1788 J. Opt. Soc. Am. A/Vol. 13, No. 9 /September 1996 Lindaas et al.

X-ray holographic microscopy by means ofphotoresist recording and

atomic-force microscope readout

Steve Lindaas and Malcolm Howells

Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720

Chris Jacobsen and Alex Kalinovsky

Department of Physics, State University of New York, Stony Brook, New York 11794-3800

Received September 12, 1995; revised manuscript received April 11, 1996; accepted April 12, 1996

We have reconstructed in-line (or Gabor) x-ray holograms at 40–50-nm resolution from a complex biologicalspecimen. The holograms were recorded as a relief pattern on photoresist with use of 1.89-nm, soft x raysfrom the X1A undulator beam line at the National Synchrotron Light Source at Brookhaven National Labo-ratory. We have improved the resolution and the fidelity and simplified the experiment compared with earlierwork by employing a special atomic-force microscope to examine and digitize the holograms. Following digi-tization the holograms were reconstructed numerically, allowing both the absorptive and phase-shifting prop-erties of the reconstructed object to be mapped. A comparison of the reconstructed images with images ob-tained from visible light and transmission electron microscopes has been made to confirm the validity of thex-ray holographic technique. The method offers promise as a technique for soft-x-ray microscopy and diffrac-tion tomography of dry and frozen hydrated specimens and for microscopy with pulsed x-ray sources. © 1996Optical Society of America.

1. INTRODUCTION

The basic ideas of holography date back to the 1948 paperby Gabor.1 The suggestion by Baez2 that these ideascould be applied to x-ray imaging followed only four yearslater. Considerable effort was made around that time torecord x-ray holograms by using a point-focus x-ray tubeand photographic film.3 However, the task proved diffi-cult, and it was some years before reconstructed images ofeven the simplest objects were obtained.4,5 It becameclear that, for effective x-ray holographic microscopy, newtechnologies would be needed, and interest in the methoddiminished.It is now clear that what was lacking in the early ex-

periments was an x-ray source of sufficient coherentpower and an x-ray recording medium of adequate reso-lution. Thus, when the prospect of x-ray laser and syn-chrotron radiation sources with many orders of magni-tude more coherent power appeared in the early 1980’s,interest in x-ray holography revived.6,7 Experimentalprograms were started at various centers,8–10 usingmostly undulator x-ray sources and polymer photoresistdetectors, and these led to the achievement of submicron-resolution reconstructed images at Brookhaven11,12 andOrsay13 and the demonstration of x-ray laser holographyat Livermore.14 High-resolution x-ray Fourier hologra-phy was also demonstrated for the first time by McNultyet al.,15,16 and the history of all these efforts was reviewedby Jacobsen.17

The original motivation for doing holography with xrays was to construct an x-ray microscope without havingto build an x-ray lens, and even though we now have

0740-3232/96/0901788-13$10.00

Fresnel-zone-plate x-ray lenses, this reasoning still has acertain validity, in part because available zone-platelenses are far from ideal. The highest-resolution oneshave efficiencies of only a few percent, and they are ex-tremely difficult to obtain. They cannot be bought com-mercially, and their production requires considerabletechnical skill and significant investment in equipmentand infrastructure. Apart from the attraction of beinglensless, x-ray holographic imaging can simultaneouslyprovide amplitude- and phase-contrast images, has aresolution competitive with that of zone plates, and hasan almost unrivaled experimental simplicity. One justexposes the resist through the sample, which, with a mod-ern undulator x-ray beam, can be placed anywhere in theroughly 1-mm2 illuminated area. Moreover, holographyhas special advantages in imaging with pulsed sourcesbecause the hologram recording step does not have to bein focus. Consequently, if the pulse is fast enough,18–20

an image can be captured at room temperature withoutradiation damage. Suitable pulsed x-ray sources basedon x-ray lasers and free-electron lasers are currently un-der development at several laboratories.21

The use of soft x rays in the spectral region 1.0–5.0 nmhas particular advantages for imaging based on their pen-etrating power and contrast mechanisms. With this inmind, a number of soft-x-ray microscopes have been con-structed for both biological and material science research.Most of these instruments use zone-plate lenses in eitherscanned-probe or full-field imaging geometry, and theirscience and technology have been reviewed in several re-cent review papers22,23 and conference proceedings.24–26

X-ray holography is, in essence, a member of this family

© 1996 Optical Society of America

Lindaas et al. Vol. 13, No. 9 /September 1996 /J. Opt. Soc. Am. A 1789

of imaging techniques, and it shares the advantages de-rived from the fundamentals of the radiation–sample in-teraction.In this paper we describe the basis for our choice of in-

line (Gabor-type) geometry for our x-ray holographic mi-croscopy system. We outline the technical requirementsfor such a system to work with good resolution with par-ticular attention to the question of hologram readout, andwe describe how we have addressed the readout problemby means of a specially developed atomic-force microscope(AFM). We explain why it was necessary to build a cus-tom AFM, and we give a description of its constructionand performance characteristics. We give a brief accountof how an x-ray holography experiment is done and showexamples of how the data from such an experiment areread by the AFM and used to produce reconstructed im-ages by digital image processing. We also discuss thecontribution that holography, practiced in this new way,can be expected to make to biological investigations in thefuture with special reference to the question of radiationdamage. We argue on the basis of evidence from electronmicroscopy that imaging at cryogenic temperatures prom-ises to allow sufficient applied dose to permit a full tilt se-ries and tomographic reconstruction of a three-dimensional image without the creation of radiationartifacts at the resolution level where soft-x-ray hologra-phy can operate.

