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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24975–24982

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Observation of the Talbot effect using broadband hard x-ray beam

Jae Myung Kim, In Hwa Cho, Su Yong Lee, Hyon Chol Kang, Ray Conley, Chian Liu, Albert T. Macrander, and Do Young Noh  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 24975-24982 (2010)
http://dx.doi.org/10.1364/OE.18.024975


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Abstract

We demonstrated the Talbot effect using a broadband hard x-ray beam (Δλ/λ ~1). The exit wave-field of the x-ray beam passing through a grating with a sub micro-meter scale period was successfully replicated and recorded at effective Talbot distance, ZT. The period was reduced to half at ZT/4 and 3/4ZT, and the phase reversal was observed at ZT/2. The propagating wave-field recorded on photoresists was consistent with a simulated result.

© 2010 OSA

1. Introduction

When a monochromatic light wave is incident on a periodic grating, the wave-field at the exit of the grating is replicated at the multiples of the 'Talbot distance', ZT, downstream of the grating. The Talbot effect, this self-imaging phenomenon of a periodic grating, was first discovered by Talbot [1

H. F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401–407 (1836).

] in 1836, and demonstrated by Lord Rayleigh a few decades later [2

L. Rayleigh, “On Copying Diffraction-Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11, 196 (1881).

]. In addition, the period in the propagating wave-field is reduced to half at 1/4 ZT and 3/4 ZT, and the phase of the grating structure reverses at 1/2 ZT. The Talbot distance ZT is given by ZT = 2a 2/λ, where a is the grating period and λ is the wavelength of the incident light. The Talbot effect is schematically illustrated in Fig. 1 .

Fig. 1 Schematic view of the Talbot effect. The grating image is replicated at the Talbot distance, ZT while its phase is reversed at ZT/2. The period of the wave-field intensity reduces to half the grating period at ZT/4 and 3ZT/4.

The Talbot effect has applications in various fields such as atomic optics [3

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995). [CrossRef] [PubMed]

], Bose-Einstein condensation [4

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, Matter-Wave-Dispersion Talbot Effect,” Phys. Rev. Lett. 83(26), 5407–5411 (1999). [CrossRef]

], and interferometry of fullerene molecules [5

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88(10), 100404 (2002). [CrossRef] [PubMed]

]. Recently, the Talbot effect in x-ray regime has also attracted a great attention in regard to applications in x-ray phase contrast imaging [6

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature 2, 258–261 (2006).

,7

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, “Hard-X-ray phase-difference microscopy using a fresnel zone plate and a transmission grating,” Phys. Rev. Lett. 103(18), 180801 (2009). [CrossRef] [PubMed]

] where the modification of the Talbot image by the phase variation in the incident wave was used to retrieve the image of an object. The Talbot effect has also great implications in optical lithography based on short wave length light such as x-rays, since patterns with nano-meter scale periods can be recorded by placing a photoresist (PR) at the Talbot distance. Talbot distance reaches up to a few millimeters when a hard X-ray beam is incident on a grating with a period of a few hundred nanometers. This distance is easily accessible for positioning and manipulating PRs.

Case et al. demonstrated the Talbot effect using a green diode laser and a grating with a 200 μm period [8

W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17(23), 20966–20974 (2009). [CrossRef] [PubMed]

], where the Talbot distance was approximately 15.4 cm. Guerineau et al. demonstrated the Talbot effect using polychromatic light in the visible regime [9

N. Guérineau, E. Di Mambro, J. Primot, and F. Alves, “Talbot experiment re-examined: study of the chromatic regime and application to spectrometry,” Opt. Express 11(24), 3310–3319 (2003). [CrossRef] [PubMed]

]. In these experiments, the grating periods were on the order of hundred micrometers. There are, however, only a few direct observations of the Talbot self-imaging under hard x-ray illumination. Cloetens et al. reported an observation of fractional Talbot images on a polymer film by using monochromatic hard x-ray [10

P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22(14), 1059–1061 (1997). [CrossRef] [PubMed]

]. The grating period used in their experiment was about 12.5 μm. To apply the Talbot effect to nano-scale lithography, reducing the wavelength of light sources and the dimension of grating period are necessary.

