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X-ray phase imaging: Demonstration of extended

conditions with homogeneous objects

L. D. Turner,1B. B. Dhal,1J. P. Hayes,2A. P. Mancuso,1K. A. Nugent,1

D. Paterson,3R. E. Scholten,1C. Q. Tran1and A. G. Peele1,4

1School of Physics, University of Melbourne, Victoria 3010, Australia

2Industrial Research Institute Swinburne, Swinburne University of Technology, Hawthorn 3122, Australia

3Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois 60439, USA

4Now at Department of Physics, Latrobe University, Victoria 3086, Australia

a.peele@latrobe.edu.au

Abstract:

consider the validity conditions for linear relations based on the transport-

of-intensity equation (TIE) and on contrast transfer functions (CTFs). From

a single diffracted image, we recover the thickness of a homogeneous object

which has substantial absorption and a phase-shift of −0.37radian.

© 2004 Optical Society of America

We discuss contrast formation in a propagating x-ray beam. We

OCIS codes: (340.7480) X-rays; (340.7440) X-ray imaging; (340.7460) X-ray microscopy

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#4387 - $15.00 USReceived 20 May 2004; revised 19 June 2004; accepted 20 June 2004

Page 2

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1.Introduction

Absorptioncontrasthasbeentheprincipalimagingmodeforx-raysforover100years,nonethe-

less there has been considerable development recently in implementing phase contrast tech-

niques. Phase contrast can be strong when absorption contrast is minimal; for instance for

low-Z materials or for high energy x-rays [1, 2, 3]. Phase contrast can also be used without de-

livering a high dose to the sample [2]. And certain phase methods require no additional optics

leading to source-limited, rather than optics-limited, resolution [4].

Demonstrated methods for obtaining the phase from an x-ray wavefield are now legion. In-

terferometric methods include the use of Bonse and Hart [5] type interferometers [6], shear-

ing interferometers [7], differential interference contrast arrangements using two zone plates

[8] or diffractive optical elements [9]. Zernike phase contrast [1] has also been demonstrated.

Other methods include refraction measurement using crystal diffraction [10], and segmented

detectors [11]. Propagation-based methods have also been developed and involve recovery of

the object phase and/or amplitude from one or more measurements of the object diffraction

pattern. Methods include in-line holography [12], iterative schemes [13] and approaches based

on solution of the equations governing the propagation of the wavefield [14, 15, 3].

In this paper we consider aspects of the latter method. Here the Fresnel integral describing

the diffracted intensity is linearized allowing a straightforward retrieval of object phase and

transmission. A first order Taylor expansion obtains the transport of intensity equation (TIE)

solution [16, 3]. Alternatively a Born-type approximation [17, 16] gives a solution identical in

form to that derived by Guigay [18, 19] in the context of electron microscopy. We will refer

to this as the contrast transfer function (CTF) solution. Both the TIE and the CTF solution can

be further simplified under the assumption of a homogeneous object [20, 16]. This permits the

thickness distribution of an object to be retrieved from a single diffracted image.

In the TIE it is the first-order Taylor expansion that restricts the validity of the solution. Other

than a requirement for paraxiality, there is no limitation on the magnitude of the phase or the

absorption. On the other hand, the Born-type approximation previously used in deriving the

CTF method can be quite restrictive. Guigay showed [19] that a less restrictive requirement

applies for a pure phase object. Like the TIE condition, this condition depends on the feature

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sizes present as well as the wavelength and propagation distance.

Here we show that the Fresnel-diffracted intensity can be linearized at a later stage of the

derivation. As a consequence we find that the less restrictive phase condition extends to weakly-

absorbing objects. In Section 2 we outline the derivations of the TIE and the CTF solutions and

their validity conditions and show how the assumption of a homogeneous object allows retrieval

of the object thickness from a single diffracted image. In Section 3 we present experimental

results demonstrating quantitative thickness retrieval under our derived validity conditions.

2.Derivations

2.1.