2. HOLOGRAPHIC GEOMETRYThe original holographic experiment of Gabor utilized anon-axis reference wave (the basic geometry is shown inFig. 1). The scheme is simple and can be implementedwithout any optics other than the source and the record-ing medium. The absence of optics is particularly per-suasive in designing an experiment in which a high-quality x-ray phase front is required, and most of thex-ray holography that has been done to date has used thisscheme. With a high-resolution detector and the meansto read it one should, in principle, realize an image reso-lution that is approximately the same as that of the de-tector, as originally predicted by Baez.2 The main limi-tations of in-line holography are (1) that the object fieldmust have high overall transparency to provide a goodreference beam from the transmitted light and (2) the so-called twin-image artifact. The latter arises as fol-lows: Assuming a reconstruction using the original ref-erence wave, the fringes recorded in the hologram diffractthe reconstructing light beam equally into the plus andminus first orders in such a way that two images areformed: one, a virtual image, at the sample position andthe other, a real image, at a distance z [Fig. 1(B)] down-stream of the hologram. At the plane of the real imagethe light diverging from the virtual image at S in Fig. 1Binterferes with the light in the zero-order (undiffracted)beam to produce a second hologram at distance 2z fromthe object, and this is mixed with the real image. Thesignal that is due to the second hologram represents acorruption of the desired real-image information that can-not be removed in any simple way. The fringes from theunwanted hologram can be seen in Figs. 8 and 10 below.We discuss ways to address this problem in Subsection6.B.

Consider a square hologram of half-width x0 at dis-tance z from the sample subtending a half-angle u asshown in Fig. 1. The numerical aperture (N.A.) of thesystem is therefore sin u, and the diffraction-limited reso-lution, dt , is l/2(N.A.), where l is the x-ray wavelength.When the sample is small compared with the hologram,the highest fringe frequencies will evidently be recordedat the edge of the hologram and the fringe frequencythere will be approximately x0/lz. In order to samplethis fringe system without loss of information, we must,according to Shannon, sample at twice the maximum fre-quency. The sampling interval Ds must therefore begiven by

Ds <lz2x0

5l

2(N.A.)5 d t . (1)

This is roughly equivalent to the well-known rule that thediffraction-limited resolution of a zone plate is approxi-mately equal to the width of its outer zone. The totalnumber of sampling intervals is N, where N 5 2x0/Ds .Thus

N 5~2x0!

2

lz5

4(N.A.)2zl

5lz

Ds2, (2)

which is 43 the number of Fresnel zones. This gives usthe size (N 3 N) of the data set that we will have to pro-cess.It is noteworthy that Fig. 1 and the above equations are

oversimplifications in the sense that we are not really freeto choose the size and the N.A. of the hologram arbi-trarily. The largest useful value of x0 (and hence of theN.A.) is the value at which the fringes just cease to be dis-cernible because of insufficient detector resolution, signal-to-noise ratio, or x-ray beam coherence.

Fig. 1. Schematic showing parameters used in our discussion onGabor holography. The top schematic (A) illustrates the record-ing scheme, while the bottom diagram (B) shows the reconstruc-tion layout. The hologram is formed by recording the fringes ofthe interference between the scattered wave and the incidentwave. The reconstructed images are formed in this case by us-ing the hologram as a diffracting structure illuminated by theoriginal reference beam.

1790 J. Opt. Soc. Am. A/Vol. 13, No. 9 /September 1996 Lindaas et al.

3. EXPERIMENTAL DESIGNCONSIDERATIONSLet us assume that we wish to make holographic imageswith transverse resolution dt by using x rays of prescribedwavelength l. By relation (1) Ds and N.A. are immedi-ately determined. However, x0 and N are not deter-mined until we choose a value for the working distance z.In one sense we would like z to be as large as possible.At a given dt and N.A. this would lead to a large holo-gram, which would make it easier to satisfy the generaltransparency requirement of in-line holography andwould also lead to a more tractable twin-image-suppression problem. However, increasing z also re-quires greater monochromaticity and spatial coherence[relations (3) and (4) below]. In addition, a larger zleads, according to Eq. (2), to a larger N, for which thereare practical computational limits. The reasoning shouldtherefore be to choose the largest N value allowed byavailable computer hardware. The value of x0 then fol-lows from relation (1), and z follows from Eq. 2.The coherence length l2/Dl must be greater than the

greatest path difference between interfering beams,which, from Fig. 1, is SB 2 AB . x0

2/2z. The monochro-maticity requirement is thus

l

Dl.

x02

2lz5

(N.A.)2z2l

5N8. (3)

From Fig. 1 it is also evident that the greatest transverseseparation of incoming rays that will later have to inter-fere is AS 5 x0 , so the coherence width (wc) must satisfy

wc . x0 5 ~N.A.!z. (4)

Using a monochromator and suitable slits, we can meetthese requirements by using almost any soft-x-ray source.However, the power remaining will be sufficient to recorda hologram in a timely fashion only if the source has highbrightness (flux per unit phase-space volume). Amongavailable continuous-wave soft-x-ray sources, undulatorscurrently provide the highest average brightness and aretherefore the source of choice for x-ray holography.The size of the source or slit needed to produce a given

coherence width at a prescribed distance is traditionallycalculated by using the van Cittert–Zernike theorem,27

which strictly applies to incoherent sources. However,given the strongly directional character of undulator ra-diation, one might suppose that an undulator source mustbe at least partially coherent. The situation has been in-vestigated by Howells and Kincaid,28 who find that in theregime of interest (small sources at great distances) thevan Cittert–Zernike theorem results can still be applied.According to the van Cittert–Zernike theorem, the de-

gree of coherence between the fields at two points in thefar field depends only on the distance between the pointsand not on their absolute positions. Thus, if the source ismultimode and unfocused, the illuminated area may bemuch greater than the coherence width and Gabor holo-grams of maximum width 2wc can be made anywherewithin that area. In fact, by using the whole beam foot-print from the multimode undulator source, we can recordmany holograms simultaneously.

