Nuclear Instruments and Methods in Physics Research A 467–468 (2001) 864–867

Toward a practical X-ray Fourier holography at high resolution

M.R. Howellsa,*, C.J. Jacobsenb, S. Marchesinia, S. Millerc,J.C.H. Spencea,c, U. Weirstallc

aAdvanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS-2-400, Berkeley, CA 94720, USAbDepartment of Physics and Astronomy, SUNY, Stony Brook, NY 11794, USA

cDepartment of Physics and Astronomy, Arizona State University, Tempe, AZ 85287, USA

Abstract

We consider the theory and data analysis in Fourier-transform X-ray holography. We also report studies and

experimental investigations of practical ways to generate a suitable holographic reference wave.# 2001 Elsevier ScienceB.V. All rights reserved.

Keywords: X-ray; Holography; Diffraction; Small-angle X-ray scattering; Pinhole arrays; Aerogels

1. Introduction

We have described the advantages and history[1–4] of Fourier-transform X-ray holography as apotential high-resolution imaging scheme in anearlier paper [5]. In this paper, we present somefurther arguments concerning the analysis of thistype of diffraction problem and the derivation ofhigh-resolution images from the measured data.We also report our experimental assessment ofrandom pinhole arrays and aerogels as possibleholographic reference objects.

2. Theory of Fourier-transform holography

Consider the plane-wave illumination of twoobjects producing exit-wave amplitudes f and

g and angular spectra F and G, respectively.We can show by standard Fourier-optics methods[6] that the intensity recorded by the detector(in the far field) is the Fourier transformhologram

IH ¼ Fj j2þ Gj j2þFG * exp2pix1 � b

lz

� �

þ F *G exp �2pix1 � b

lz

� �;

where the complex phase factors result from thespacing of the two objects, b, according to the shifttheorem. Normally, we would regard one of theobjects (say g) as the reference object and seek tomake a reconstruction by illuminating the holo-gram by the original reference wave from g andback propagating the result to the object plane.We generalize that here slightly and use the wavefrom another object g0 which may or may notequal g [7]. The resulting complex amplitude c in

*Corresponding author. Tel.: +1-510-486-4949; fax: +1-

510-486-7696.

E-mail address: mrhowells@lbl.gov (M.R. Howells).

0168-9002/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 4 9 8 - 3

the sample plane is then

c xð Þ ¼ g0 *Rff þ g0 *Rgg þ f x � bð Þ*Rgg0

þ f * �x � bð Þ*ðg0*gÞ;

where Rgg0 is the cross correlation of g and g0, etc.and * indicates convolution. If we take g and g0 tobe delta functions or at least single sharp peaksthen the usual four holographic image terms, theautocorrelation functions of f and g and the trueand conjugate images, are visible in c. Since this isan off-axis holographic scheme the twin imageswill be separated if the original objects aresufficiently separated.

3. Reconstruction of high resolution images:

deconvolution

We are interested in ways to achieve a high-resolution image without having to realize a point-like reference object. Therefore, following Stroke[7], we first note that the interesting true-imageterm is convolved with Rgg0 . Thus if Rgg0 issufficiently point-like, then a high-resolution imageimmediately results. One interesting case is wheng ¼ g0 and g is a random array of pinholes or othersmall objects. In this case, Rgg0 reduces to theautocorrelation of a single object and a high-resolution true image again results. (g*g

0 does notsimilarly reduce and the conjugate image does notreconstruct.) There may be other ways in whichRgg0 can be made small, however, we are alsointerested in the general case when g is known butRgg0 is not small. In such a case a high-resolutionimage can still be recovered if a deconvolution canbe implemented.The standard way to do this starting from c

would be to calculate F�1fF cð Þ=GG0* g. How-ever, we can do the same thing more directly bytaking F�1fIH=G* g. Evidently the power spec-trum of g should be non-zero at all frequenciesx1=lz for which we desire information. Unfortu-nately this is unlikely to happen because, even forobjects with the right spectral coverage likerandom pinhole arrays, the power spectrum (i.e.the diffraction pattern) will be a speckle patternwith many zeros. The way around this, which alsoallows some correction for spectral distortion due

to noise, is to use a Wiener filter [8]. The recoveredimage is then

cr xð Þ ¼ F�1 IH1

G*

Gj j2

Gj j2þF

� �;

where F is the ratio of the noise power to signalpower. This provides a pathway to high resolutionfor reference objects which are known. In princi-ple, this can also include unknown referenceobjects because one can holograph a pair ofcomplex objects, a reference and a sample, placedwith sufficient separation, and recover the struc-ture of both via the iterative Fourier-transformalgorithm [9]. A diffraction tomography experi-ment could then proceed by rotating the samplethrough at least 1808 while keeping the referenceobject fixed in the position in which its exit-facewave function was measured.