WebeginwiththeFouriertransform,F[·],oftheintensityI(r,z)obtainedunderFresneldiffrac-

tion of an object-plane wavefield f(r) = f(r,z = 0) with wavelength λ [18]:

?+∞

−∞

The transverse spatial coordinates and their corresponding Fourier conjugates are given by r

and u respectively. To obtain the TIE solution we Taylor expand the wavefield to first order,

f(r+λzu/2) = f(r)+1

TIE and CTF

F[I(r,z)] =

f∗(r+λzu/2)f(r−λzu/2)exp(−2πir·u)dr.

(1)

2λzu·∇f(r),

(2)

where ∇ is the gradient operator in the transverse plane. Substituting into Eq.(1) gives

∇·(I(r,z)∇φ(r,z)) = −2π

where φ is the phase of the wavefunction f. The validity condition for the TIE solution is

therefore that the higher order Taylor expansion terms can be disregarded:

λ

∂

∂zI(r,z),

(3)

?????

∞

∑

j=2

1

j!

?1

2λzu·∇

?j

f(r)

?????? 1.

(4)

This condition can always be satisfied by choosing a sufficiently small propagation distance,

while requiring no approximation regarding the magnitude of the amplitude or phase.

The previous approach to obtain the CTF solution was to write the object wavefunction in

terms of its absorption µ and phase φ components such that:

f(r) = f0exp(−µ(r)+iφ(r)),

where I0= |f0|2is the intensity of the plane wavefield incident on the object. Then the Born-

type approximation of µ ? 1 and |φ| ? 1 was made so that

f(r) = f0(1−µ(r)+iφ(r)).

Substituting Eq.(6) into Eq.(1) and retaining µ and φ to first order, obtained Guigay’s result

?δ(u)−2cos(πλzu2)F[µ(r)]+2sin(πλzu2)F[φ(r)]?

in which δ(u) denotes the Dirac delta distribution. When the object is pure phase (µ = 0),

Guigay [19] also showed, by substituting Eq.(5) into Eq.(1), that the corresponding form

of Eq.(7) can be obtained if, for all r,

(5)

(6)

F[I(r,z)] = I0

(7)

|φ(r+λzu/2)−φ(r−λzu/2)| ? 1.

(8)

This is sometimes referred to as the slowly-varying phase condition. It should be noted that

the displacement vector λzu, over which points in the phase should be similar, is a function of

propagation distance z and of spatial frequency u.

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2.2.

Consider the case of an optically thin and homogeneous object for which

µ(r) = kβT(r)

where k = 2π/λ, the refractive index is n = 1−δ +iβ and T is the thickness of the object. We

substitute into Eq.(1), re-factor and make a first-order Taylor expansion of T yielding

?+∞

−∞

If we assume that |λzu·∇φ(r)| ? 1 then we can expand the second exponential to first order,

F[I(r,z)] = I0

−∞

and then applying the Fourier derivative theorem F[∇f(r)] = 2πiuF[f(r)] we obtain:

TIE for a homogeneous object

and

φ(r) = −kδT(r),

(9)

F[I(r,z)] = I0

exp(−2kβT(r))exp(−ikδλzu·∇T(r))exp(−2πir·u)dr.

(10)

?+∞

exp(−2kβT(r))(1−ikδλzu·∇T(r))exp(−2πir·u)dr

(11)

F[I(r,z)] = I0F[exp(−2kβT(r))]

?

1+δ

βλzu2

?

.

(12)

This may be solved for the thickness T:

T(r) = −

1

2kβlnF−1

?

β

β +δπλzu2F

?I(r,z)

I0

??

.

(13)

The validity condition on the thickness T is similar to that for the TIE in Eq.(4).

2.3.

Here we begin by substituting Eq.(5) into Eq.(1):

?+∞

−∞

×exp(−2πir·u)dr.