4. PHOTORESIST RECORDINGThe holograms reported here were recorded by using thephotoresist poly(methyl methacrylate) (PMMA) as the de-tector. PMMA is the highest-resolution organic materialcommonly used as a photoresist, and it has been used towrite 10-nm isolated lines by electron-beam litho-graphy.29 This suggests that similar resolution can beachieved by using soft x rays. In addition, the x-ray de-tective quantum efficiency of PMMA has been estimatedas 10% (Ref. 30), although we believe that there may bepossibilities for improvement in that figure. PMMA iscomposed of long-chain molecules, and when ionizing ra-diation (.4.3 eV) is absorbed, bonds are broken along themain chain as well as between the main chain and sidegroups. The recording of x rays below a certain dose op-erates by reducing the local molecular weight by these re-actions, thereby increasing the dissolution rate in the wetdeveloper. We have begun to characterize the sensitivityof PMMA as an x-ray detector31 with the aim of determin-ing optimum exposure/development levels.After development a surface relief map is produced

with valleys corresponding to high exposure and moun-tains to low exposure. However, the amount of resist re-moved is not a linear function of the x-ray exposure. Thedependence of the depth, td , of resist removed by the de-veloper as a function of absorbed dose D is well approxi-mated by

td 5 ThR0S DD0Dg

5 kDg, (5)

where T is the development time, h is the developer con-centration, R0 has a value in the range 10–100 nm/s fortypical conditions with D0 5 10

4 gray, and g . 2 (Ref.32). We use this relationship with a constant k in an at-tempt to linearize the x-ray recording during data reduc-tion (Subsection 7.E).

5. READOUT OF THE HOLOGRAM BY ANATOMIC-FORCE MICROSCOPEA. RequirementsSince the hologram exists as a relief pattern, the AFM isthe ideal measurement tool because it measures relief di-rectly. It also allows a hologram to be recorded on a rigidsubstrate (rather than a thin membrane) and has the ad-ditional benefits of being nondestructive and of providingdirect (one-step) digitization of the hologram.We noted in Section 2 that the hologram must be digi-

tized with a pixel size (Ds) no larger than the desired im-age resolution dt . Numerical calculations of the opticalperformance of Fresnel zone plates (which are a type ofhologram) have shown that zones must be correctlyplaced to an accuracy of approximately 1/3 of the finestzone width.33,34 Thus the measured position of hologramdigitization points must be accurate to Ds/3. This re-quirement must be met over the entire hologram width2x0 , leading to a fractional position-error tolerance ofDs/6x0 5 1/3N. For example, to achieve 20-nm resolu-tion at a 500-mm working distance with l 5 2 nm, wewould have N 5 2500, which imposes an absolute accu-racy requirement of approximately 1 part in 104.

Lindaas et al. Vol. 13, No. 9 /September 1996 /J. Opt. Soc. Am. A 1791

B. Principle of OperationThe AFM operates by raster scanning a fine tip across thesurface to be measured. The tip is attached to the end ofa miniature cantilever that has a suitably small springconstant (Fig. 4 below). In the operating mode we em-ploy the scan proceeds by detecting the displacement ofthe cantilever, due to the tip–sample interaction, and ap-plying a feedback-controlled displacement of the sampleto maintain a constant cantilever deflection (i.e., constantforce). Near a surface the major forces acting on the tipare the van der Waals force between the tip and the sur-face and the spring force of the cantilever. The operationof an AFM can be based either on the very-short-range re-pulsive van der Waals force (contact mode) or on thelonger-range, but weaker, attractive force (noncontactmode).35 We have operated in contact mode exclusively,using cantilevers with spring constants of k , 0.01 N/m,so the force exerted on the sample by the tip was typicallyless than 1029 N. Under these conditions the boundariesof the scan areas could not be seen on subsequent largerscans, leading us to believe that no surface damage oc-curred. This observation is in agreement with the expe-rience of practitioners of x-ray contact microscopy whoalso use an AFM in contact mode to scan PMMAsurfaces.36,37

There are many methods to determine the cantileverdeflection. The most common, and the one that we use, isan optical lever. Light from a laser is reflected off thecantilever’s back surface into a split photodiode. Bend-ing the cantilever moves some light from one half of thephotodiode to the other. The difference between the pho-todiode currents can thus provide the feedback signal tothe sample driver (Fig. 4 below), which moves the sampleso as to null the difference signal.Most commercial AFM’s use piezoelectric translators to

provide the scan motion. Calibration curves of the scanfield are then used to reduce nonlinearities to a level ac-ceptable for most applications. In one ‘‘linearized’’ com-mercial AFM system that we checked we measured ;1%field distortions, and we believe that this is typical.These problems are overcome in our AFM system by us-ing capacitance micrometers to index accurately the x–yscan motion by means of a closed-loop feedbacksystem38,39 as described in Subsection 5.C.

C. High-Linearity x–y Scanning StageWe have used a monolithic x–y motion stage with flex-ural hinges (Fig. 3 below) to scan the sample in x and ywhile the tip is held fixed. Figure 2 shows one bendingelement from the stage. It is constructed from two can-tilevers each of length l/2 joined by a rigid center piece oflength L. For this element the displacement h due to aforce F is

h 5l2

4EI S l3 1 L2 DF, (6)where E is Young’s modulus and I is the section momentof inertia. For a beam of rectangular cross-sectionI 5 wt3/12, where w is the width and t is the thicknessof the beam. This allows the spring constant, k 5 uF/hu,to be easily calculated.

The x and y linear-motion flexures are each composedof four pairs of these elements, as shown in Fig. 3. Thetwo classical rectilinear motion mechanisms40,41 are lo-cated one inside the other. By this design we are able toachieve extremely good orthogonality and independenceof the x and y drives. Since the stage is not compact, itsthermal expansion during a scan must be considered.We discuss this further in Subsection 5.E.The design parameters for our custom aluminum stage

are given in Table 1. Using these values in Eq. (6), wecalculate an effective spring constant of 5.5 3 105 N/m,which is in good agreement with the measured value of(6.0 6 0.5) 3 105 N/m. Using an accelerometer, we mea-sured a resonant frequency of 150 Hz for the inner (x)axis, which is safely above our piezo’s maximum slew rateof 20 Hz and in good agreement with the value of 140 Hzcalculated from the above value of the spring constantand the estimated mass of the moving part of the stage.

