Page 1
Some simple rules for contrast, signaltonoise
and resolution in inline xray phasecontrast
imaging
Timur E. Gureyev *, Yakov I. Nesterets, Andrew W. Stevenson, Peter R. Miller,
Andrew Pogany and Stephen W. Wilkins
CSIRO Materials Science and Engineering, Private Bag 33, Clayton South 3169, Australia
* Corresponding author: Tim.Gureyev@csiro.au
Abstract: Simple analytical expressions are derived for the spatial
resolution, contrast and signaltonoise in Xray projection images of a
generic phase edge. The obtained expressions take into account the
maximum phase shift generated by the sample and the sharpness of the
edge, as well as such parameters of the imaging setup as the wavelength
spectrum and the size of the incoherent source, the sourcetoobject and
objecttodetector distances and the detector resolution. Different
asymptotic behavior of the expressions in the cases of large and small
Fresnel numbers is demonstrated. The analytical expressions are compared
with the results of numerical simulations using Kirchhoff diffraction theory,
as well as with experimental Xray measurements.
©2008 Optical Society of America
OCIS codes: (340.7440) Xray imaging; (100.2960) Image analysis; (110.2990) Image
formation theory; (110.4980) Partial coherence in imaging.
References and links
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Page 2
16. A. Krol, R. Kincaid, M. Servol, J.C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E.
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1. Introduction
Xray phasecontrast imaging is a rapidly developing technique that shows great promise in
medical, scientific and industrial applications [1]. Among different types of Xray phase
contrast imaging, the socalled inline imaging method is the simplest in principle and the
easiest in practical implementation [24]. It typically involves an Xray source that can
provide high degree of spatial coherence in the incident illumination, while high chromatic
coherence is often not required [3]. Suitable sources that have been used in this imaging
modality include synchrotron insertion devices [2, 4], microfocus laboratory sources [3], and
more recently plasma Xray sources generated either by highpower optical lasers [5] or X
pinch [6]. It has been shown that inline phasecontrast imaging is capable of providing
greatly enhanced image contrast compared to conventional absorptionbased imaging,
especially for hard Xrays (~10100 keV) and samples consisting predominantly of lowZ
chemical elements. As a consequence, this technique is considered particularly promising for
medical diagnostic applications [3, 7].
It is wellknown [8, 3, 9] that inline Xray phase contrast in the case of weakly absorbing
samples and moderate propagation distances is proportional to the second derivatives
(Laplacian) of the projected electron density distribution in the sample. As a consequence,
this imaging method is particularly sensitive to edges and interfaces in the sample providing a
natural edgeenhancement effect in the images. From the point of view of most applications
this edge enhancement is considered the main advantage of the method. Therefore, it is
particularly important to find quantitative dependencies of the degree of edge enhancement,
which is conventionally characterised by suitably defined maximum image contrast, spatial
resolution and signaltonoise ratio in the vicinity of the geometric image of the edge, as a
function of the relevant properties of the source, the sample, the detector and the imaging
layout. Some results in this direction have been obtained previously (see e.g. [10] and
references therein). However, the need still exists in simple analytical expressions describing
the main characteristics of the images of edgelike features, which on one hand are general
enough to cover most experimental conditions of interest, and on the other hand are simple
enough to allow one to easily estimate the expected degree of edge enhancement without
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resorting to computer simulations. The present paper aims at providing such simple formulae
which can be used for rapid evaluation of inline Xray imaging conditions.
The structure of the paper is as follows. In the next section we derive analytical
expressions for contrast, spatial resolution and signaltonoise ratio (SNR) in inline images of
a generic phase edge. In the case of large Fresnel numbers (short propagation distances), the
formulae are derived for both monochromatic and polychromatic incident radiation, while in
the case of small Fresnel numbers (long propagation distances) only monochromatic incident
radiation is treated. In section 3 we analyze the expressions describing the contrast, spatial
resolution and SNR and present simple "rules" for estimation of these quantities in inline
images. Section 4 contains the results of numerical simulations verifying the accuracy of the
proposed "rules", while in section 5 we describe the experimental tests performed with an in
line imaging system based on a laboratory microfocus Xray source and Imaging Plates.
Finally, brief conclusions are given in section 6.
2. Analytical formulae for inline phase contrast
In this section we derive analytical expressions for contrast, spatial resolution and SNR in in
line images of an edgelike feature in a nonabsorbing object. We consider three different
cases: (1) monochromatic incident radiation and short propagation distances; (2)
polychromatic incident radiation and short propagation distances; and (3) monochromatic
incident radiation and long propagation distances. We show that our analytical formulae
obtained in these three different cases are consistent which each other, as well as with the
expected qualitative physical behavior of Xray image contrast in the relevant regimes.
Let a sample be located immediately before the "object" plane z = 0 transverse to the
optic axis z (Fig. 1). The sample is illuminated by an Xray beam emanating from an
extended spatially incoherent source located near the point
ray transmission through the sample can be characterised by the complex transmission
function
),,(
ν
yxQ
,
] exp[
μ ϕ −≡
iQ
, where (x, y) are the Cartesian coordinates in the object
plane and ν is the radiation frequency (
/
λν
c
=
transmitted beam is registered by a positionsensitive detector located immediately after the
"detector" plane
2
Rz =
.
1 R
−
z
=
. We assume that the X
)2 /(
π
kc
=
, where k is the wavenumber). The
Fig. 2. Phase shift distribution in the object
plane after transmission of the incident plane
Xray wave through a blurred phase edge.
1 0.50
x
0.51
1
0.75
0.5
0.25
0
ϕ/ϕmax
σ
Fig. 1. Inline imaging geometry.
Let us consider a simple but instructive example. Consider a generic edgelike feature in
a nonabsorbing sample (μ = 0), in a vicinity of which the distribution of transmitted phase
can be modeled as (Fig. 2)
)(
ν
  )
x
)(*(),
ν
,(
maxobj
ϕϕ
PHyx
−=
, (1)
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where the edge is assumed to be extended along the y coordinate, ϕ max(ν) is a constant
representing the local maximum of the absolute value of the phase shift,
Heaviside "step" function (which is equal to 0 for negative x and is equal to 1 for positive x),
)]2 /()( exp[) 2 (),(
obj objobj
σ πσ
yxyxP
+−=
is a function describing the "sharpness" of the
edge and the asterisk denotes twodimensional convolution.
Let
(exp{ )]( 2 [),,(
syssys
xMMyxP
πσ−=
function (PSF) of the imaging system referred to the object plane (we assume for simplicity
that the PSF is the same at all Xray energies),
variance of the PSF, σsrc and σdet are the standard deviations of the source intensity
distribution and the detector PSF, respectively, and
magnification. The particular form of the above expression for σsys is a direct consequence of
the projection imaging geometry (Fig. 1).