Assuming both real and imaginary parts of the exponential are small, we expand, noting the

Fourier transforms:

F[I(r,z)]/I0= δ(u)−F[µ(r+λzu/2)+µ(r−λzu/2)]

+iF[φ(r−λzu/2)−φ(r+λzu/2)].

Applying the Fourier shift theorem F[f(r−a)] = exp(−2πia·u)F[f(r)] to each term and

rearranging recovers Eq.(7). If the object is homogeneous then, substituting Eq.(9), we can

retrieve the thickness T from a single diffracted image:

1

−2k(δ sin(πλzu2)+β cos(πλzu2))F

The linearizing assumption made in obtaining Eq.(15) is that

2µ(r) ? 1

These conditions on the validity of Eq.(7) are much less stringent than previously realised.

While the object must be weakly absorbing it need not be non-absorbing, which is less restric-

tive than the pure phase assumption made by Guigay. Furthermore, the weak phase condition

required in the Born-type approach is here relaxed to the slowly-varying condition Eq.(8).

Assuming object homogeneity allows us to incorporate the effect of object absorption. Con-

sequently the contrast transfer function, which is the denominator term of Eq.(16), is non-zero

at u = 0. Nulls in the CTF at frequencies above u = 1/√2λz may be avoided by incorporating

additional images at different propagation distances [21, 22], or simply by Tikhonov regularisa-

tion [23]. We find that Tikhonov regularisation leaves only minor artefacts at high frequencies

whereas assuming a pure phase object introduces intractable low-frequency instability.

Derivation of extended validity CTF solution

F[I(r,z)] = I0

exp(−µ(r+λzu/2)−µ(r−λzu/2)+i(φ(r−λzu/2)−φ(r+λzu/2)))

(14)

(15)

T(r) = F−1

??I(r,z)

I0

−1

??

.

(16)

and

|φ(r+λzu/2)−φ(r−λzu/2)| ? 1.

(17)

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0

-4

-2

2

105/CTF (m)

0

1

234

4.75

6cm

Effective propagation distance z

0

100µm

-4.9

CTF

TIE

-5

Fig. 1. Inverse of the contrast transfer functions for the TIE (blue dotted line) and CTF (red

line) forms calculated for an infi nite grid with spatial feature size of 1.9µm and the exper-

imental conditions. The green dashed line is at the experimental distance. The embedded

movie shows the inverse CTFs as shown here with a slider indicating the z position corre-

sponding to diffraction distance. Also shown in the movie is a plot of an input amplitude

(green) and the amplitude retrieved using either the CTF (red) or TIE (blue) methods for

the indicated propagation distance. The CTF method correctly accounts for the contrast re-

versals that arise on propagation. The TIE method should only be applied for z closer than

the fi rst contrast reversal; it may retrieve inverted amplitudes if applied at greater z.

3.Experimental results

Laser ablation [24] was used to etch a grid of lines in a polyimide film (composition

C22H10N2O4and density 1.45gcm−3). The lines were measured by atomic force microscope

(AFM) to be approximately 1.9µm apart and 90nm high and were superimposed on an 80µm

square of 650nm in height.

The experiments were performed at a wavelength of 0.436nm and with z = 0.0475m, where

z is as defined above. Through the thickest part of the object the transmission exp(−2µ) is

98.7% and the phase-shift φ is −0.37radian. Accordingly, neither the Born-type approximation

nor the phase-only requirement are met. However, for most positions and spatial frequencies

the slowly-varying phase condition Eq.(8) is obeyed.

Figure 1 shows a plot of the inverse of the contrast transfer function from the CTF solu-

tion Eq.(16) for the 1.9µm grid features. A contrast transfer function can also be defined, in

the weak absorption limit, for the TIE solution Eq.(13) and its inverse is also shown in Fig. 1.

Inspection shows that the TIE solution should yield grid features reversed in contrast compared

to the CTF solution. No contrast reversal is expected for the 80µm square.