Fig. 2. Schematic of one bending element from the stage.When used in the whole stage the hinge is designed to translatealong F without rotation. Table 1 lists the hinge dimensions.

Fig. 3. Custom stage built for the atomic force microscope(AFM). The inner stage is the fast (x) axis. The two piezos areoriented orthogonal to each other. A, B, and C are through holesfor the tripod-mounted cantilever deflection sensing unit, while Sis the mounting hole for the z piezo.

Table 1. Design Values for the Aluminum Stage’sFlexure Dimensionsa

t (thickness) (mm) 1.42w (depth) (mm) 28.7L (mm) 22.25l (mm) 5.7

aRefer to Fig. 2 for a schematic of the hinge.

1792 J. Opt. Soc. Am. A/Vol. 13, No. 9 /September 1996 Lindaas et al.

In order to attain good absolute positioning accuracy,we used Queensgate Instrument piezo drive systems withbuilt-in indexing.42 The system generates an indexingsignal derived from a capacitive sensor consisting of twoparallel plates whose separation is determined by mea-suring their capacitance with an ac bridge. The Queens-gate devices that we use have a departure from linearityof approximately 4 parts in 104 over a 75-mm range.However, the voltage-position curve is sufficiently stableand reproducible to yield an absolute positioning accuracyof ;1 nm or 1 part in 7.5 3 104. The electronic readoutnoise is 0.005 nm/AHz, i.e., 0.5 nm at the piezo system’spresent bandwidth of 10 kHz.

D. Atomic-Force Microscope System Architecture andControlThe stage scans the sample in x and y below a fixed tipmounted in a commercial AFM cantilever-deflection sens-ing unit that we have interfaced to our computer. Thesample is mounted on the scanning stage by means of an-other piezo translator (the z piezo depicted in Fig. 4),which drives it toward or away from the tip under feed-back control.Therefore there are three principal elements to con-

trol: (1) the raster scan (x and y axes), (2) the sample’sposition relative to the tip (through the z piezo), and (3)the data acquisition. To control the x and y scans, weuse a 16-bit digital-to-analog converter (DAC) with twooutput channels programmed to produce staircase func-tions. We require the 16-bit accuracy to take full advan-tage of the Queensgate piezo’s indexing resolution (;1nm) and range (;75 mm).The commercial cantilever-deflection sensing unit de-

livers the amplified sum and difference signals from thesplit photodiode. The integral of the difference signal isused to generate the z-piezo voltage and is also the signalused to record the surface height at each pixel.A 16-bit analog-to-digital converter (ADC) (with eight

input channels) is used to collect the value of thefeedback-controlled z-piezo drive voltage for each pixel aswell as environmental data (air and stage temperaturesalong with humidity) after each scanned row. Both the

Fig. 4. Schematic of our custom AFM’s interface. Forces actingon the probe tip cause the cantilever to bend, which is monitoredby a sensor. The signal is then used in the z-drive feedback,which regulates the sample–tip force.

DAC and the ADC have sufficient internal memory sothat the computer is involved at only the beginning andthe end of a scan line.Interfaces were constructed to both the commercial de-

flection sensing head and the Queensgate piezo drivers.The hardware is controlled by means of an IEEE 488.2 in-terface to an IBM RS/6000 UNIX workstation. By usinga UNIX workstation to control our system, we are able toacquire virtually any size data set (a flexibility that com-mercial AFM’s do not normally have). In addition, theextra computational power and memory management ca-pabilities of our IBM RISC machine are highly advanta-geous when reconstructing holograms. For example,even though a single 2048 3 2048 complex floating-pointarray occupies 64 Mbytes of computer memory, we canstill reconstruct a hologram that is encoded with this pre-cision.

E. Atomic-Force Microscope EnvironmentTo achieve atomic resolution, most AFM’s are designed tobe light and stiff, giving them high resonant frequenciesand therefore good isolation from mechanical noise. OurAFM has less need of such isolation, since atomic-resolution imaging is not a goal of the present work (nor apossibility given the 0.5-nm position noise of the capaci-tance micrometers). Instead, our requirement is to meetthe measurement requirements outlined in Subsection5.A over an AFM scan time that can be as long as 30 min(e.g., 2048 3 2048 pixels scanned unidirectionally with a0.1-ms pixel dwell time). With the stage now used,which has mechanical paths of approximately 20 cm be-tween the AFM head and the sample scan mechanism,the stage temperature should remain constant to approxi-mately 0.1 °C during a scan. Therefore, in order to mini-mize drifts that would misplace pixels from their theoreti-cal positions and therefore lead eventually to imageaberrations, a large thermal mass is favored. We use aninsulated enclosure to provide acoustic, thermal, and op-tical isolation of the AFM during operation, and the scanis abandoned if a stage-mounted thermocouple indicatestemperature changes that exceed our limits. The AFM isalso mounted on an air table and rests on a stack of steelplates with Viton spacers for vibration damping.

F. Performance TestsWe have imaged gratings that were well characterized in-dependently, and by this means we have so far demon-strated that the scan linearity of our AFM is 0.05% or bet-ter. We believe that we will be able to demonstrate theexpected value of 0.01% when we have implemented anindexed z piezo (e.g., by using a capacitance micrometer).The best measure of performance is the reconstructed im-age quality from actual holograms. Figure 5 is a Fresnelhologram of a gold wire (12 mm in diameter) with a lineplot taken from the indicated region. The fringes thatcan be seen at the edge of this scanned hologram and thesuccessful reconstruction of the other holograms to givehigh-quality images lead us to believe that the AFM andthe stage are operating as designed.

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6. NUMERICAL RECONSTRUCTIONA. TheoryIn principle, we could reconstruct the hologram encodedin the PMMA by placing it in the original illuminatingwave and observing the real image a distance z down-stream. To mimic this numerically, we have to modulatea plane wave by the recorded hologram function and nu-merically propagate it by a distance z to calculate the in-tensity distribution at the real-image plane. This pro-duces the same result that we would obtain in theequivalent laboratory process: the true image plus a sec-ond hologram of the sample at the distance 2z (see Sec-tion 2).We propagate the incident wave c0(x, y: 0) modulated

by the function g(x, y) to the image plane (i) by

c~xi , yi : z ! 5 E E2`

`

c0~x, y: 0 !g~x, y !