)(xH
is the
22212
−
)]}(2 /[)
2
sys
2212
My
σ+
−
be the pointspread
2
det
22
src
222
sys
) 1
−
()(
σσσ
−−
+=
MMMM
is the
121
/ )(
RRRM
+=
is the geometric
2.1. Monochromatic nearfield inline contrast for a phase edge
It is well known that at sufficiently short propagation distances the spatial distribution of the
spectral density,
),,,(
ν
zyxS
, in inline images can be described by the Transport of Intensity
equation (TIE) [8, 11, 12]. Let us substitute Eq. (1) into the monochromatic TIE for pure
phase objects [3, 12],
)],(*),
ν
,()/(1 )[(
ν
),
ν
,,(
sys
2
in
2
2
yxPyxkRSMR MyMxS
ϕ∇
′
−=
−
, (2)
where
distribution in the image plane
("defocus") distance. As the derivative of the Heaviside function is the Dirac deltafunction,
we can write:
(*)( *
objmaxsys
δϕϕ
x
PxP
∂−=∇
where
)(
sys obj
M
M
σσσ+≡
. Substituting this expression into Eq. (2), we obtain
)(
inν
S
is the spectral density of the incident beam,
),,,
is the effective propagation
(
2ν
RyxS
M
is the spectral density
2
Rz =
and
RR
/
2
=
′
)2 /()]2 /(exp[)*
3
M
max
2
M
2
sys
2
πσϕσ
xxP
−=
,
222
)]}2 /(exp[  )2 /(1){(
ν
),
ν
,,(
2
M
2
max
3
M
in
2
2
xxkRSMR My MxS
σϕπσ−
′
−=
−
. (3)
15 105051015
0.8
0.9
1.0
1.1
1.2
+σM
σM
S/Sin
x, μm
NF=5
NF=1
NF=0.2
Fig. 3. Plot of the spectral density distribution in the vicinity of the geometric
image of the edge feature for different values of the Fresnel number. The
hatched area indicates the location of the object.
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From Eq. (3) we can easily find the locations,
the spectral density in the vicinity of an edge. These positions correspond to the centre of the
"positive" and "negative" Fresnel fringes near the geometric shadow of the edge (Fig. 3) (note
that the TIE allows for only one positive and one negative Fresnel fringe).
In the monochromatic case we define the image signal, Σ(ν, A), as the absolute value of
the difference between the image spectral density distribution and the corresponding spectral
density, S0(ν) ≡ M 2 Sin(ν), in the background image (without the edge feature), integrated
over the area A = 2Ma×MLy in the image plane, where Ly is the "length" of the edge feature in
the object plane in the direction parallel to the edge and
the x coordinate. Note that Ly is assumed to be sufficiently large (in particular, it is much
larger than σsys and σobj). Using Eq. (3) we obtain:
M
x
σ
±=
, of local maximum and minimum of
),(
aa
−
is a vicinity of the edge along
()
. )]
2
exp(1 [)(
ν

2
)(
ν
2d
2
exp
2
)(
ν
)(
ν
2d)(
ν
,
ν
,,),(
ν
2
M
2
max in
2
M
2
2
0
3
M
max002
M
y
Ma
∫
y
Ma
∫
−
Ma
y
a
σ
k
R
LSx
M
x
M
x
k
R
MLSxSRyxSMLA
ϕ
πσ
σ
σπ
ϕ
−−
′
=
′
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
′
−×
′′
=
′
−
′′
≡Σ
The noise is then calculated assuming the Poisson statistics as
where
y
aLSAD
)(4),(
ν
=
is the sum of the total Xray spectral densities incident on the
region of interest in the images with and without the edge feature. The SNR is then equal to,
R
LSASNR
νϕ
πσ
),(
A SNR ν
as a function of a reaches its maximum at
approximately equal to the width of the first Fresnel fringes in the TIE regime, it is natural to
choose
M
a
σ
2
=
for the calculation of the SNR corresponding to the phase contrast generated
by the edge. Such a choice leads to almost maximal SNR, while also being convenient for
practical evaluation of SNR in experimental phasecontrast images. We therefore define
) 4 ,()(
yM
MLM SNRSNR
×≡
σνν
and obtain:
),(
ν
),(
ν
ADAN
=
,
inν
)]
2
162
exp(1 [)(
2
])(
ν
[),(
ν
2
M
σ
2
2 / 1
−
max
2/ 1
in
M
y
a
σ
a
k
−−
′
=
. It can be verified that
M
a
. 2
≅
. As the quantity
M
σ
2
is
TIE
)(
ν
 )](
ν
[)(
ν
max
12 / 1 TIE
1
TIE
ϕ
−
F
=
NDCSNR
, (4)
where
yMLSD
σ
)
ν
(
ν
(8)
in
TIE
=
is the corresponding total incident Xray spectral density,
RkN
MF
′
=
/
2
σ
is the minimal Fresnel number (which corresponds to the size, 2
M
σ
, of the
smallest resolvable detail in the image [13]) and
absolute constant.
We define the
)([),(
min maxmax
νν
SSAC
−=
maximum and minimum of the image spectral density in region A. Given the locations,
x
σ±=
, of local maximum and minimum of spectral density in the vicinity of an edge and
using Eq. (3) we obtain the following expression for the maximum contrast
)4 ,()(
maxmax
yM
MLMCC
×≡
σνν
inside the area A = 4MσM × MLy :
0862. 0])2 ( 4/[)e1 (
2/ 12
1
≅−=
−
π
C
is an
maximum
/[)](
ν
contrast
S
+
in
, where
a region
S
of
and )
ν
an image
are the
as
)](
ν
)(
ν
min max
S
)(
ν
(
minmax
S
M
TIE
2/ 1
−
TIE TIE
3max

1
2
TIE
max
)](
ν
[ )(
ν
)(
ν
)(
ν
−
F
==ϕ
DSNRCNCC
, (5)
where
Equation (5) describes the dependence of the contrast on such parameters as the wavelength
2420 . 0e) 2 (
1/2
2
≅=π
C
and
806 . 2)]e1 (e/[4
2 1/2
3
≅−=
C
are absolute constants.
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of the incident radiation, effective defocus distance and the spatial resolution of the imaging
system. This equation indicates a very simple behavior of the image contrast in the TIE
regime, i.e. the contrast is directly proportional to the maximum phase shift and to the inverse
of the Fresnel number.
Equation (5) gives a convenient indicator of the validity of the TIE approximation that
was used for its derivation. It is known [3, 12] that Eq. (2) is valid only if the corresponding
image contrast is weak, i.e.