The experiments were performed at beamline 2-ID-B at the Advanced Photon Source. A

beam of 2844eV (∆E/E ? 10−3) x-rays with FWHM size 1.5×0.5mm illuminated a 160µm

diameter gold zone plate with an outer zone width of 50nm. The focal length was 18.4mm

and a 10µm diameter order sorting aperture was placed at a distance of 17.2mm from the zone

plate. A 30µm wide gold beamstop blocked the zero order beam. This zone plate configuration

provided a point source of illumination at a distance of R1= 54.7mm from the object. After

passing through the object the beam was allowed to propagate through a He-filled flight tube a

distance of R2= 363mm onto a crystal scintillator which was imaged through a 20× objective

byaCCDcamerawith13.5µmpixels.Thisexpandingbeammodeisanalyzedusingtheparallel

beam mode derivations described above using the conversion [4]

1

M2I∞

where M = (R1+R2)/R1is the magnification, IR1(r,R2) is the expanding beam diffraction-

IR1(r,R2) =

?r

M,R2

M

?

,

(18)

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0.0

1.0

2.5

02040 6080100µm

4550 55 6065µm

560

nm

600

680

Height

µm

AFM

CTF

TIE

a

a

b

Fig. 2. (a) CTF-retrieved thickness map for the square with grid lines.(b) Column-average

of retrieved thickness for the grid pattern in the region shown in (a) for the TIE solution

(blue) and the CTF solution (red). The AFM result (green) shows excellent agreement.

AFM measurements also confi rm the presence of grid lines outside the square. These are

not a retrieval artefact, unlike the circular fringes around the contaminant at centre right.

Thecontaminating material presumablyviolatestheassumptionofanhomogeneous object.

pattern due to a point source a distance R1behind the object and I∞is parallel beam diffraction-

pattern. The object thickness map was retrieved from the measured diffraction pattern using

both the CTF solution Eq.(16) with Tikhonov regularization, and the TIE solution Eq.(13),

and is shown in Fig. 2(a). We line average the essentially one-dimensional image in the region

outlined in white. Both CTF and TIE methods retrieve the mean height of the 80µm square as

610±50nm, concurring with the AFM result of 650nm. However, the TIE retrieval grossly

underestimates the height of the grid pattern and, as predicted in Fig 1, reverses the contrast

(Fig. 2(b)). As expected from the validity conditions for Eq.(16), the CTF retrieval of the grid

pattern (Fig. 2(b)) is in excellent agreement with the AFM measurement of 90nm.

4. Conclusions

We have explored the validity conditions for the linear CTF expression relating object phase-

shift and absorption to the contrast of the Fresnel diffraction-pattern. The linear expression is

found not to be restricted to weakly phase-shifting objects: it applies to a substantially wider

class of objects which show weak absorption and slowly-varying phase.

Ifanobjectismadeofonematerialwithknowncomplexrefractiveindex,theCTFexpression

maybeinvertedtoretrievetheobjectthicknessfromasinglediffractedimage.Wedemonstrated

an example where the CTF solution could correctly retrieve thickness features of a weakly-

absorbing object with large, but slowly-varying, phase-shift. Thickness features at two well-

separated spatial frequencies were retrieved by the CTF solution while the TIE retrieval was

valid only at the lower spatial frequency. These results augur well for wider applications of the

CTF technique such as imaging cold atom clouds [25, 26] and plasmas [27].

Acknowledgments

The authors acknowledge Australian Research Council Fellowships (LDT, APM: Australian

Postgraduate Awards, AGP: QEII Fellowship, KAN: Federation Fellowship). This work was

supported by the Australian Synchrotron Research Program, which is funded by the Common-

wealthofAustraliaunder theMajor National Research Facilities Program. UseoftheAdvanced

Photon Source was supported by the U.S. D.O.E., Basic Energy Sciences, Office of Science

under Contract No. W-31-109-Eng-38. We thank A. Cimmino for providing the AFM results,

D. Paganin for useful discussions and an anonymous referee for insightful comments.

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#4387 - $15.00 USReceived 20 May 2004; revised 19 June 2004; accepted 20 June 2004