3 h~xi 2 x, yi 2 y: z !dxdy. (7)

The propagator function h( ) is given in the Rayleigh–Sommerfeld formulation43 as

h~xi 2 x, yi 2 y: z ! 5il

exp~2i2pr/l!r

cos u, (8)

Fig. 5. Gabor hologram from a 12-mm-diameter gold wire. Theplots show the average of the 30 lines outlined in the image.Fringes are visible out to the edge of the field of view. The plotsshow a fringe positioning accuracy of ,0.1 mm and a height reso-lution of approximately 1 nm.

where r2 5 (xi 2 x)2 1 (yi 2 y)

2 1 z2 and r is the dis-tance from a point in the hologram plane to a point in theimage plane, u is the angle between the direction of r andthe axis, and cos u 5 z/r is the obliquity factor. Applyingthe convolution theorem and specializing to the case ofplane-wave illumination, we can express Eq. (7) as

c~xi , yi : z ! 5 F 21$G~ fx , fy!H~ fx , fy : z !%, (9)

where we use F to represent a Fourier transform, so thatG( fx ,fy) 5 F $g(x,y)%, etc., and fx is the spatial fre-quency corresponding to x, etc. The propagator functioncan be expressed exactly in transform space (Ref. 44, Ap-pendix 1) as

H~ fx , fy : z ! 5 expF2i2p zl ~1 2 fx2 2 fy2!1/2G5 expS 2i2p zl D expF iplz~ fx2 1 fy2!

1ipl3z4

~ fx2 1 fy

2!2 1 •••G . (10)Ignoring the constant phase factor and using the Fresnelapproximation, we can write

H~ fx , fy : z ! . exp@iplz~ fx2 1 fy2!#. (11)

This approximation is usually valid in the regime inwhich we operate; however, the additional terms shownin Eq. (10) are used in our reconstruction algorithm iftheir effect is significant.

B. Twin-Image ProblemThe twin-image artifact was explained in Section 2. Theconventional approach to eliminating it is to use iterativephase-retrieval algorithms such as those developed byGerchberg and Saxton45 and Fienup.46 Applications ofthis type of algorithm to x-ray holography have been re-ported in a preliminary way by us47 and by Koren et al.48

The algorithms work by propagating a complex wave fieldback and forth between the object and hologram planes.Constraints are applied at each plane. At the hologramplane the intensities are constrained to equal the mea-sured ones; while, at the object plane, the constraintmight be that the empty parts of the field of view areforced to have a transparency of unity (the finite-supportconstraint). The intention is that the procedure shouldconverge toward a unique object transparency functionthat satisfies the object-plane constraints and that dif-fracts an incoming plane wave into the measured holo-gram. We have implemented an algorithm consisting ofcombinations of the algorithms of Gerchberg and Saxtonand of Fienup. The simplest form of this approach, inwhich only the finite-support constraint is applied at theobject plane, has been successful in removing the twin-image signal when the object is small and well isolated.It is less successful in cases in which there are manystrong scatterers inside or just outside the hologram area.This is a complex problem and is the subject of continuingresearch on which we will report more fully in the future.We believe that the problem will yield to further efforts inall cases in which the sample is sufficiently sparse for in-line holography to work at all.

1794 J. Opt. Soc. Am. A/Vol. 13, No. 9 /September 1996 Lindaas et al.

7. EXPERIMENTAL DETAILSA. Synchrotron-Radiation Beam LineFor the work reported here we used the 8-cm-period un-dulator at the X1A beam line49,50 at the National Syn-chrotron Light Source at Brookhaven National Labora-tory, which has a brightness more than 10 orders ofmagnitude higher than that of a typical microfocus x-raytube. Figure 6 is a schematic of the optical layout of theX1A beam line. In the latter part of 1996 we plan to con-tinue our experimental program on an even brighter un-dulator source at the Advanced Light Source at LawrenceBerkeley National Laboratory.While wet biological specimens are best studied by us-

ing ‘‘water-window’’ soft x rays with 2.3 nm , l , 4.4nm, the studies reported here were on dry specimens.For this case the use of water-window wavelengths is lessimportant, and we used mostly l 5 1.89 nm. For a typi-cal sample–hologram spacing of ;500 mm and using rela-tions (1) and (3), we see that to obtain 40-nm resolution inthis case, we require spatial coherence over a transversedistance x0 * 15 mm and l/Dl * 80. In the actual ex-periment we had l/Dl . 400, a horizontal coherencewidth of 50 mm, and a vertical coherence width *200 mm.The flux delivered to the sample for in-vacuum experi-ments was ;105 photons/s/(mm)2 per 100 mA of storedelectrons in the synchrotron storage ring. We could havechosen to concentrate the beam into a smaller area withoptics, but this would not have made the exposure timeshort enough to avoid radiation damage while it wouldhave made targeting more difficult and risked complica-tions because of imperfect optics.

B. Photoresist HandlingWe used a solution of 9% by weight PMMA (molecularweight 5 9.7 3 105 Daltons) in chlorobenzene spun at4.5 k rpm onto glass substrates, resulting in ;1-mm films.The photoresist film was baked at 150 °C for 2 h. Thebake serves to outgas adsorbed gases and maximizesPMMA chain scission relative to cross-linking uponirradiation.31 The glass substrates were cut from micro-scope slides and cleaned in preparation for spinning. Wefound that high relative humidity (.50%) frustrated thePMMA film’s adhesion to the glass substrate.The photoresist was developed by immersion in a mix-

ture of 1 part methyl isobutyl ketone to 5 parts isopropylalcohol for 5–30 s, briefly rinsed in isopropyl alcohol, anddried with a filtered jet of nitrogen gas. The first few(typically five) Fresnel diffraction fringes (*500-nmwidth, ;200-nm depth) could be seen in the optical micro-scope and were used to assess hologram exposure.

