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Optics and Lasers in Engineering 48 (2010) 760–765

Contents lists available at ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

A super-resolution Fresnel zone plate and photon sieve

Jia Jia �, Changqing Xie, Ming Liu, Lixi Wan

Institute of Microelectronics, Chinese Academy of Sciences, Beijing, PR China

a r t i c l e i n f o

Article history:

Received 12 August 2009

Received in revised form

8 March 2010

Accepted 12 March 2010Available online 1 April 2010

Keywords:

Photon sieve

Fresnel zone plate

Super-resolution

Diffractive optics

66/$ - see front matter & 2010 Elsevier Ltd. A

016/j.optlaseng.2010.03.007

esponding author.

ail address: jiajia@ime.ac.cn (J. Jia).

a b s t r a c t

Realizing a smaller or sharper diffractive center spot is a valuable research aim in soft X-ray focus and

other related research applications. Fresnel zone plates (FZP) and photon sieves (PS) are often used to

focus the X-rays or other wavelength light at present. Here, we show that combination of a super-

resolution phase mask (SPM) and an FZP (or PS) as one diffractive optical element can realize a smaller

or sharper diffractive center spot without significantly increasing the fabrication difficulty. All these

diffractive phase elements can be applied to beam shaping, mask-less lithography, energy congregation

in high power lasers, soft X-rays focus, and any other field that requires a smaller or sharper diffractive

center spot.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Focusing X-ray radiation [1] is a crucial procedure in manymodern physics experiments, and the design of optical deviceshas attracted considerable interest over the past 10s years.The variety includes X-ray mirrors to concentrators in the formof tapered capillaries [2], compound refractive lenses [3], Fresnelzone plates [4] (FZP), and photon sieves [5] (PS). The spatialresolution that can be achieved with these devices is usuallydependent on the width of the outermost zone, and is thereforelimited by the smallest structure (20–40 nm) that can befabricated by lithography today [6,7]. On the other hand, themost severe problem with refractive optics for focusing electro-magnetic radiation is absorption. The FZPs provide an effectivesolution to absorption problem, but progress is still desired alongthe following three directions: (1) overcoming the limitation ofthe spatial resolution given by the width of the zone plate’soutermost zone; (2) elimination of unwanted diffraction orders;and (3) reduction of the scattering intensity arising from the edgeof the finite-sized zone plate.

There are two kinds of FZPs: amplitude FZPs and phase FZPs [8].In a phase FZP, the opaque zones are replaced by transmittingzones of optical thickness l/2, equivalent to a phase change of p.The phase FZP greatly improves the efficiency. A modified FZP,such as a composite zone plate [8], can also improve the resolutionwithout the loss of diffractive energy. A PS [5] with the appropriatedistribution of pinholes over the Fresnel zones can focus the softX-rays effectively, with a sharpening of the focal spot combinedwith an effective suppression of higher orders (of what) and a

ll rights reserved.

finite-size effect. Some modified PSs [9–13] and new appli-cations [13–15] have been developed.

The super-resolution technique is a promising method toachieve a diffraction spot smaller than the Airy spot when asuper-resolution pure-phase mask is employed. The first sig-nificant study of super-resolution was done by di Francia [16].Since then, much attention has been devoted to the design of thesuper-resolution phase mask (SPM) for its applications inenhancing the resolution of con-focal systems [17], increasingstorage in optical disk systems [18], etc. Further details on super-resolution technology and its applications can be found in Refs.[19–21]. The super-resolution technique using the phase plate hasa set of mathematical tools to design the structure of the SPM.These phase plates have the same annular phase distribution.Thus, we can regard the super-resolution technique as themodulation between the annular phase plate and the Fourierdiffraction field (which is also the far-field diffraction or theFraunhofer diffraction). The super-resolution technique can alsoachieve a smaller diffractive spot [22] or beam shaping [23],especially in far-field beam shaping.