1)(
max
<<ν
C
, or, according to Eq. (5),
necessary TIE validity condition is complementary to another commonly used necessary
condition,
1
>>
F
N
, which does not depend on the magnitude of the phase shift. It was
demonstrated in recent numerical simulations [13] that the TIE approximation does break
down if one increases the maximum phase shift while keeping all other imaging parameters
constant, therefore the condition
1
>>
F
N
alone cannot be sufficient for the validity of the
TIE. It is also easy to show that when
(
max
νϕ
sufficient for the validity of the TIE (see section 2.4 below). On the other hand, it is known
that the TIE approximation is valid if and only if the transmission function satisfies the
following condition:
 /  ),(
σλν
r
RQ
M
∇ <<
′
∇
order Taylor approximation for Q [13]. For the phaseedge object the latter condition is
equivalent to
max
})(  max{1,
σνϕλ
R
<<
′
})(  max{1,
ϕ >>
′F
N
, where
2
σ πσ
N
MF
=
′
validity of the TIE, Eq. (2). As,
obj
[
σσ
≡
M
condition
})( max{1,
ϕ >>
F
N
is necessary, but not always sufficient for the validity of
the TIE. In particular, an inline image of a sharp edge (with a small σobj) may not be possible
to adequately describe using the TIE, even if the spatial resolution of the imaging system is
very low (σsys(M) is large). Numerical simulations suggest that in such cases the convolution
with the broad PSF of the imaging system may wash out highorder Fresnel fringes in the
image, but the remaining first Fresnel fringe may become asymmetric, which obviously
cannot be described by the TIE [10].
The spatial resolution of the imaging system shown in Fig. 1 can be conveniently
estimated from the lateral spread of an image of a straight edge (for which σobj<< σsys). For
large Fresnel numbers
F
N′ the inline image is described by Eq. (2), where the lateral
spreading is obviously determined by the convolution with the PSF of the imaging system. If
we exclude the influence of the object properties, then
following expression for the finest achievable spatial resolution
TIE
)( 
maxνϕ >>
F
N
. This
1)
<<
, the condition
)( 
maxνϕ >>
F
N
is not
 ),
ν
(
2
r
Q
, which allows one to use the first
obj
/(
σ
M
.
λ
′
Therefore,
, is necessary and sufficient for the
the condition
maxν
)
obj
σ
+
R
obj
2 / 12
sys
2
)](
σ
≥
M
, then
F
′
F
NN
≥
, and the
maxν
)(
sysM
σ
M
σ
=
and we obtain the
)(2)(
sys
TIE
min
Mx
σ
=Δ
. (6)
One can see that the TIE validity condition, i.e.
limit on the spatial resolution consistent with the use of the TIE, Eq. (2). Indeed, it implies in
particular that if σobj<< σsys , then
λσ
R′
>>
sys
width of the first Fresnel zone, which is also equal to the width of the first Fresnel fringe in an
image of a sharp edge in an ideal imaging system with the deltafunctionlike PSF. In a
system with the finite PSF width, σsys>0, the width of Fresnel fringes depends on σsys as well.
Note also that the TIE approximation allows for existence of only a single Fresnel fringe near
the geometric image of an edge (this can be easily seen from the mathematical structure of
Eq. (2)), so the spatial resolution of the TIEbased imaging is naturally associated with the
width of the first Fresnel fringe. As will be shown explicitly in section 2.4 below, the width
of the first Fresnel fringe reduces to the value given by Eq. (6) under the condition
λσ
R′
>>
sys
, i.e. when NF >>1.
})(  max{1,
>>
maxνϕ
′F
N
, imposes a
. The expression
λ
R′ is recognized as the
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Equation (6) also indicates that an improvement in the spatial resolution of inline phase
contrast imaging in the TIE regime can only be achieved by deconvolving the system PSF
from the registered images, which would allow one to eliminate (at least partially) the effect
of the image blurring due to the finite source size and detector resolution.
2.2. Polychromatic nearfield inline contrast for a phase edge
In the case of polychromatic incident radiation and large Fresnel numbers, equations similar
to Eqs. (4)(6) can be obtained by integrating the spectral densities over the frequencies
according to
∫
,,(),(
yxSyxI
, where I is the corresponding timeaveraged intensity.
The polychromatic TIE for pure phase objects is [14]
′′
=
′′
νν d )
] ),(* d ),
ν
,()(
ν
1 [
in
),,(
sys
1
ϕ
in
2 1
in
2
2
yxPyxkSIRIMR MyMxI
ν
−−
∫
∇
′
−=
. (7)
Substituting the expression
integrating over the area 4σM × Ly we obtain the following expression for the signaltonoise
ratio in the polychromatic case:
)2 /( )]2 /(exp[*
3
M
max
2
M
2
sys
2
πσϕσϕ
xxP
−=∇
into Eq. (7) and
max
22 / 1
)
TIE
1
TIE
(
ψσ
RDC SNR
M
′
=
−
, (8)
where
yML
d
ν
ID
σ
in
)
νϕν
TIE
8
=
is the corresponding total incident Xray intensity,
()(
1
in
1
in
ψ
−
∫
≡
kSI
is the "generalized eikonal" [14] of the transmitted polychromatic
∫
kSI
0862 . 0]) 2 ( 4/[)e 1 (
≅−=π
is an absolute constant.
As under the assumed approximations the positions,
Fresnel fringes are independent from the Xray frequency, we can find an expression for the
maximum image contrast from Eq. (7):
wave in the object plane,
d)(
ν
 )(
ν

1
maxin
1
inmax
νϕψ
−
≡
is the spectrally averaged
maximum eikonal and
2 / 12
1
−
C
M
x
σ±=
, of the centre of the first
TIE2 / 1
−
TIE
3 max
2
2
TIE
max
)(
SNRDCRCC
M
−
=
′
=ψσ
, (9)
where
2420. 0 e) 2 (
1/2
2
≅=π
C
and
806
>>
. 2
′ R
)]e
σ
M
1 (e /[4
21/2
3
≅−=
C
are absolute constants. The
ψπλ
corresponding TIE validity condition is
∫
S cI
.
Equation (6) for the limit on the spatial resolution in TIEbased imaging remains valid in
the polychromatic case.
} ), /(2max{ /
max obj
σ
, where
d)(
ν
1
in
1
in
ννλ
−
≡
2.3. Optimization of SNR in nearfield inline imaging
If one is interested in optimizing the conditions for Xray inline imaging in the sense of
maximizing the SNR for an image of a phase edge at a fixed level of incident intensity, then,
according to Eq. (8), one has to maximize the quantity
∫
−
′
=
ν
d
ν
(
ϕν
(
σ
γ
)  )
1
maxin
2/3
M
2/ 1
L
TIE
kS
R
SNR
y
, (10)
where
SNR optimization problem assumes that the incident intensity in the region of interest of the
object plane is kept constant (this is generally consistent with the assumption of a fixed dose).