C. Sample MountingDepending on the sample we used either a Si3N4 windowor a standard electron microscope grid as a holder. Aspacer plate was then used to set an approximate value ofthe object-to-hologram distance z. The frame of theSi3N4 window (or the grid) carrying the objects was thenclamped to one side of the spacer, and the photoresist sub-strate was clamped to the other side (Fig. 7). This en-sured a rigid mechanical connection between the objectand the holographic recording medium, allowing high-

contrast fringes to be obtained without any harmful vi-bration effects.

D. Recording and ReadoutThe object–hologram packages were aligned to the x-raybeam in a vacuum chamber, and the holograms were re-corded over exposure times of one-third to a few minutes.A pressure of approximately 1022 Torr was maintained inthe vacuum chamber, which was separated from thevacuum of the beam line by an x-ray transparent 0.1-mm-thick silicon nitride window. The exposure time (0.3–3min) was chosen to deliver ;6 3 107 photons/(mm)2 to thephotoresist [105 photons/(40 nm)2, giving 1% shot-noisestatistics if the resist detective quantum efficiency is

Fig. 6. Optical layout of the X1A beam line at the National Syn-chrotron Light Source. The numbers correspond to the distancein meters from the optical element to the undulator center, whichhas a horizontal source size of 390 mm and a vertical source sizeof 18 mm (both rms half-widths). In the horizontal plane (A),the exit slit (width dh) of a spherical grating monochromator isused as a spatial filter for coherent illumination of the hologramlocated a distance lh 5 1.98 m away. In the vertical plane (B), acylindrical mirror produces a vertical source of height dv 5 23mm at a distance lv 5 7.63 m downstream of the sample.

Fig. 7. The photoresist used to record the hologram is rigidlymounted to the specimen support by using a spacer to set thespecimen-to-hologram distance z.

Lindaas et al. Vol. 13, No. 9 /September 1996 /J. Opt. Soc. Am. A 1795

10%]. Photon fluxes were monitored and controlled byusing a retractable calibrated aluminum photodiode.The PMMA recordings were measured with an AFM

pixel size Ds over an area of (2x0)2. For N 5 1024,

l 5 1.89 nm, and z . 500 mm we use Ds . 32 nm and2x0 5 33 mm. From relation (1) this leads to an expectedresolution of 40 nm.

E. Data Reduction and Image ReconstructionSince PMMA is not a linear recording medium, we mustfind a mapping from measured resist thickness to the in-cident hologram irradiance I(x, y). Equation (5) indi-cates that the remaining thickness of developed resist,t(x, y), can be written in the form

t~x, y ! 5 t0 2 k@I~x, y !#g, (12)

where g . 2 for PMMA and k is a constant. We cantherefore write I(x, y) as

I~x, y ! 5 F t0 2 t~x, y !k G1/g

. (13)

In an AFM readout that includes unexposed areas such asthe shadows of grid bars we can take t0 to be the maxi-mum resist thickness. Without loss of generality the pa-rameter (1/k)1/g can be adjusted to set the average valueof I(x, y) equal to unity. In an optical reconstruction of alinearly recorded amplitude hologram, a wave field withuniform phase and an amplitude equal to the square rootof the recorded hologram irradiance would be launchedfrom the hologram toward the image plane. Hence weset the hologram function g(x, y) of Eq. (7) to beAI(x, y) with a phase of zero.To reconstruct the hologram, we then propagate the

wave field g(x, y) a distance z by using Eq. (9). Thepropagation involves a Fourier transform operation ong(x, y), an array multiplication with H( fx , fy : z), andan inverse Fourier transform operation.Before the final calculation is carried out, a small sub-

region of this wave field is extracted and used to form re-constructed images at a variety of values of z about theexpected value. From the sharpness of these images (us-ing larger subregions as the true z is approached) the bestz value is obtained.

Fig. 8. Line trace across the edge of a reconstructed hologram ofa diatom with each pixel indicated by a hatch mark. We take asa measure of our resolution the distance it takes to go from 20%to 80% of the maximum amplitude. This distance is 1 or 2 pix-els, each of which is 31 nm. The ripples are due to twin-imagenoise and are expected from theoretical calculations (see thetext).

8. RESULTSAlthough our long-term aim is to use x-ray holography toimage thick hydrated samples, the present investigationwas carried out with dried samples. One was a diatom,which we used, like many before us, because its steplikefeatures are suitable for assessing the imaging systemresolution. The other, which was intended to show thecapability to image more complex biological objects, was adried-cell preparation that was sufficiently thin and ra-diation hard to allow TEM examination for comparativemicroscopy. Although the dry cell was useful in allowingcomparative microscopy, it is not the type of object forwhich the soft-x-ray holography technique is being devel-oped. As explained in Section 10, the plan is to acquire acapability to make three-dimensional images of samples,whose total thickness (sample plus water) may be up to10 mm, which would therefore be well beyond the range ofuseful electron microscopy.

A. Diatom Hologram: Demonstration of ResolutionThe resolution test using diatoms was carried out atl 5 2.23 nm with an x-ray exposure of 20 s delivering1.5 3 107 photons/mm2. The photoresist was developedfor 30 s and then digitized by using the AFM with a pixelsize of 31 nm. The hologram was reconstructed at a dis-tance of z 5 460 mm without twin-image suppression.Figure 8 shows a line scan across the reconstructed im-

age of a part of the diatom resembling a parallel-sidedopaque strip. The strong ringing signals on both sides ofboth edges are due to twin-image noise and appear as pre-dicted by the theoretical calculations that have been donefor this type of object.51 In spite of the harm done to theoverall fidelity of the image by the twin-image noise, it isstill possible to estimate the system resolution from thesharpness of the step. We make the conservative as-sumption that the object step is ideally sharp and that allof the finite width of its image is due to blurring by theimaging system. Taking the 20%–80% step height, wearrive at a value of ;40 nm for the resolution.It is of interest to understand what determines this