In this study, we report an observation of the Talbot effect using a broadband hard x-ray beam, which is more efficient for lithography than a monochromatic beam since the exposure time on a PR could be reduced significantly using a high flux broadband x-ray beam. We used a specially designed x-ray grating with a sub-micrometer scale period to illustrate the Talbot effect. A simulation based on the Fresnel diffraction theory predicted the experimental observations quite well.

2. Experimental setup

The experiment was performed with synchrotron radiation at beamline 5C1 at Pohang Light Source (PLS) in Korea. The electron beam energy was 2.5 GeV and the nominal bending magnetic field was 1.32 Tesla. No monochromator or focusing element was involved in the beamline. To illustrate the Talbot effect, we used a specially prepared sectioned WSi2 /Si multilayer grating. WSi2 and Si layers were alternately grown by sputter deposition on to a silicon substrate to prepare the grating. It was then sectioned vertically into small slices and polished until the section width reaches to about 20 μm. Detailed process of the grating preparation was reported previously [11

C. Liu, R. Conley, A. T. Macrander, J. Maser, H. C. Kang, M. A. Zurbuchen, and G. B. Stephenson, “Depth-graded multilayers for application in transmission geometry as linear zone plates,” J. Appl. Phys. 98(11), 113519 (2005). [CrossRef]

,12

H. C. Kang, G. B. Stephenson, C. Liu, R. Conley, R. Khachatryan, M. Wieczorek, A. T. Macrander, H. Yan, J. Maser, J. Hiller, and R. Koritala, “Sectioning of multilayers to make a multilayer Laue lens,” Rev. Sci. Instrum. 78(4), 046103 (2007). [CrossRef] [PubMed]

]. Shown in Fig. 2(a) is a schematic illustration of the experimental set up.

Fig. 2 (a) Schematic illustration of the experimental configuration. (b) SEM image of the sectioned Si/WSi2 multilayer grating. The bright and dark regions represent WSi2 and Si respectively. (c) Transmittance through the transparent Si and the opaque WSi2 regions. (d) Broken red line represents the spectral density of the bending magnet and the solid line represents the spectral density of the polychromatic beam incident on the multilayer mask.

Figure 2(b) shows a scanning electron micrograph (SEM) of the grating. The period of the sectioned multilayer grating corresponding to the thickness of a single WSi2 /Si pair was 676 nm. The fraction of Si and WSi2 was incidentally 57.1% and 42.9% respectively. The ratio of section depth to grating period was about 30, which is essential for sufficient x-ray contrast. The transmittance of the white x-rays through the relatively transparent Si regions and the opaque WSi2 regions of the grating was calculated using the tabulated x-ray attenuation factor [13

Filter Transmission, http://henke.lbl.gov/optical_constants/filter2.html

], and illustrated in Fig. 2(c).

The spectrum of the x-ray beam incident on the grating shown in Fig. 2(d) was estimated by considering preferential absorption of low energy x-rays by the Be-windows together with the spectral density calculated using the bending magnet parameters [14

K. Lee and S. S. Lee, “Deep X-ray mask with integrated actuator for 3D microfabrication,” Sens. Actuators A Phys. 108(1-3), 121–127 (2003). [CrossRef]

]. Two Be-windows of 254 μm thick were placed upstream of the grating, separating the beamline from the front-end and the experimental station. The spacing between the last Be window and the grating was 48.5 cm. The wavelength at the maximum flux in the spectrum occurred at 1.2Ǻ, and the spectral width (full width at the half maximum: FWHM) was about 1.2Ǻ. Long wavelength components were attenuated by the Be-windows while short wavelength components were limited by the critical energy set by the bending magnet parameters.