This implies, for example, that if the sourcetoobject distance is varied from
2 / 1
−
in
2
)4 )(e1 (
−
−≡
I
πγ
is a constant. We should emphasize that this formulation of the
) 0(
1 R
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to
provided that the source intensity stays the same, etc. For a given feature with fixed
parameters σobj and Ly, the maximization can be achieved by improving the spatial resolution
of the imaging system (decreasing σsys(M)), increasing the defocus distance R' and adjusting
the incident spectrum in favor of the energies with larger values of
incident spectrum is fixed as well and only the geometric parameters of the imaging system
can be varied, then the quantity to be maximized is
1
) 0 (
1
) 1 (
1
RRR
Δ+=
, then the exposure needs to be increased by the factor
2) 0 (
1
) 1 (
1
)/(
RR
k / )(
maxνϕ
. If the
4/32
det
2
src
22
obj
22 / 12/3
M
TIE
] ) 1
−
([
) 1
−
(
σσσ
γ
σ
γ
++
′
=
′
′
=
MMM
MRR
SNR
, (11)
where
=
′
γ
monotonically increases as the sourcetodetector distance R increases, or as the source size
and detector PSF decrease. The only nontrivial dependence is that on magnification. Note
that
0)(
=
M SNR
at both limits, M = 1 and M = ∞, according to Eq. (11). Therefore, the
function
)(
SNR
has a maximum at some intermediate value of magnification, M = Mopt,
which can be found using Eq. (11).
Note that the optimal value of magnification will be independent of the total sourceto
detector distance, and will be determined only by parameters σobj, σsrc and σdet, i.e. by the
sharpness of the edge, the Xray source size and the spatial resolution of the detector system.
It can be shown that when
detsrc
σσ=
, Mopt is always equal to 2. However, when
Mopt can take different values. Consider, for example, the parameters used in our numerical
and experimental tests later in this paper (these parameters correspond to an inline system
with a laboratory microfocus Xray source and Imaging Plates as a detector), where
m
μσ
7 . 1
and
m
μσ
5 . 42
. Then, taking e.g.
that Mopt = 18. On the other hand, under conditions more typical for synchrotron experiments,
where one may have
m
μσ
5 .42
,
σ
7 . 1
one obtains Mopt = 1.059. These values of the optimal magnification are fairly consistent with
typical experimental experience.
The above results can be easily generalized further for the case of partiallycoherent
Schelltype incident illumination using the approach developed in [15].
It is also reasonably straightforward to account for Xray absorption in the feature,
provided the absorption is weak, so that the approximation
be used at all frequencies ν present in the incident Xray spectrum, where
maximum Xray attenuation in the feature [16].
) 1
−
ν
d
/(
221
ν
(
=+=
MMRR
ϕ
RR
is the total sourcetodetector distance and
∫
−
ν
(
γ
)  )
1
maxin
2 / 1
Ly
kS
is a constant. Obviously, in the TIE regime this SNR
TIE
TIEM
det src
σσ≠
,
src=
det=
m
μσ
7 . 0
obj=
, one obtains from eq.(11)
src=
m
μ
det=
, and assuming the same
m
μσ
7 . 0
obj=
,
)(
ν
)
is the
2
maxνμ
1 )](
ν
2exp[
max
(
max
μμ−
2
≅−
can
2.4. Fresnelregion inline contrast for a phase edge
In this section we derive analogues of Eqs. (4)(6) in a more general form, which are not
limited to the validity region of the TIE. Let the phase ϕ satisfy the Guigay condition [17]:
1  ),
ν
(),
ν
(
<<−
−+
ϕϕ
rr
, (12)
for all
B with the centre at the origin of coordinates in the Fourier space and radius ρ,
)2 , min(
obj sys
ρρρ =
, where ρsys and ρobj are the respective radii of the smallest circles outside
which the Fourier transform of the system's PSF and the transmission function are negligibly
small in magnitude. In the case of a plane monochromatic incident wave and the object plane
),(
yx
=
r
in the object plane and all
ρ
rr
) 2/( λ
R′
±=
±
with
),( ηξ=
ρ
from the circle
ρ
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phase satisfying Eq. (12), the 2D spatial Fourier transform of the spectral density in the
detector plane,
xiRS
(2 exp[),,, (ˆ
2
ξπνηξ
∫∫
the following form [15, 17]:
yxRyxSy
d d ),
ν
,,( )]
2
η+−=
, can be expressed in
)},,(
ˆ
),
ν
, ( ˆ
ϕ
)](sin[),( ){(
ν
),
ν
,/,/ (ˆ
S
sys
22
in2
MPRSRMM
ηξηξη
+
ξ πληξδηξ
′
+=
, (13)
where we also assumed for simplicity that
generalization of Eq. (2), with Eq. (2) formally obtainable from Eq. (13) by means of
replacing the sine function by its argument followed by the inverse Fourier transform. The
main advantage of Eq. (13) over Eq. (2) is that Eq. (13) is not limited to the "nearfield", i.e.
the condition
})(  max{1,
ϕ >>
′F
N
is not necessary for the validity of Eq. (13). Instead,
Eq. (13) can be derived under condition (12) which effectively allows the phase function to
consist of two components, one of them being large in magnitude, but slowly varying, with
the second one being small in magnitude, but possibly rapidly varying [13]. Let us consider
the small rapidly varying component first. The function
<
′
≤
′
<<
F
NRλσε
1  ),(
0
<< ≤ εϕ
yx
. Therefore, the inline phase contrast, including the Talbot effect, for the
latter function can be described by Eq. (13) [19]. On the other hand, it is obvious that such
effects cannot be generally described by Eq. (2). Thus, for the small phase functions, such
that
1)( 
max
<<νϕ
, the condition

ϕ >>
′F
N
TIE. Now consider the large slowly varying component of a phase function satisfying
condition (12). For such functions condition (12) implies that
by definition the spatial Fourier spectrum of the slowly varying component is confined to a
small circle,
obj
ρ
B
, around the origin of coordinates in the Fourier space, such that
λρ
R′
<<
, hence
1
>>
′F
N
. In this case the sine function in Eq. (13) can be replaced by
its argument, and Eq. (13) reduces to the TIE, Eq. (2). These examples agree well with the
statement presented earlier in this paper that the necessary and sufficient condition for the
validity of the TIE can be expressed as
max{1,
>>
′F
N
Having confirmed that Eq. (13) represents a nontrivial extension of Eq. (2), we proceed
with deriving explicit expressions for the spatial resolution, contrast and SNR in inline
images of the edgelike feature defined by Eq. (1) under conditions (12). Substituting Eq. (1)
into Eq. (13) and taking inverse Fourier transform we obtain that
1) , 0 , 0 (
sys
P
ˆ
=
M
. This equation represents a
maxν
)/ sin(),(
0
σεϕ
xyx
=
, with such
parameters ε and σ that
1) /(
2
, obviously satisfies condition (12), because
)(
maxν
is not sufficient for the validity of the
)( 
maxνϕ >>
′F
N
. Moreover,
) /(1
2
obj
})( 
maxνϕ
.
]} ),/([ / 2 )(
ν
1){(
ν
),
ν
,,(
max in
2
2
FFMF
NnxFnSMRMy MxS
σπϕ
−=
−
, (14)
∫
0
−−=
−
F
x
FF
tNtNtNxF
212
d)]2/(arctan ) 2/ 1sin[() 2/ exp(),(
, (15)
where
understood in the context of Eq. (14), if one considers the energy conservation requirement
together with the obvious property
),(
F
NxF
=
density in the vicinity of a geometric image of the edge calculated in accordance with
Eqs. (14)(15) are shown in Fig. 3 for the following values of the relevant parameters: M = 1,
1
ϕ
,
1
=
M
σ
μm, NF = 0.2, 1 and 5.
