot generated by a conventional FZP.


of rings and openings, does not differ significantly from theperformance of a conventional FZP. The most significant in-fluence comes from a random azimuthal phase that helps toavoid deterministic interference patterns around the focus.Here, we also pointed out the usefulness of the well-knownOTF theory for the analysis of FZPs. The virtue of the OTFconcept lies in the possibility to perform the analysis basedon the calculation of an autocorrelation, which is relativelyeasy to carry out for binary structures.

We conclude with two further remarks: as an additional de-gree of freedom, one may use the widths of the zones in orderto reduce sidelobes and even obtain a Gaussian focal spot[16]. Finally, as mentioned at the beginning, analysis of themechanical performance of the different types of zone platestructures is an interesting goal for further investigation.

ACKNOWLEDGMENTSThe authors thank Qing Cao (Shanghai University), JimFienup (University of Rochester), and JürgenMohr (KarlsruheInsitute of Technology, Germany) for interesting discussions.Furthermore, useful feedback from the reviewers is highlyappreciated.

REFERENCES1. A. Heuberger, “X-ray lithography,” J. Vac. Sci. Technol. B 6,

107–121 (1985).2. W. Chao, B. D. Harteneck, J. A. Liddle, and D. T. Attwood, “Soft

x-ray microscopy at a spatial resolution better than 15 nm,”Nature 435, 1210–1213 (2005).

3. D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation:

Principles and Applications (Cambridge University, 1999).

4. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compoundrefractive lens for focusing high-energy x-rays,” Nature 384,49–51 (1996).

5. G. Schmahl and D. Rudolph, “High-power zone plates as imageforming systems for soft x-rays,” Optik 29, 577–585 (1969).

6. J. Als-Nielsen, D. Jacquemain, K. Kjaer, F. Leveiller, M. Lahav,and L. Leiserowitz, “Principles and applications of grazingincidence x-ray and neutron scattering from ordered molecularmonolayers at the air-water interface,” Phys. Rep. 246, 251–313(1994).

7. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S.Harm, and R. Seemann, “Sharper images by focusing soft x-rayswith photon sieves,” Nature 414, 184–188 (2001).

8. G. Andersen, “Large optical photon sieves,” Opt. Lett. 30,2976–2978 (2005).

9. Q. Cao and J. Jahns, “Focusing analysis of the pinhole photonsieve: individual far-field model,” J. Opt. Soc. Am. A 19,2387–2393 (2002).

10. I. Mitsuishi, Y. Ezoe, M. Koshiishi, M. Mita, Y. Maeda, N. Y.Yamasaki, K. Mitsuda, T. Shirata, T. Hayashi, T. Takano, andR. Maeda, “Evaluation of the soft x-ray reflectivity of microporeoptics using anisotropic wet etching of silicon wafers,” Appl.Opt. 49, 1007–1011 (2010).

11. A. W. Lohmann, Optical Information Processing (TU IlmenauUniversity, 2006).

12. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–242(1964).

13. J. Jahns and S. Helfert, Introduction to Micro- and Nanooptics

(VCH-Wiley, 2012).14. J. Jahns and S. J. Walker, “Two-dimensional array of diffractive

microlenses fabricated by thin film deposition,” Appl. Opt. 29,931–936 (1990).

15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical

Functions with Formulas, Graphs, and Mathematical Tables

(Dover, 1964).16. Q. Cao and J. Jahns, “Modified Fresnel zone plates that produce

sharp Gaussian focal spots,” J. Opt. Soc. Am. A 20, 1576–1581(2003).


diffraction theory for azimuthally structured fresnel zone

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