In this paper, we use SPM technology to modify the FZP and PS.Super-resolution technology is also applied to these devices. Wewill demonstrate that the super-resolution FZP and PS can beused to focus the X-rays effectively to achieve a smaller orsharper diffractive spot in the far field without increasing thedifficulty of the fabrication of the super-resolution FZP and PS. Infact, we combined the FZP (or PS) and the SPM into one device.We call it super-resolution Fresnel zone plate (SFZP) (or super-resolution photon sieve (SPS). Super-resolution technology pro-duces many kinds of phase zones masks, here we choose only the2-zone or the 3-zone binary or multiple phase zone masks tosimulate the super-resolution FZP (or PS) due to their ease offabrication.

ARTICLE IN PRESS

J. Jia et al. / Optics and Lasers in Engineering 48 (2010) 760–765 761

2. Theoretical background

In the standard optical configurations shown in Fig. 1, weconsider the diffraction at a transparency with the transmittancedistribution function (TDF) t(x, y) in the plane z¼0 (plane 1).Illuminating it with a plane, the linearly polarized mono-frequency wave of the wavelength is l and the wave field isEe(x, y). To begin, immediately behind the plane where z¼0, thefield E(x, y)¼t(x, y)Ee(x, y) is present. A field E (x0, y0, z) in plane 3 isobtained given by the Kirchhoff diffraction integral [24,25]:

Eðx0,y0,zÞ ¼1

il

Z þ1�1

Z þ1�1

Eðx,yÞ1

rexpðikrÞcosðn,rÞdx dy ð1Þ

For large distances from the diffraction plane (the far field orthe Fraunhofer region) and for a finite size of the diffractionpattern in x and y, we obtain

Eðx0,y0,zÞ ¼expðikzÞ

ilzexp i

plzðx02þy02Þ

h i

�

Z þ1�1

Z þ1�1

Eðx,yÞexpð�2pix0x

lzþ

y0y

lz

� �dx dy ð2Þ

The diffraction pattern at the point (x0, y0, z) in plane 3 is givenby the intensity I¼9E (x0, y0, z)92. In Fig. 1, neglecting the bold partin plane 2, the diffraction pattern in the plane z¼z is

Iðnx,nyÞ ¼1

l2z2jF½Eðx,yÞ�ðnx,nyj

2 ð3Þ

F is the Fourier transformation. Plane 1 is the FZP or PS andplane 3 is the diffractive field in the focal plane accordingly. Forthe X-ray focus, a smaller diffraction spot is a valuable researchaim. In this paper, we applied super-resolution technology to get asmaller or sharper diffractive spot than that of conventional FZPsor PSs. Plane 2 in Fig. 1 shows an SPM with the transmittancephase distribution S(x, y). Then, the diffraction far field in plane 3is the Fraunhofer diffraction of E(x, y)¼S(x, y)t(x, y)Ee(x, y). Thediffraction pattern in plane 3 is

Iðnx,nyÞ ¼1

l2z2jF½Eðx,yÞ�ðnx,nyj

2

¼1

l2z2jF½Sðx,yÞtðx,yÞEeðx,yÞ�j2

¼1

l2z2jF½Sðx,yÞ�F½tðx,yÞ�F½Eeðx,yÞ�j2 ð4Þ

So without the modulation function in plane 2, the diffraction farfield is the convolution between the Fourier transformation of the

PS or FZP

Plane 3

SPM

Plane 1Plane 2

S (x,y)

Focal Plane

τ (x,y)

Fig. 1. Illustration of diffractive optical element and optics rays. Plane 1 in black is

the diffractive optical elements such as the FZP or PS with TDF t(x, y). Plane 2 in

bold black is the SPM with TDF S(x, y). Plane 2 is adjacent to plane 1. Plane 3 in

black is the focal plane of the FZP or PS.

transmittance of the FZP (or PS) in plane 1 and the Fouriertransformation of the incident light. In diffraction applications, ifthe incident light is a plane wave, the aperture is limited and itsFourier transformation is not a d function, but the Fouriertransformation of the circle aperture of the FZP (or PS). So if weput an SPM in plane 2, the diffractive field is the convolutionbetween the Fourier transformation of the transmittance of the FZP(or PS) and the Fourier transform of the SPM. Since the Fouriertransformation of the SPM is smaller than the Fourier transformationof the aperture of the FZP (or PS) (that is the airy spot of theaperture), a smaller or a sharper diffractive spot can be realized. Thus,the SPM can realize a smaller or a sharper diffraction spot in X-rayapplications. The Fourier transformation of the SPM can be applied torealize a smaller center spot than that of the same circle aperture.