The locations, xm, of local maxima and minima of the spectral density are defined by the
equation
0] ),/( )[(
=∂
FFMmx
NnxF
σ
, i.e.
2
1
−
F
+=
F
Nn
and we used the fact that
0),(
=∞
F
NF
(the latter fact can be easily
),(
F
NxF
−−
). Typical profiles of the spectral
max=
)arctan 2 (
F
1
−
F
±
m
+±=
FM
NmNnx
πσ
,
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,...2
±
, 1
±
σ
, 0
=
m
. The locations of the first extrema to the left and right of the edge are
11
0
arctan)(
−
F
−
F
±
+±=
FM
NNNx
, hence the width of first Fresnel fringe is approximately
11arctan)(2
−
F
−
F
+=Δ
function, Pobj, we obtain an expression for the limit of spatial resolution,
FM
NNNx
σ
. Excluding the contribution of the sample unsharpness
11
sysmin
arctan)()(2)(
−
F
−
F
+=Δ
F
NNNMx
σ
. (16)
Note that the above definition of spatial resolution assumes that the higherorder Fresnel
fringes are sufficiently weak to be neglected, and thus the spatial resolution is determined
primarily by the width of the first Fresnel fringe. When
we obtain from Eq. (16) that
2)(
sysmin
x
σ
≅Δ
resolution for a sharp edge as defined by Eq. (6). At the opposite limit, when
π
≅
F
N
and we obtain another wellknown result [20, 18],
1
>>
F
N
, then
11
arctan
−
F
−≅
F
NN
and
)(
M
, which gives the limit of the spatial
1
<<
F
N
, then
2/ arctan
1
−
λ
′
π
(
σ
2
RNMx
F
==Δ
−1
sys
far
min
) 2/)()(
. (17)
Equation (17) also implies that at small Fresnel numbers a significant improvement in spatial
resolution can potentially be achieved using phase retrieval, i.e. by numerically reconstructing
the distribution of transmitted phase in the object plane from the registered intensity
distribution in the image plane [13, 15].
As before, we define the local contrast in the vicinity of the edge as the ratio of the
difference and the sum of spectral densities at the first maximum and minimum. Note that
this definition of contrast assumes that the higherorder Fresnel fringes are weaker than the
first fringe, and thus the local contrast is determined by the first Fresnel fringe. Taking into
account Eq. (14) and the expression for
for the contrast:
±
0 x given above, we obtain the following expression
)(
ν
  )
F
, arctan() (1 /2)(
ν
max
14 / 12
max
ϕπ
FFF
NNNFNC
−−
+=
. (18)
For large values of the
≅
Fresnel
−
number we have
1 arctan
1→
−
FF
NN
,
)2 /() 1 ( )]2 /(arctan ) 2/ 1 sin[(
221
FFF
NtNtN
−
−
, and taking into account the value of the
definite integral
2 / 1
−
1
0
22
e d )1 )(2/ exp(
=−−
∫
ttt
, we obtain that the expression given by
Eq. (18) for
<<
F
N
1
>>
F
N
coincides with that given by Eq. (5). At the opposite end, when
π
→
, and taking into account the value of the definite
1
, we have
2/ arctan
1
FFF
NNN
−
integral
asymptotically tends to the constant value
4876 . 0d )]1 )(4/sin[(
0
∫
1
2
≅−
tt
π
, we obtain that when
0
→
F
N
, the contrast
)(
ν
4876 . 0)(
ν =
max
far
max
ϕ
C
. (19)
We define the signal as above, i.e. as the absolute value of the difference between the image
intensity distribution and the corresponding flat field intensity, S0(ν) ≡ M 2 Sin, integrated over
the area corresponding to the first Fresnel fringes:
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()
. d  ),(/8)(
ν
)(
ν
d  ]),/([/8)(
ν
)(
ν
d)(
ν
,,
1
00
0
arctan2
0
3
F
max in
2
0
max in
2
2
0
tNtFnLS
xNnxFnLSxSyxS ML
FF
NN
FMy
x
FFMFy
Mx
Mx
y
∫
∫∫
−
++
−
=
=−
′′
≡Σ
πσϕ
σπϕν
The corresponding Poisson noise is
incident Xray spectral density integrated over the area corresponding to the first Fresnel
fringe in the images with and without the edge feature. The signaltonoise,
then equal to
)(ν
DN =
, where
)(
ν
8)(
ν
in0
SLxD
y
+
=
is the
N SNR
/
Σ≡
, is
)(
ν
 )](
ν
[)(
ν
max
2 / 1
ϕα
D SNR
F
N
=
, (20)
where
∫
−
F
−−
F
=
1
arctan2
0
2 / 112 / 1
F
d ) , (
t
) arctan8 (
F
F
NN
FFN
tNFNNn
πα
is a positive value depending
only on the Fresnel number. Therefore, the signaltonoise is directly proportional to the
square root of the integrated Xray intensity incident on the feature and to the maximum phase
shift produced by the feature. It can be easily verified that for large values of the Fresnel
number,
])2 ( 4 /[) 1 (
FN
Ne
F
πα
−≅
, and hence Eq. (20) coincides with Eq. (4). When
∫ ∫
t
NF
2 / 12
−
0
→
F
N
, then
1325 . 0dd)]1)(4/ sin[( ) 4/ 1 (
2
00
2
≅−→
xt
x
πα
, and Eq. (20) reduces to
2 / 1 farfar
maxmax
2 / 1 farfar
)](
ν
)[(
ν
2717 . 0)(
ν
 )](
ν
[ 1325 . 0)(
νϕ
DCD SNR
≅≅
, (21)
where
λ
′
ν
(
RLSD
y
=
in
far
4)
. Therefore, the quantity
)(
ν
)](
ν
[ )(
ν
)(
1
max
2 / 1
−
γ
−
≡
CDSNRNF
, (22)
tends to approximately 0.2717 when
−→
F
N
γ
of
)(
F
N
γ
for a wide range of Fresnel numbers and found that
between the two limits, 0.2717 and 0.3564.
One can obtain analogues of the above equations for the contrast, spatial resolution and
signaltonoise in the polychromatic case integrating the spectral density over the frequencies
according to
∫
,,(),(
yxSyxI
. The polychromatic analogue of Eq. (14) is the
following equation for the timeaveraged intensity of a projection image of a purephase edge
feature:
0
→
∞
F
N
→
. One can see from Eq. (5) that
3564. 04/ )e 1 (e)(
22 / 1
≅
−
when
F
N
. We have calculated numerically the value
(
N
γ
)
F
varies slowly in
=
νν d )
νσ
/(
πν
(
ϕν
( d ]
F
),[ / 2 ) )),,(
maxinin2
2
FMF
NnxFnSIRMyMxIM
∫
−=
. (23)
However, unlike the TIE case considered above, here the dependence on the wavelength
(
νλ
/ c
=
) cannot be generally factored out, unless the spectrum is so narrow that the
dependence of
)/(2
λπσ
RN
MF
′
=
on ν can be neglected. In the latter case it is trivial to
obtain exact analogous of Eqs. (16)(21) with the maximum monochromatic phase shift,
)(
ϕ
, replaced by its average value over the spectrum,
If such a simplification is impossible, the dependence of contrast and other image parameters
on the wavelength spectrum may become rather complicated [18].