The wave-field intensity of the x-ray beam through the grating was recorded on a PR (ZEP520A-7, Zeon corporation) spin-coated on Si (001) substrates. The thickness of the PR was set to 250 nm. The grating-to-PR gap varied from 200 μm to 6 mm, and the exposure was adjusted at each gap distance to obtain images with the clearest contrast. The PR exposed by x-rays was developed in ZED-N50 for 40 sec, and rinsed in isopropyl alcohol for 60 sec.

3. Simulation of the Talbot effect under broadband x-ray illumination

In order to understand the propagation of the broadband x-ray beam through a periodic grating structure, we carried out a simulation of the wave-field propagation using the experimental parameters given in the previous section. The wave-field at a position (X, Y) in the downstream of a delta-function like monochromatic source located at origin is given by the impulse response function of the Fresnel diffraction,
h(X,Y;   λ)= e ikz iλz e ik 2z ( X2+ Y2),
(1)
where, z is the distance from the grating to the observed position, and k represents the wave vector amplitude, 2π/λ [15

J. W. Goodman, Introduction to Fourier optics (Roberts & Company, 2005).

]. When a grating, described by a grating function f(x,y;λ),is placed in the beam, the wave-field at a distance z downstream of the grating can then be calculated by convolving the impulse response function given in Eq. (1) and the grating function,

g(X,Y;   λ)= e ikz iλz ∫ ∫ f(x,y;λ) e ik 2z [ ( X−x)2+ ( Y−y)2]dxdy.
(2)

The grating function f(x,y;λ) represents the amplitude of the wave-field at the exit surface of the grating. It can be evaluated by considering the grating structure and the transmission coefficient through the structure. In the case when a broadband beam is incident on the grating, the total wave-field through the grating can be obtained by summing the contribution of each wavelength component given in Eq. (2).
E total(X,Y)= ∫ B(λ)g ( X,Y;   λ) dλ,
(3)
where, B(λ) represents the spectral amplitude. The intensity recorded on is then given by,
〈 I(X,Y)〉= ∫ | B(λ)|2 | g(X,Y;   λ)|2 S(λ) dλ.
(4)
where S(λ) represents the effectiveness of x-rays on a PR in relation to the chemical modification of PRs. In most PRs, Auger electrons and photo electrons generated from a substrate are the major sources for breaking polymer chains rather than electrons generated by direct absorption of x-ray photons [16

K. Suzuki, and B. W. Smith, Microlithography: Science and Technology (CRC Press, 2007).

]. Therefore, we used the linear absorption coefficient of the substrate Si [17

X-ray Form Factor, Attenuation, and Scattering Tables, http://physics.nist.gov/cgi-bin/ffast/ffast.pl?Z=14&Formula=&gtype=5&range=S&lower=2.4848&upper=24.848&density=2.33

] as S(λ).

The simulated wave-field intensity profile downstream of the grating calculated from Eq. (4) using the experimental grating structure and the spectral density | B(λ)|2shown in Fig. 2(d), is illustrated in Fig. 3(a) . We defined the effective Talbot distance ZT eff by identifying the distance where the period becomes half in the simulation as ZT eff/4. ZT effwas 5.76 mm. For comparison, a simulated intensity profile under a monochromatic beam at an averaged wavelength λ avdefined by,
λ av= ( ∫ λ | B(λ)|2T(λ)S(λ) λ2 dλ)/ ( ∫ | B(λ)|2T(λ)S(λ) λ2 dλ),
(5)
is also shown in Fig. 3(b). Here, T(λ)is the transmittance through the transparent Si regions. λ av calculated using Eq. (5) was 1.495 Ǻ.

Fig. 3 Simulated wave-field intensity downstream of a periodic grating under broadband illumination (a) and monochromatic illumination (b). We used the grating structure and the spectral density used in this experiment. For the monochromatic case, the averaged wavelength obtained from the spectral density was used.