3. Simulation, design, and experiments

Fig. 2 shows the circle aperture and 3 kinds of (SPM) profiles[26]. The SPM consists of several circular phase zones with differentzone radii and zone phases. The parameters of the 2-zone binarySPM are a¼0.34, j1¼0, and j2¼p; the parameters of the 3-zonebinary SPM are a¼0.09, b¼0.36, j1¼0.9p, j2¼0, j3¼0.9p; theparameters of the 3-zone multiple SPM are a¼0.09, b¼0.36, j1¼0,j2¼0.06p, and j3¼0.86p. For definition of parameters, one cansee Ref. [26]. All these SPMs are designed according to rule [26]:Gp0.8 when the center spot energy reaches the maximum. Fig. 3 isthe diffraction pattern with a circle aperture and a 2 zone or 3 zoneSPM. The TDF of the FZP (or PS) is the multiplication of itself by thecircle aperture function accordingly. Our aim is to apply these SPMsto the FZP (or PS) to search for a smaller or sharper diffractive centerspot.

In this paper, we propose the center spot intensity ratio (CSIratio) and the first zero to describe the center spot compressioneffect when comparing SPM to the FZP and PS. The CSI ratio isdefined as the ratio between the intensity of the center main spotand the intensity of all diffractive fields. That is the CSI ratiomeans the ratio between the center diffractive spot energy andthe all diffraction field energies. A larger CSI ratio means moreenergy has been suppressed into the central diffractive spot. Thefirst zero is defined as the first zero position of the intensity fromthe center point in the diffraction field. A small first zero indicatesa smaller central diffractive spot.

3.1. Super-resolution amplitude Fresnel zone plate

Fig. 4 is the profile of a conventional FZP and 3 kinds of SPMsapplied to FZP. The TDF of every device is the TDF of theconventional FZP multiplied by the TDF of the circle aperture inFig. 2(a), the 2-zone binary SPM in Fig. 2(b), the 3-zone binary SPMin Fig. 2(c), and the 3-zone multiple SPM in Fig. 2(d), respectively.They are called a conventional FZP(FZP) in (a), a 2-zero binarysuper-resolution FZP(2-zero binary SFZP) in (b), a 3-zero binarysuper-resolution FZP(3-zone binary SFZP) in (c), and a 3-zonemultiple super-resolution FZP(3-zone multiple SFZP) in (d) in Fig. 4.The conventional FZP design parameters are wavelengthl¼0.6328mm, focal length f¼100,000mm, and zone numberF¼50. To ease fabrication, when we apply the SPM to the FZP,we do not strictly conform to the radial values of a and b in the SPMif these values are in the middle of the ring range of the FZP. Weadjust the a and b values to the nearest edge of the ring of the FZP,and thus avoid changing the super-resolution effect significantly.

Fig. 5 presents the theoretical far-field intensity distribution ofthe FZP in solid line, of the 2-zone binary SFZP in big dashed line,of the 3-zone binary SFZP in small dashed line, and of the 3-zonemultiple SFZP in dash–dot line.

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X

Y

1 0 1

1

0

1

X

Y

1 0.34 0 0.34 1

1

0.34

0

0.34

1

π 0

X

Y

10.3

60.0

90 0.09

0.36 1

1

0.36

0.090

0.09

0.36

1

0.9π0.9π0

Apodized PS

X

Y

10.3

60.0

90 0.09

0.36 1

1

0.36

0.090

0.09

0.36

1

00.06 π0.86π

Fig. 2. Profiles of the circle aperture and 3 kinds of SPM. The phase of the circle aperture is 0. The parameters of the 2-zone SPM in (b) are a¼0.34, j1¼0, j2¼p; the

parameters of the 3-zone binary SPM in (c) are a¼0.09, b¼0.36, j1¼0.9p, j2¼0, j3¼0.9p; the parameters of the 3-zone multiple SPM in (d) are a¼0.09, b¼0.36, j1¼0,

j2¼0.06p, j3¼0.86p. a, b are the radii of the super-resolution phase zones and are normalized.