2
maxν
in max in max
/d)(
ν
  )(
ν

IS
νϕϕ
∫
=
.
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3. Simple rules for estimation of contrast, SNR and spatial resolution
In this section we analyze the expressions obtained in the previous section for the contrast,
SNR and spatial resolution in inline images of nonabsorbing edgelike feature, and suggest
some simple rules for evaluation of these principal image parameters.
The graph of the value of
)2 /()(
sys min
σ
x
Δ
as a function of the inverse of Fresnel number
calculated in accordance with Eq. (16) is presented in Fig. 4. As one can see from Fig. 4, the
resolution values tend to a finite limit equal to
agreement with Eq. (6), while for small Fresnel numbers the spatial resolution becomes
proportional to the square root of the inverse Fresnel number, eq.(17), which corresponds to a
straight line in the logarithmic plot. Furthermore, one can see that the following "rule of
thumb", which simply combines Eq. (6) and Eq. (17) and gives a reasonably good
approximation for the spatial resolution of inline imaging across a large range of Fresnel
numbers.
Rule 1 (spatial resolution). The spatial resolution in inline imaging of a phase edgelike
feature satisfying Eq. (12) can be estimated as the largest of the two values, namely the width
of the PSF of the imaging system,
)(2
σ
, and the width of the first Fresnel zone,
)(2
sysM
σ
for large Fresnel numbers in
sysM
λ
R′ .
10
2
10
1
10
NF
0
10
1
10
2
1
10
(Δx)min / (2σsys )
1
Fig. 4. Plot of the spatial resolution in inline imaging as predicted by Eq. (16)
(solid line), Eq. (17) (dotted line) and Eq. (6) (dashdotted line).
The graph of the value of
calculated in accordance with Eq. (18) is presented in Fig. 5. One can see that the image
contrast as a function of the inverse Fresnel number displays a behavior which is
complementary to that of the spatial resolution, i.e. at large Fresnel numbers the image
contrast is directly proportional to the inverse Fresnel number in agreement with Eq. (5),
while at small Fresnel numbers the contrast asymptotically converges to a constant value in
accordance with Eq. (19).
The behavior illustrated by Fig. 4 and Fig. 5 indicates the existence of a tradeoff between
the image contrast and resolution as a function of inverse Fresnel number, i.e. as a function of
the wavelength of the radiation, effective propagation distance and the width of the PSF of the
imaging system. Therefore, the issue of optimization of experimental conditions in inline
imaging becomes important and nontrivial [18, 21, 10]. However, one can see that the
)(
ν
  / )
ν
(
maxmax
ϕ
C
as a function of the inverse of Fresnel number
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following "rule of thumb", which simply combines Eq. (5) and Eq. (19), gives a reasonably
good approximation for the contrast of an inline image of a phase edge across a large range
of Fresnel numbers.
Rule 2 (image contrast). The maximum contrast in an inline image of a phase edgelike
feature satisfying Eq. (12) can be estimated as the product of the maximum absolute phase
shift generated by the feature,
)(
ϕ
, and the smallest of the two values,
4876 . 0
.
maxν
F
N
/ 0.2420
and
10
2
10
1
10
NF
0
10
1
10
2
10
3
10
2
10
1
10
0
0.4876
Cmax / ϕ max
1
Fig. 5. Plot of the maximum image contrast in inline imaging as predicted by
Eq. (18) (solid line), Eq. (19) (dotted line) and Eq. (5) (dashdotted line).
The graph of the value of
Fig. 6.
)(
F
N
γ
calculated in accordance with Eq. (22) is presented in
105
103
101
101
103
105
NF
1
0
0.1
0.2
0.3
0.2717
0.4
0.3564
0.5
0.6
SNR / [Cmax D1/2]
Fig. 6. Plot of the ratio of the SNR to the product of the maximum image
contrast and the square root of the integral Xray intensity incident on the
region of interest.
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One can see from Fig. 6 that
0.3564, i.e. it does not change significantly over the full range of Fresnel numbers. Therefore,
we can formulate the following "rule of thumb".
Rule 3 (signaltonoise). The signaltonoise in an inline image of a phase edgelike
feature satisfying Eq. (12) can be estimated as the product of approximately 0.3 times the
image contrast (see Rule 2) and the square root of twice the integrated Xray intensity
incident on the feature.
)(
F
N
γ
varies slowly in between its two limits, 0.2717 and
4. Numerical results
In this section we present results of numerical simulation of inline image contrast for a pure
phase edge obtained using the Kirchhoff diffraction theory. These results are compared with
the predictions given by the analytical formulae derived in section 2 above. For reader's
convenience, the following table summarizes the validity conditions required for various
imaging regimes considered below.
Table 1. Validity conditions for various approximations.
Approximation Validity condition
"Exact" (Kirchhoff diffraction theory)
λ >>
′ R
"TIE" (Transport of Intensity equation)
λ >>
′ R
,
objmax
})(
ν
 max{1,
λ
′
σ
M
σϕ
R
<<
"WO" (Weak object approximation)
λ >>
′ R
,
1  ),
ν
(),
ν
(
<<−
−+
ϕϕ
rr
"Far" (WO in the far Fresnel region)
λ >>
′ R
,
1  ),
ν
(),
ν
(
<<−
−+
ϕϕ
rr
,
1
<<
F
N
As the simulated sample we used an edge of a polyethylene (C2H4, density = 1 g/cm3) sheet of
thickness 100 μm, with the unsharpness parameter σobj=0.7 μm. The source was modeled as
an Xray tube with a tungsten anode operated at Ep=50 kVp, and with 0.3 mm thick Be
window. We also assumed that the lower Xray energies were filtered out using a 1 mm thick
Al filter. The normalized Xray spectrum incident on the sample is shown in Fig. 7.
0 10 2030 4050
0.00
0.01
0.02
0.03
S(E)
E, keV
Fig. 7. Incident Xray spectrum used to calculate the values in Tables 2 and 3.
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3 March 2008 / Vol. 16, No. 5 / OPTICS EXPRESS 3236
Page 15
The maximum absolute value of the generalized eikonal corresponding to the chosen sample
and the Xray spectrum was found to be equal to
assumed 4 μm (FWHM), the detector resolution was 100 μm (FWHM). We modeled an X
ray system with a fixed sourcetodetector distance R=2 m. In the simulations we changed the
value of magnification (M) by changing the sourcetoobject distance, the effective defocus
distance R' was changing accordingly. The comparison between the results obtained by
simulating the images with the help of Kirchhoff integrals and those obtained using the
analytical formulae (Eqs. (6), (8) and (9)) derived in the preceding sections are presented in
Tables 2 and 3. It is easy to see that the agreement between the two sets of results is quite
good, with the accuracy of the analytical results improving as the necessary validity condition
for the TIE,
}  ),/(2 max{ /
maxobj
ψπλσσ
>>
′ R
M
Å 2947 . 0
max=
ψ
. The source size was
, becomes better satisfied.