value. The hologram was scanned on an AFM grid spac-ing of 31 nm, so the resolution is close enough to the ex-pected diffraction limit given by relation (1) that we can-not rule out the possibility that there is higher-resolutioninformation in the hologram that is lost as a result of thechoice of Ds . One way to address this question is to scana typical hologram area at a much smaller pixel size andexamine the power spectrum. Such an examination re-veals that the power spectrum of a hologram scannedwith (10 nm)2 pixels rolls off to white noise at ;25 mm21,suggesting that information is encoded at the 20-nm-resolution level. However, to utilize this information, wewould have to scan with a finer grid and a larger area,which would challenge both the power of our computerand the temperature stability of our AFM. Thus we be-lieve that the resolution of the holograms and the recon-structions that we have made in this study represents thelimitations imposed by our present experimental appara-tus and not a fundamental limit of the technique or theresist.

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B. Holographic Microscopy of a Dried CellTo demonstrate x-ray holography with a biologicalsample, we used NIL8 hamster neural fibroblasts grownin culture for 1–2 days on a carbon-stabilized Formvarfilm that had been deposited on a gold electron microscopegrid. The individual cells adhered well to the grid andspread out so that they were typically 1–2 mm thick whenwet. To prepare the cells for imaging, they were glut-araldehyde fixed and critical point dried.The cells were x-ray imaged at l 5 1.89 nm with an es-

timated dose of 7.5 3 105 gray. The photoresist was de-veloped for 10 s and digitized with a step size of 31 nm.The hologram was reconstructed at a distance of z 5 415mm without twin-image suppression.After reconstruction of the hologram we took compari-

son pictures by using other types of microscopes. First, avisible-light microscope VLM with a 1003, N.A. 5 0.9 drylens was used to image the cell in reflected differential in-terference contrast. Then the cell was carbon coated andimaged in a JEOL 1200 TEM at an accelerating voltage of100 keV and magnifications of 20003–10,0003. Theelectron microscope delivered high-contrast images ofthin regions of this dry sample.Figure 9 is an overall view of the cell’s pseudopod,

while Figs. 10–13 show various subregions at highermagnification.

9. DISCUSSIONThe examples shown in Figs. 9–13 demonstrate that wecan produce holographic images of complex biologicalsamples without use of prior knowledge. The similarityof the holographic images to the TEM and the VLM im-ages gives us confidence that the entire x-ray holographicprocedure is working correctly. However, the x-ray re-constructions would benefit from twin-image reduction.For example, the ripples parallel to the main stem in thex-ray image in Fig. 10 are absent in the two other imagesand are clearly twin-image noise. Nevertheless, the truesample features (within the resolution limit) are evidentlyrendered faithfully in the x-ray images. Furthermore,there is a good deal of information that is absent from theVLM images that is successfully resolved in the x-rayones. In particular, the tendril seen in the x-ray imagein Fig. 10 cannot be seen in the optical micrograph, whilethe TEM image provides clear confirmation that it is notan artifact. Another region of interest is the central areashown in Fig. 12. This is a region of the cell that violatesthe requirement for high average transparency of the ho-logram area. However, comparison with the TEM imageagain shows that the features in the reconstructed holo-graphic image are not artifacts. This validation of ourx-ray holographic method is encouraging and leads us toconsider its future potential in more detail.

10. LIMITATIONS ON IMAGING AS ARESULT OF RADIATION DAMAGEOur broad intention for the future is to use holographicmicroscopy to image hydrated biological specimens. Ide-ally one would like to be able to image specimens in theirnatural wet state with resolution much higher than that

of the light microscope and to make three-dimensionalimages by means of a tilt series with no limit on the num-ber of exposures. Unfortunately, all of the currentlyavailable high-resolution imaging methods that can col-lect true three-dimensional information extendingthroughout the volume of the sample require the use ofsufficiently penetrating ionizing radiation. This neces-sarily imposes limitations that result from radiation dam-age.The radiation dose required to form a sub-100-nm-

resolution, soft-x-ray image in absorption contrast using

Fig. 9. Visible-light micrograph (top) and reconstructed x-rayhologram (bottom) of a critical point dried NIL8 hamster neuralfibroblast that was grown in culture on a film supported by a goldmesh. Much of the cell is out of this field of view; the nucleus isbeyond the upper right boundary of this image. Many intercel-lular organelles are shown at the upper right corners, and fur-ther structures are shown within the pseudopod at center rightin the images. The boxed areas are shown in greater detail invisible-light, x-ray holographic, and transmission electron micro-graphs as follows: A: Fig. 10; B: Fig. 11; C: Fig. 12; andD: Fig. 13.

Lindaas et al. Vol. 13, No. 9 /September 1996 /J. Opt. Soc. Am. A 1797

Fig. 10. Visible-light, x-ray holographic, and transmission electron micrographs of a long protrusion from the NIL8 cell shown in regionA of Fig. 9. The x-ray image shows clearly the organelles at upper right in the image, and the very small tendril at lower left, which isdifficult to see in the visible-light micrograph.

even the most dose-efficient technique is in the104–105-gray range.22,52 Hydrated biological specimens atroom temperature suffer mass loss and shrinkage at thisdose level over time scales of seconds or longer even whenthey are chemically fixed.53,54 These effects are due toreactions consequent upon the radiolysis of water. As-suming that the detective quantum efficiency of thePMMA resists is 10%, doses of the order of 106 gray perhologram will be required. Therefore we cannot makeeven single holograms of natural hydrated samples with-out significant radiation damage. This is one reasonwhy, for our present round of experiments, we have con-centrated on the use of dehydrated specimens.One might suppose that the dose required to make a

tilt series of, say, N members would be N times greaterthan that for a single two-dimensional image. However,it can be shown on information-theory grounds55,56 thatthe dose needed for definition of a single voxel with givenresolution and statistical accuracy in a single view doesnot need to be increased if the voxel becomes part of acomplex object and many views are taken so as to recon-struct the same voxel to the same resolution and statisti-cal accuracy by tomographic methods. Some of theidealizations involved in arriving at this conclusion mightbe difficult to realize in a practical experiment, but it sug-gests that something near 107 gray may be required for atomography experiment.There are various strategies available to reduce radia-

tion damage to wet specimens. One is the use of radicalscavengers.57 These materials combine with the highlyreactive free radicals formed by the radiolysis of waterand thereby reduce the rates of certain indirect forms ofradiation damage. However, this approach does not pre-vent direct damage effects (i.e., ones not involving radi-cals) and would not be expected to extend the damage re-sistance of natural biological material to 107 gray.A more effective approach is fixing and/or drying.