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nor

mal

ized

inte

nsity

Radial Distance

Airy pattern2-zones binary SPM3-zones binary SPM3-zones multiply SPM

Fig. 3. Theoretical far-field intensity distribution of the circle aperture in solid line

and of the super-resolution phase zones diffraction. The 2-zone binary SPM is in

big dashed line; the 3-zone binary SPM is in small dashed line; and the 3-zone

multiple phase SPM is in dash–dot line. The super-resolution patterns generated

by the 2-zone and 3-zone plates are quite close to each other.

J. Jia et al. / Optics and Lasers in Engineering 48 (2010) 760–765762

The CSI ratios of the FZP, the 2-zone binary SFZP, the 3-zonebinary SFZP, and the 3-zone multiple SFZP are 0.9849, 0.7796,0.8309, and 0.8050, respectively. The first zeros of the FZP, the

2-zone binary SFZP, the 3-zone binary SFZP, and the 3-zonemultiple SFZP are 13, 11, 11, and 11, respectively. The unit is thediffractive point in the diffractive field in plane 3 in Fig. 1. The unitis 1.7371mm. Fig. 5 and these data suggest that the SFZP canactually realize a smaller or sharper diffractive spot than the FZP.Though having a smaller diffractive center spot, some energy haschanged into the first diffractive rings. Of these three kinds ofSFZPs, the 3-zone binary SFZP shows the best performance.

3.2. Super-resolution photon sieve

Fig. 6 is the profile of the conventional PS and 3 kinds of SPMsapplied to PS. The total TDF of every device is the TDF of theconventional PS multiplied by the TDF of the circle aperture inFig. 2(a), the 2-zone binary SPM in Fig. 2(b), the 3-zone binarySPM in Fig. 2(c), and the 3-zone multiple SPM in Fig. 2(d). They arecalled a conventional PS(PS) in (a), a 2-zero binary super-resolution PS(2-zero binary SPS) in (b), a 3-zero binary super-resolution PS(3-zone binary SPS) in (c), and a 3-zone multiplesuper-resolution PS(3-zone multiple SPS) in (d) in Fig. 6.The conventional PS design parameters are wavelength l¼0.6328mm, focal length f¼100,000mm, zone number F¼50, andk15¼d/w¼1.5. The transmission window5 is f(R)¼1�R2/RT

2,where R is the radial radius and RT the maximum radius of PS.To ease fabrication, when we apply the SPM to the PS, we do notstrictly conform to the radial values of a and b in SPM if thesevalues are in the middle of the pinhole range of the PS. We adjust

ARTICLE IN PRESS

Y (μ

m)

-1500

-1000 -50

0 050

010

0015

00

-1500

-1000

-500

0

500

1000

1500

Y (μ

m)

-1500

-1000 -50

0 050

010

0015

00

-1500

-1000

-500

0

500

1000

1500

Y (μ

m)

-1500

-1000

-500

0

500

1000

1500

Y (μ

m)

-1500

-1000

-500

0

500

1000

1500

X (μm) X (μm)

-1500

-1000 -50

0 050

010

0015

00-15

00-10

00 -500 0

500

1000

1500

X (μm) X (μm)

Fig. 4. The profile of the conventional FZP and 3 kinds of SPM applied to the FZP. The total TDF of the devices is the TDF of the conventional FZP multiplied by the TDF of the

circle aperture, 2-zone binary SPM, 3-zone binary SPM, and 3-zone multiple SPM, respectively. They are named as the conventional FZP(FZP) in (a), 2-zero binary super-

resolution FZP(2-zero binary SFZP) in (b), 3-zero binary super-resolution FZP(3-zone binary SFZP) in (c), and 3-zone multiple super-resolution FZP(3-zone multiple SFZP) in