Table 2. Comparison of the image characteristics obtained with Kirchhoff diffraction theory ("Exact ") and those
obtained using analytical formulae derived in the present paper, Eqs. (6), (8)(9) ("TIE"), using Iin = 10 ph/μm2,
Ly = 1 mm and R = 2 m. The source size was 4 μm (FWHM) and the detector resolution was 100 μm (FWHM).
Contrast, %
Resolution, μm
Exact
SNR
M
max
ψ
′
σ
M
σ
R
obj
NF R', mm
Exact TIE TIE Exact TIE
2
1.01 142.0 500 0.76 0.79 43.28 42.52 3.54 3.67
5
0.641 36.6 320 2.81 3.06 17.81 17.26 8.44 9.07
10
0.603 18.2 180 5.60 6.15 9.48 9.14 12.30 13.26
25
50
100
0.76
1.209
2.244
12.3
16.0
27.8
76.8
39.2
19.8
8.46
6.78
3.98
9.08
7.02
4.04
5.08
4.06
3.77
4.91
3.99
3.74
13.56
9.75
5.51
14.34
10.0
5.56
Table 3. Comparison of the image characteristics obtained with Kirchhoff diffraction theory ("Exact ") and those
obtained using analytical formulae derived in the present paper, Eqs. (6), (8)(9) ("TIE"), using Iin = 10 ph/μm2 and
Ly = 1 mm and R = 2 m. The source size was 100 μm (FWHM) and the detector resolution was 4 μm (FWHM).
Contrast, %
Resolution, μm
Exact
SNR
M
max
ψ
′
σ
M
σ
R
obj
NF
R', mm
Exact TIE TIE Exact TIE
1.32 0.672 46.2 367.3 2.23 2.42 21.43 20.80 7.38 7.87
1.16 0.607 24.4 237.8 4.18 4.59 12.59 12.16 10.59 11.41
1.08 0.621 14.7 137.2 6.94 7.61 7.44 7.17 13.51 14.52
1.04
1.02
1.01
0.775
1.229
2.265
12.4
16.2
28.0
74.0
38.4
19.6
8.46
6.70
3.94
9.06
6.93
4.00
4.99
4.05
3.76
4.83
3.98
3.74
13.47
9.62
5.46
14.18
9.85
5.51
In the next set of calculations we compared the results of analytical formulae, Eqs. (16),
(18) and (20), with the corresponding results obtained using Kirchhoff diffraction theory,
under the conditions corresponding to a broad range of Fresnel numbers including the small
ones, where Eqs. (6)(9) based on the TIE approximation are no longer valid. Here we used
monochromatic incident radiation with energy E = 30.78 keV (the average energy of the
spectrum in Fig. 7), λ = 0.4 Å. Compared to the first set of calculations, we also decreased
the value of the maximum phase shift to ϕ max = 1 rad (which corresponds to the thickness of
polyethylene of 25.66 μm) in order to satisfy conditions (12) across the whole considered
range of Fresnel numbers. We kept the magnification constant at M = 25.3, which led to the
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constant value of the standard deviation of the system PSF, σsys = 2.341 μm. The propagation
distance was varied between 2 m and 1024 m, which corresponded to the Fresnel numbers
between 12.28 and 0.024. The corresponding results are presented in Table 4. In the same
table we also present for comparison the results obtained with Eqs. (9) and (6) (large Fresnel
numbers, or TIE approximation), and with Eqs. (17) and (19) (small Fresnel numbers). One
can see that Eqs. (16) and (18) give the values which agree very well with the results obtained
using Kirchhoff diffraction theory across the whole considered range of Fresnel number
values. As expected, the results given by Eqs. (6) and (9) and Eqs. (16) and (19) agree well
with the Kirchhoff diffraction results for large and small Fresnel numbers, respectively. One
can also easily verify that the simple Rules 1 and 2 formulated above give rather good
estimation of the spatial resolution and contrast, respectively, in the considered examples.
Table 4. Comparison of the image contrast and spatial resolution obtained with Kirchhoff diffraction theory ("Exact")
with those obtained using analytical formulae derived in the present paper. Analytical results are given for the general
formulae, Eq. (16) and (18) ("WO"), as well as for the limiting cases of very large Fresnel numbers, Eq. (5) and
Eq. (6) ("TIE") and very small Fresnel numbers, Eq. (17) and (19) ("Far"). The source size was 4 μm (FWHM), the
detector resolution was 100 μm (FWHM), λ = 0.4 Å, ϕmax = 1 rad and M = 25.3.
Contrast, %
Resolution, μm
Exact
R, m
R', m NF
Exact WO TIE Far WO TIE Far
2
4
8
16
32
64
128
256
512
1024
0.08
0.15
0.30
0.61
1.22
2.43
4.86
9.72
19.44
38.88
12.28
6.14
3.07
1.54
0.768
0.384
0.192
0.096
0.048
0.024
1.96
3.90
7.61
14.10
23.24
32.28
38.67
42.35
44.23
45.21
1.97
3.92
7.68
14.37
23.88
33.33
40.15
44.21
46.42
47.58
1.97
3.94
7.88
15.76
31.53
63.05
126.11
252.22
504.43
1008.87
4.90
4.94
5.08
5.51
6.72
9.19
13.19
18.98
27.18
38.71
4.90
4.93
5.06
5.49
6.74
9.30
13.36
19.26
27.58
39.28
1.75
2.47
3.50
4.95
6.99
9.89
13.99
19.78
27.98
39.57
48.76
4.89
5. Experimental results
In order to further test some of the formulae presented here we performed Xray imaging
experiments with an object consisting of two sheets of 100 μm thick polyethylene. These
sheets were overlapped and the edges (vertical and horizontal) crossed on the optic axis of the
Xray imaging instrument. The sheets in the sample were arranged in a crossed fashion in
order to enable testing of the images of the vertical as well as the horizontal edges. The X
rays were produced by a Feinfocus FXE225.20 microfocus source with a cylindrical W
reflectionbased target and 250 μm Be window. The source was operated at 30 kVp and
50 μA. The sourcetoobject distance R1 and objecttodetector distance R2 were varied, but
R = R1+R2 was fixed at 2 m (magnification M = R / R1). The detector was Fuji Imaging Plates
(20 cm x 25 cm; FDLURV), scanned with a Fuji BAS5000 scanner (using 25 μm pixel size).
Under these experimental conditions, the object yielded images which possessed both
absorption and phase contrast, but were dominated by the latter in the form of characteristic
single blackwhite fringes (a typical image is presented in Fig.8). The images were subjected
to flatfield correction and then analysed to provide contrast and resolution values. The
observed (experimental) contrast values were obtained from the difference between the
maximum and minimum intensity values divided by the sum of these quantities; the observed
resolution values were obtained from the spatial separation of lines of maximum and
minimum intensity, referred to the object plane. The observed data values, in both vertical
and horizontal directions, are listed in Table 5.