These treatments lead to a considerable increase in radia-tion tolerance as reviewed, for example, by Kirz et al.22

However, resistance still does not extend to 107 gray, andvarious artifacts may be introduced.It appears that the only way that a hydrated sample

can be made to tolerate the doses needed for x-ray holo-graphic tomography is cooling to liquid-nitrogen tempera-ture. It has been shown that 5-nm structural details in

biological samples at liquid-nitrogen temperature surviveat least until 108 gray.58 By cooling a sample at a suffi-ciently rapid rate (approximately 104 °C/s for one in abuffer solution), amorphous ice rather than crystalline iceis formed, which preserves the sample’s morphology.Moreover, there is already extensive experience in doingelectron microscopy on cryofixed samples, and commer-cially built systems are available for solving the practicaldifficulties of cooling the sample and keeping it cool whileintroducing it into the vacuum.We therefore believe that high-resolution x-ray holo-

graphic tomography can be implemented by means ofcryofixation. Therefore, taking into account that soft xrays have much higher penetration than the electronsnormally used for imaging and that they have high natu-ral image contrast and low background signals (i.e., neg-ligible bremsstrahlung and multiple scattering), we mayconclude that soft-x-ray cryotomography is feasible andpromises a unique capability. Moreover, it appears thatthis capability is well suited for studying whole-cellpreparations.Although we have been developing this argument for

x-ray holographic imaging, the same general reasoningalso shows that soft-x-ray cryotomography based onscanned-probe or full-field imaging schemes also promisessomewhat similar capability.

11. POSSIBILITIES FOR BETTERRESOLUTIONIn order to improve image resolution, we are making ef-forts on several fronts. One strategy is to record holo-grams at a reduced specimen-to-hologram distance z.This should permit the use of a smaller grid spacing, Ds ,without the need for a larger array size. In the longerterm we will work toward better thermal stability of theAFM scanning stage (Invar construction instead of alumi-num) and an ability to process larger arrays. We are alsostudying ways to improve the signal-to-noise ratio of therecording. This could be based on reducing the PMMAnoise by improved preparation and development tech-niques or on increasing the signal by an increase of theexposure. The prospects for improving the fundamentalinformation-limited resolution are restricted by its verysteep dependence on the dose. One can show59,60 that

Fig. 11. Visible-light, x-ray holographic, and transmission electron micrographs of the edge of the NIL8 cell at region B of Fig. 9. Thex-ray image resolves the edge structures from this region as well as internal voids that are difficult to visualize in the visible-light mi-crograph.

Fig. 12. Visible-light, x-ray holographic, and transmission electron micrographs of an organelle at region C in the middle of the NIL8cell pseudopod of Fig. 9. This area is well within the pseudopod in a region where the x-ray holographic technique should have troubleworking well. However, the x-ray hologram reconstructs well and clearly shows structure not easily visualized in the visible-light im-age.

Fig. 13. Visible-light, x-ray holographic, and transmission electron micrographs of the edge of the NIL8 cell at region D of Fig. 9.

1798 J. Opt. Soc. Am. A/Vol. 13, No. 9 /September 1996 Lindaas et al.

the dose varies inversely as the sixth power of the resolu-tion. Thus, once we reach a true limit to the resolutionattainable at our present exposure levels, the additionalimprovement available by increasing the exposure will besomething less than a factor of 2.

12. CONCLUSIONIn conclusion, we believe that we have now demonstratedthat soft-x-ray holographic imaging utilizing atomic-forcemicroscope readout is a promising new form of imaging inbiological research. Furthermore, we have argued thatthe way to obtain both single and multiple x-ray holo-graphic imaging of hydrated biological material withminimal radiation and other artifacts is by imaging atcryogenic temperatures. We believe that the way is nowopen for cryotomography of whole-cell preparations of to-tal thickness up to approximately 10 mm. We are con-structing a new experimental system, to be used initiallyat the X1A beam line at Brookhaven, with the goal of de-veloping such a technique so as to produce unique infor-mation in the regime of samples that are too thick forelectron microscopy but possess interesting structuresthat are too small for observation in visible light.

ACKNOWLEDGMENTSWe thank Ilan Spector for providing to us the NIL8 cellline, Vivian Oehler for maintaining and preparing the cellcultures, and Sue Wirick for assisting at the X1A beamline. We also acknowledge valuable conversations withour colleagues Robert Glaeser and Kenneth Downing atthe Donner Laboratory at Lawrence Berkeley NationalLaboratory. This research was supported in part by theAlexander Hollaender Distinguished Postdoctoral Fellow-ship Program (S. Lindaas) sponsored by the Office ofHealth and Environmental Research of the Departmentof Energy and administered by the Oak Ridge Institutefor Science and Education, by Department of Energygrant DE-FG02-89ER60858, by the National ScienceFoundation under grant BIR 91-12062, and by Presiden-tial Faculty Fellow award RCD 92-53618 (C. Jacobsen).Holography experiments were carried out at the NationalSynchrotron Light Source, which is supported by the De-partment of Energy under grant DE-AC02-76CH00016.

Note added in proof: Our cryo-holography apparatus,mentioned in Section 12, has been completed and used torecord successfully cryo-holograms of malarial infectedred blood cells. These holograms are of sufficient qualityto yield good reconstructions. This work will be reportedon in the near future.

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