(d), respectively. (a) is the conventional FZP while white is transparent and black is opaque. In (b), black and grey outside the FZP is opaque, grey zone with radius bigger

than 0.34 is phase p, grey zone with radius smaller than 0.34 is opaque, and white is transparent with phase 0. In (c), black and grey outside the FZP is opaque, grey zone

with radius bigger than 0.36 is phase 0.9p, grey zone with radius smaller than 0.36 and bigger than 0.09 is opaque and white in this area is transparent with phase 0, black

with radius smaller than 0.09 is phase 0.9p. In (d), black and grey outside the FZP is opaque, grey zone with radius bigger than 0.36 is phase 0.86p, grey zone with radius

smaller than 0.36 and bigger than 0.09 is opaque and white in this area is transparent with phase 0.06p, white with radius smaller than 0.09 is transparent with phase 0.

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

Inte

nsity

(arb

.uni

ts)

FZP2 zones SFZP3 zones binary SFZP3 zones multiply SFZP

Radial Distance (1.7371μm)

Fig. 5. The theoretical far-field intensity distribution of the FZP is in solid line, of

the 2-zone binary SFZP is in star dashed line, of the 3-zone binary SFZP in circle

dashed line, and of the 3-zone multiple SFZP in dash–dot line.

J. Jia et al. / Optics and Lasers in Engineering 48 (2010) 760–765 763

the a and b values to the nearest edge of the pinhole of the PS, andthus avoid changing the super-resolution effect significantly.

Fig. 7 presents the theoretical far-field intensity distribution ofthe PS in solid line, of the 2-zone binary SPS in big dashed line, ofthe 3-zone binary SPS in small dashed line, and of the 3-zonemultiple SPS in dash–dot line.

The CSI ratios of the PS, the 2-zone binary PS, the 3-zone binaryPS, and the 3-zone multiple PS are 0.9137, 0.5033, 0.6916, and0.5846, respectively. The first zeros of the PS, the 2-zone binaryPS, the 3-zone binary PS, and the 3-zone multiple PS are 17, 11,13, and 12, respectively. The unit is the diffractive point in thediffractive field in plane 3 in Fig. 1. The unit is 1.7371mm. Fig. 7and these data show that the SPS can actually realize a smaller orsharper diffractive spot than the PS. Though having a smallerdiffractive center spot, some energy has changed into the firstdiffractive ring. Of these three kinds of SPSs, the 3-zone binary SPSshows the best performance.

This paper presents some simple experiments to support thesimulation and design. The test optical diagrams are shown inFig. 8. We have fabricated the diffractive optics elements PS andSPS. And the far field diffractive intensity photos are listed inFig. 9. We have put the PS and SPS at the ZP and PS position. From

ARTICLE IN PRESS

Y (μ

m)

-1500

-1000 -50

0 050

010

0015

00

-1500

-1000

-500

0

500

1000

1500

Y (μ

m)

-1500

-1000 -50

0 050

010

0015

00

-1500

-1000

-500

0

500

1000

1500

Y (μ

m)

-1500

-1000

-500

0

500

1000

1500

Y (μ

m)

-1500

-1000

-500

0

500

1000

1500

X (μm) X (μm)

-1500

-1000 -50

0 050

010

0015

00-15

00-10

00 -500 0

500

1000

1500

X (μm) X (μm)

Fig. 6. The profile of the conventional PS and 3 kinds of SPM applied to the PS. The TDF of each device is the TDF of the conventional PS multiplied by the TDF of the circle

aperture, 2-zone binary SPM, 3-zone binary SPM, and 3-zone multiple SPM. They are called the conventional PS(PS) in (a), the 2-zero binary super-resolution PS(2-zero

binary SPS) in (b), the 3-zero binary super-resolution PS(3-zone binary SPS) in (c), and the 3-zone multiple super-resolution PS(3-zone multiple SPS) in (d). (a) is the

conventional PS while white is transparent and black is opaque. In (b), black is transparent with phase p, grey is opaque, and white is transparent with phase 0. In (c), black

is transparent with phase 0.9p, grey is opaque, white is transparent with phase 0. In (d), black is transparent with phase 0.86p, grey is opaque, white is transparent with

phase 0 and light grey is transparent with phase 0.06p.