4. Practical reference objects

We have made experimental tests of pinholearrays and silica aerogels as candidate referenceobjects. The pinholes were about 10 nm indiameter and were produced by etching damagetracks made in 5 mm-thick mica plates by a 4MeVa-particle beam. The aerogels were of pure silica,3–4% dense and with peak pore size of 10–20 nm.We know, based on published hard X-ray small-angle patterns [10], that such materials haveabout the right spatial-frequency coverage,0.1–0.001 (A�1, for high-resolution X-ray imaging.Ours were provided by A. Hunt of the LawrenceBerkeley National Laboratory Division of Envir-onmental Energy Technology. The choice ofpinholes was based on the theoretical argumentsadvanced in the previous section plus the promiseof easy measurement by standard scanned-probeor electron-microscope methods.

5. Experiments and results

The candidate reference objects were illumi-nated by unmonochromatised undulator radiation(‘‘pink beam’’) on beam line 9.0.1 at the AdvancedLight Source at the Lawrence Berkeley NationalLaboratory. The beam was restricted by filters,

M.R. Howells et al. / Nuclear Instruments and Methods in Physics Research A 467–468 (2001) 864–867 865

apertures and a beam stop just in front of the CCDdetector so that in the absence of diffraction by thesample, the count rate recorded by the detectorwas essentially zero. The beam consisted essen-tially of 100% 560 eV X-rays (for maximumabsorption by mica) in the third undulatorharmonic with coherence length of 120 wavesand nominally perfect spatial coherence.We were not able to observe the expected

diffraction pattern and wave-guide effects fromthe pinhole arrays. This may have been due toblockage of the holes by contamination, althoughwe did apply a standard countermeasure to that (a10�4 Torr background pressure of oxygen). Ourobservation of the holes by electron microscopywas necessarily done on a thin surface flake ofmica which also leaves open the possibility of onlypartial penetration by the etchant. Reciprocityarguments suggest that alignment of the holes isnot critical for pinholes of this size (diameter=5l).On the other hand, the aerogels gave good soft-X-

ray scattering patterns with a spatial frequency cutoffcorresponding to about 20 (A resolution (Fig. 1).

6. Discussion and conclusion

Although it is reassuring that aerogel referenceobjects seem to work well, we have to take into

account that the scattering strength per unitvolume is much reduced due to the low density.The requirement is that the integrated scatteringstrength of the reference should be at least as highas that of the biological sample material relative toits background of water. To help in understandingthis issue we show in Fig. 2 the scattering strengthof three important materials expressed in electronsper (A3. The weaker scattering of the aerogelscompared to the plotted values for normal silica,could be compensated in part by loading theaerogel with high Z material during manufactureand by simply making the reference object larger.Although estimates of the attainable scatteringstrength seem to rule out 2D arrays of goldspheres, another interesting approach wouldbe to construct a 3D array of such spheresby means of a variant of the normal labelingprocess.Overall, we believe that it is practical to build

suitable reference objects for X-ray Fourierholography. Once this is done, we also see apathway to making a high-resolution deconvolu-tion of the image at each view direction of thetilt series in spite of the inevitable zeros inthe reference object diffraction pattern.

Fig. 1. Measured soft X-ray scattering pattern from a silica

aerogel.

Fig. 2. Scattering strength of three interesting materials for

designing a Fourier-holography experiment. Curves are calcu-

lated from tabulated atomic-scattering-factor data.

M.R. Howells et al. / Nuclear Instruments and Methods in Physics Research A 467–468 (2001) 864–867866

Acknowledgements

We wish to thank Arlon Hunt for supplying theaerogel samples, Tom Baer and Steven Kevan forhelping us to get beam time and the US Depart-ment of Energy Office of Environmental andBiological Research for support under contractDE-FG02-89ER60858.

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