The major features of the Talbot effect were still present even in the broadband illumination shown in Fig. 3(a), although the detailed interference structure of the Talbot carpet shown in Fig. 3(b) was smeared out. As the quarter of the effective Talbot distance ZT effwas approached, the intensity in the transparent regions became weaker while the opaque regions became brighter both in Figs. 3(a) and 3(b). The fine structure presented in Fig. 3(b), however, were not present in Fig. 3(a). The line profile at ZT eff/4 showed that the period in the wave-field intensity was reduced to half the grating period. Downstream of this distance, the intensity in the opaque regions kept increasing, and the phase of the wave-field intensity was reversed at the half Talbot distance ZT eff/2. The grating image was replicated at this distance, but the bright and dark regions are reversed.

In the case of monochromatic illumination shown in Fig. 3(b), the wave-field intensity from ZT eff/2 to ZT eff was almost exactly opposite of the intensity from 0 to ZT eff/2. The period in the wave-field intensity became half at 3  ZT eff/4 and the grating structure was completely replicated at ZT eff. On the other hand, in the case of broadband illumination, the intensity in the opaque regions decreased, but did not disappear completely even at the Talbot distance. Therefore, the replication of the grating structure at ZT eff was incomplete as shown in Fig. 3(a).

The presence of the key features of the Talbot effect using a broadband polychromatic x-ray beam in the simulation is rather surprising but understandable. A broadband x-ray beam generated from a typical white x-ray beamline with Δλ/λ about 100% can still exhibit every feature of the Talbot effects, since the spectral distribution is narrow enough.

4. Experimental results and discussion

Figure 4 shows SEM images of the x-ray wave-field intensity at several distances d downstream of the grating recorded on the PR. The bright regions, where the PR material survived, represent the opaque low wave-field intensity regions, and the dark regions correspond to the high intensity regions. At z = 200 μm, the image recorded on the PR shown in Fig. 4(a) replicated the grating structure shown in Fig. 1 well. The period of the wave-field intensity in the image was 676 nm, exactly matching the grating period. This distance was much shorter than the effective Rayleigh distance a2/ λ av∼997 μm estimated using the averaged wavelength, 0.14 nm with the opening size a corresponding to the Si layer thickness of 386 nm. The PR was located well in the near field region and the recorded image represented the shadow of the grating closely.

Fig. 4 Wave-field intensity recorded on PR at 0.2 mm(a), 1.4 mm(b), 2.8 mm(c), 4.2 mm (d), 5.6 mm (e) downstream of the grating. On the right is the simulated result for comparison.

The grating structure was replicated quite clearly as z approached 2.8 mm for the first time as shown in Fig. 4(c), although a broadband x-ray beam was used. Experimentally, this distance corresponds to half the Talbot distance, since the grating structure was replicated first at half the Talbot distance as shown in the simulation results. A series of measurements were performed around twice this distance to find out the Talbot distance experimentally, which turned out to be 5.6 mm. This is very close to the effective Talbot distance estimated in the simulation, 5.71 mm. Figure 4(e) shows an SEM image of the wave-field intensity recorded on the PR at 5.6 mm. The grating structure was mostly replicated. At this position, however, a dark line signifying high wave-field intensity showed up in the middle of the bright stripe. This was consistent with the simulation where the intensity in the opaque regions did not disappear completely at the Talbot distance.

It is interesting to note that the wave-field intensity at a distance corresponding to a quarter and three quarters of this distance, 1.4 mm and 4.2 mm respectively, was quite different from the grating structure. As shown in Figs. 4(b) and 4(d), the period of the recorded intensity was reduced to 338 nm, half the original grating period. We think that this phenomenon was due to the relatively narrow spectral distribution of the x-ray beam used in this experiment as noted in the description of the simulation.

In fact, the grating structure replicated at 2.8 mm, which we consider to be effectively half the Talbot distance, shown in Fig. 4(c), is not exactly the same as the near field image shown in Fig. 4(a). Upon close inspection, we noticed that the number of lines in the image was smaller by one in Fig. 4(c) than in Fig. 4(a). This is in fact due to the phase reversal expected at this distance, which is illustrated clearly in Figs. 5(a) , 5(b) where we placed the two intensity profiles next to each other. The bright and dark regions are reversed in the two images. The phase reversal is also well illustrated in the line profiles of the simulated result shown in Figs. 5(c) and 5(d). The phase reversal at the half the Talbot distance is due to the interference of the light wave emanating from each transparent Si region that becomes destructive in the opening positions and constructive in the closed positions in the downstream region. The edge diffraction effect appeared near the edge of the grating in the simulation, was not recorded well in Fig. 5(b). We think that the contrast was not enough to record the detailed features related to the edge diffraction.