0 5 10 15 20 25 30 35 40 45 500

50

100

150

200

250

300

350

400

450

Radial Distance (1.7371μm)

Inte

nsity

(arb

.uni

ts)

PS2 zones SPS3 zones binary SPS3-zones multiply SPS

Fig. 7. The theoretical far-field intensity distribution of the PS is in solid line, of

the 2-zone binary SPS is in big dashed line, of the 3-zone binary SPS in small

dashed line, and of the 3-zone multiple SPS in big dash–dot line, respectively.

Fig. 8. The optical diagram to test my design diffractive optics elements. We put

PS and SPS to the position of ZP or PS. The diffractive optics elements far-field

intensity photo have been captured in computer and listed in Fig. 9.

J. Jia et al. / Optics and Lasers in Engineering 48 (2010) 760–765764

Fig. 9, we can see that the diffractive spot size of SPS actuallybecomes smaller than PS does, which proves that our simulationand design are correct.

4. Discussion and conclusion

Essentially, SFZP and SPS are diffractive phase elements. Forfabrication details, refer to Ref. [27]. In this paper, we combine theSPM and the conventional FZP and PS as one diffractive phaseelement to get the SFZP and SPS without increasing the difficulty

ARTICLE IN PRESS

Fig. 9. The far field diffractive optics intensity photo of diffractive optics elements: common PS(a), 2-zone binary SPS in (b), and 3-zone binary SPS (c). From (a) to (c), we

can see that the diffractive spot size of SPS actually becomes smaller than common PS does, which supports that the simulation and design are correct.

J. Jia et al. / Optics and Lasers in Engineering 48 (2010) 760–765 765

of fabrication. In simulation, we chose the incident lightwavelength l¼632.8 nm to verify that these devices can actuallymake the diffractive center spot smaller or sharper in the focalplane (or far field or Fraunhofer region). However, these devicescan also focus soft X-rays if we adjust the design parameters tothe soft X-ray wave band and fabricate it. These devices overcomethe limitation in spatial resolution of the width of the zone-plateoutermost zones (or the diameter of the photon sieve outermostpinhole).

For ease of fabrication, when we applied the SPM to the FZPand PS, we did not strictly conform to the radial values of a and b

in the SPM if this value was in the middle of the ring range of theFZP or the pinhole range of the PS. We adjusted the a and b valuesto the nearest edge of the ring of the FZP or the edge of the pinholeof the PS and thus avoided changing the super-resolution effectsignificantly.

Furthermore, the design parameter of the SPM in this paper isto conform to the rule: Gr0.8 when the center spot energyreaches the maximum. G is the normalized spot size and isdefined as the ratio of the first zero position of the super-resolution diffractive pattern to the first position of the Airydiffractive pattern. In application, we can design other SPMsconforming to other rules and apply these SPMs to FZPs and PSs tosatisfy the specific demand.

We have simulated the intensity of the diffractive fieldof the FZP, SFZP and PS, SPZ. Detailed comparisons have beenpresented above. Because the SPM design parameters are Gr0.8,SFZP and SPS show a smaller first zero (approximately 0.8in simulation accuracy) than those of the FZP and PS, respectively.Though having realized a smaller diffractive center spot, theSPZP and SPS have a lower peak intensity of the diffractivecenter spot than the FZP and PS, respectively, due to someenergy changing into the side-lobe from the diffractive centerspot.

All these diffractive phase elements can be applied to beamshaping, mask-less lithography, energy congregation in highpower lasers, soft X-rays focus, and any other field that requiresa smaller or sharper diffractive center spot.

Acknowledgments

The author acknowledges the support of special funds formajor state basic research (973) projects (2007CB935302) and thenational S&T major project with the contract No. 2009zx02038.

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