Fig. 5 Recorded wavefield intensity in the near field region (a) and at half the Talbot distance (b). The number of line patterns is reduced by one in (b) due to phase shift of electric field. The observed phase shift is consistent with the line profile of the simulated intensity profile in the near field (c) and at half the Talbot distance (d). Dashed line indicates the relative phase shift between the two regions.

It is interesting that the interference effect was observed even under broadband white x-ray illumination. We note that the key features of the Talbot effect, i.e. the periodic replication of the grating structure at half the Talbot and the Talbot distances were exhibited even under the broadband illumination.

5. Conclusion

We have observed the Talbot effects using a broadband x-ray beam generated from a synchrotron beamline. The grating structure is well replicated at the Talbot distance. The half period image and the phase shifted images were recorded at fractional Talbot distances. A simulation based on the Fresnel diffraction theory predicts the experimentally observed wave-field propagation well. We believe that high flux broadband x-rays generated from a bending magnet synchrotron beamline can be utilized in hard x-ray lithography utilizing the observed Talbot effect. Our simulation suggests that hard x-ray lithography with a resolution close to 10 nm is possible with the multilayer grating technology. As feature size decreases to 10 nm, it is impractical to place a PR in the shadow near field regime. Therefore utilizing the Talbot effect observed in this experiment is critical. The broad band polychromatic x-ray beam at synchrotron is much more intense than monochromatic beam.

Acknowledgements

This work was supported by National Core Research Center grant (No. R15-2008-006-00000-0) and general research program (No. 2010-0023604) provided by National Research Foundation (NRF) of Korea. We also acknowledge the support from GIST through ‘Photonics 2010’ project. Work at Argonne was supported by the Department of Energy, Office of Basic Energy Science under contract DE-AC-02-06CH11357. Work at Brookhaven was supported by the Department of Energy, Office of Basic Energy Sciences under contract DE-AC-02-98CH10886.

References and links

1.

H. F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401–407 (1836).

2.

L. Rayleigh, “On Copying Diffraction-Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11, 196 (1881).

3.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995). [CrossRef] [PubMed]

4.

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, Matter-Wave-Dispersion Talbot Effect,” Phys. Rev. Lett. 83(26), 5407–5411 (1999). [CrossRef]

5.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88(10), 100404 (2002). [CrossRef] [PubMed]

6.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature 2, 258–261 (2006).

7.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, “Hard-X-ray phase-difference microscopy using a fresnel zone plate and a transmission grating,” Phys. Rev. Lett. 103(18), 180801 (2009). [CrossRef] [PubMed]

8.

W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17(23), 20966–20974 (2009). [CrossRef] [PubMed]

9.

N. Guérineau, E. Di Mambro, J. Primot, and F. Alves, “Talbot experiment re-examined: study of the chromatic regime and application to spectrometry,” Opt. Express 11(24), 3310–3319 (2003). [CrossRef] [PubMed]

10.

P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22(14), 1059–1061 (1997). [CrossRef] [PubMed]

11.

C. Liu, R. Conley, A. T. Macrander, J. Maser, H. C. Kang, M. A. Zurbuchen, and G. B. Stephenson, “Depth-graded multilayers for application in transmission geometry as linear zone plates,” J. Appl. Phys. 98(11), 113519 (2005). [CrossRef]

12.

H. C. Kang, G. B. Stephenson, C. Liu, R. Conley, R. Khachatryan, M. Wieczorek, A. T. Macrander, H. Yan, J. Maser, J. Hiller, and R. Koritala, “Sectioning of multilayers to make a multilayer Laue lens,” Rev. Sci. Instrum. 78(4), 046103 (2007). [CrossRef] [PubMed]

13.

Filter Transmission, http://henke.lbl.gov/optical_constants/filter2.html

14.

K. Lee and S. S. Lee, “Deep X-ray mask with integrated actuator for 3D microfabrication,” Sens. Actuators A Phys. 108(1-3), 121–127 (2003). [CrossRef]

15.

J. W. Goodman, Introduction to Fourier optics (Roberts & Company, 2005).

16.

K. Suzuki, and B. W. Smith, Microlithography: Science and Technology (CRC Press, 2007).

17.

X-ray Form Factor, Attenuation, and Scattering Tables, http://physics.nist.gov/cgi-bin/ffast/ffast.pl?Z=14&Formula=&gtype=5&range=S&lower=2.4848&upper=24.848&density=2.33

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.1970) Diffraction and gratings : Diffractive optics
(050.5080) Diffraction and gratings : Phase shift
(110.6760) Imaging systems : Talbot and self-imaging effects
(340.7450) X-ray optics : X-ray interferometry

ToC Category:
X-ray Optics

History
Original Manuscript: September 15, 2010
Revised Manuscript: October 27, 2010
Manuscript Accepted: October 28, 2010
Published: November 15, 2010

Citation
Jae Myung Kim, In Hwa Cho, Su Yong Lee, Hyon Chol Kang, Ray Conley, Chian Liu, Albert T. Macrander, and Do Young Noh, "Observation of the Talbot effect using broadband hard x-ray beam," Opt. Express 18, 24975-24982 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24975


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References

  1. H. F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401–407 (1836).
  2. L. Rayleigh, “On Copying Diffraction-Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11, 196 (1881).
  3. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995). [CrossRef] [PubMed]
  4. L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, Matter-Wave-Dispersion Talbot Effect,” Phys. Rev. Lett. 83(26), 5407–5411 (1999). [CrossRef]
  5. B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88(10), 100404 (2002). [CrossRef] [PubMed]
  6. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature 2, 258–261 (2006).
  7. W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, “Hard-X-ray phase-difference microscopy using a fresnel zone plate and a transmission grating,” Phys. Rev. Lett. 103(18), 180801 (2009). [CrossRef] [PubMed]
  8. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17(23), 20966–20974 (2009). [CrossRef] [PubMed]
  9. N. Guérineau, E. Di Mambro, J. Primot, and F. Alves, “Talbot experiment re-examined: study of the chromatic regime and application to spectrometry,” Opt. Express 11(24), 3310–3319 (2003). [CrossRef] [PubMed]
  10. P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22(14), 1059–1061 (1997). [CrossRef] [PubMed]
  11. C. Liu, R. Conley, A. T. Macrander, J. Maser, H. C. Kang, M. A. Zurbuchen, and G. B. Stephenson, “Depth-graded multilayers for application in transmission geometry as linear zone plates,” J. Appl. Phys. 98(11), 113519 (2005). [CrossRef]
  12. H. C. Kang, G. B. Stephenson, C. Liu, R. Conley, R. Khachatryan, M. Wieczorek, A. T. Macrander, H. Yan, J. Maser, J. Hiller, and R. Koritala, “Sectioning of multilayers to make a multilayer Laue lens,” Rev. Sci. Instrum. 78(4), 046103 (2007). [CrossRef] [PubMed]
  13. Filter Transmission, http://henke.lbl.gov/optical_constants/filter2.html
  14. K. Lee and S. S. Lee, “Deep X-ray mask with integrated actuator for 3D microfabrication,” Sens. Actuators A Phys. 108(1-3), 121–127 (2003). [CrossRef]
  15. J. W. Goodman, Introduction to Fourier optics (Roberts & Company, 2005).
  16. K. Suzuki, and B. W. Smith, Microlithography: Science and Technology (CRC Press, 2007).
  17. X-ray Form Factor, Attenuation, and Scattering Tables, http://physics.nist.gov/cgi-bin/ffast/ffast.pl?Z=14&Formula=&gtype=5&range=S&lower=2.4848&upper=24.848&density=2.33

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