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Fig. 8. An image of a phantom used in the experimental tests acquired at R1 = 10 cm.
Table 5. Observed and calculated (corresponding to refined parameters in leastsquares analysis) values of contrast
and resolution, for both vertical and horizontal directions.
R1, cm
R2, cm
NF Cobs, %
Ccalc, %
(Δx)obs, μm
(Δx)calc, μm
vertical
5
10
20
40
60
195
190
180
160
140
32.1
19.2
16.0
22.3
34.1
6.9 (0.5)
9.6 (0.4)
10.7 (0.4)
9.3 (0.3)
7.5 (0.2)
5.7
9.5
11.4
8.1
5.3
10.0 (1.0)
10.0 (1.0)
13.0 (1.0)
20.0 (1.0)
27.0 (4.0)
9.2
9.9
12.4
19.6
27.8
horizontal
5
10
20
40
60
195
190
180
160
140
40.2
23.1
17.8
23.2
34.6
4.9 (0.3)
7.5 (0.4)
8.4 (0.3)
7.3 (0.5)
5.6 (0.4)
4.5
7.8
10.2
7.8
5.2
9.0 (2.0)
10.0 (1.0)
13.0 (1.0)
20.0 (1.0)
29.0 (3.0)
10.3
10.9
13.2
20.0
28.0
The analysis of the experimental data in Table 5 was performed by nonlinear leastsquares
refinement using a modification of the LevenbergMarquardt algorithm [22, 23]. The
experimental data is interpreted in terms of Eq. (5) for the contrast values and using 2σM for
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the resolution values, i.e. monochromatic formulae. Correlation matrices and estimated
standard deviations (esds) for refinedparameter values were calculated [24, 25], as was
Hamilton’s Rfactor RH [26] as a measure of agreement between theory and experiment. All
of the observations (contrast and resolution) were weighted equally in the analysis. A total of
six parameters could be refined (although the limited data set and correlations between
parameters precluded all six being varied simultaneously): σobj; σsrc(vert); σsrc(horiz); σdet;
effective Xray energy E; polyethylenesheet thickness t. Whilst the σvalues associated with
the object and the detector could justifiably be treated as being the same in both vertical and
horizontal directions, this was not true for the Xray source (see below) and so two parameters
were required.
Table 6. Refined parameter values from leastsquares analysis of experimental data. Values in italics (and
without an esd) were fixed. The values in square brackets are expressed as FWHM rather than σ (FWHM ≈
2.35σ) for convenience. RH is Hamilton’s Rfactor and provides a measure of the agreement between theory and
experiment [26].
σobj, μm
σsrc(vert), μm
σsrc(horiz), μm
σdet, μm
E, keV
<λ, Å>
ϕ max, rad
t, μm RH, %
4.0
[9.4]
2.0 (0.6)
[4.7 (1.3)]
3.1 (0.4)
[7.3 (1.0)]
44 (1)
[104 (2)]
14.7 (0.4)
<0.84 (0.05)>
7.54 (0.38)
100
6.8
Table 6 provides the parameter values obtained by a leastsquares refinement in which the
blur of the polyethylene edges was fixed at σobj = 4 μm and the thickness was fixed at
t = 100 μm. This value of σobj was arrived at by trialanderror and, inasmuch as the
corresponding FWHM value is of order 10% of t, is physically reasonable. The value of RH is
consistent with a good fit to the experimental data. As the value of E was refined, the value of
phase shift per unit length, being energy dependent, had to be changed accordingly. These
values were calculated, for polyethylene (C2H4; ρ = 0.923 g/cm3), using Xray data from [27].
The agreement between the observed and calculated contrast and resolution values (see Table
5) is in general quite good, with the contrast values showing the expected peak as a function
of R1 (with R being fixed), and the resolution values increasing monotonically.
The microfocus Xray source was operated at a power of 1.5 W for which the
manufacturer specifies that the source size is a minimum, nominally 4 μm. It is also widely
acknowledged that, because of the geometry of the target and the incident electron beam
(target angle 22.5°), the effective Xray spot as viewed along the optic axis is elliptical in
shape. The disposition of the Xray tube used here was such that the minor (major) axis of the
ellipse was vertical (horizontal). The resolution of standard imaging plates has been estimated
to be 150 μm FWHM at best (see e.g. [28]). Whilst the pixel size is determined by the raster
step size and the size of the laser focus in the imagingplate scanner, the resolution is largely
determined by the scattering of the laser beam inside the phosphor layer. The Fuji Imaging
Plates that were used here were special plates developed for electron microscopy. The
phosphor layer is thinner (110 μm) than for standard plates, which resulted in higher
resolution (but lower sensitivity), and it contained a blue pigment which helped to reduce the
scattering of the laser beam. Amemiya [29] has indicated that these plates have “a spatial
resolution of slightly less than 100 μm”, in excellent agreement with the detectorparameter
value obtained in Table 6. Finally, the effective Xray energy (E) value given in Table 6 is
entirely consistent with an Xray tube operating at 30 kVp with minimal beam hardening from
the tube window and object. In summary, the leastsquares analysis has provided a good fit to
experimental contrast and resolution data, and the refinedparameter values are all physically
reasonable and in accord with expectations.
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6. Conclusion
We have derived simple analytical expressions ("rules of thumb") for the maximum contrast,
spatial resolution and signaltonoise in the vicinity of the geometric image of a phase edge
like feature in a sample. The expressions depend on the sharpness of the edge and the
maximum phase shift generated by the feature, as well as on the PSF of the imaging system,
the sourcetoobject and objecttodetector distances, and the wavelength spectrum. The main
factors determining the characteristics of the image are the maximum absolute phase shift,
ϕ max, and the Fresnel number,
RkN
MF
=
/
σ
variance
M
σ
equal to the sum of the variance of the PSF of the imaging system,
and that of the unsharpness of the edge,
obj
σ
. The spatial resolution behaves quite differently
for large and small Fresnel numbers. In the case of large Fresnel numbers (short propagation
distances) the spatial resolution is simply equal to the width of the PSF of the imaging system,
)(2
σ
, while for small Fresnel numbers (large propagation distances) the resolution is
′
2
, which is defined with respect to the total
2
)(
2
sysM
σ
,
2
sysM
equal to the width,
signaltonoise are always directly proportional to the maximum absolute phase shift. At
short effective propagation distances R', when the corresponding Fresnel number is large, the
maximum image contrast is inversely proportional to the Fresnel number, while at long
propagation distances (small Fresnel numbers) the maximum contrast asymptotically tends to
the limit 0.4876ϕ max, which does not depend on any parameters other than the maximum
absolute phase shift. The signaltonoise behaves similarly, as it is proportional to the product
of maximum image contrast and the square root of the integrated Xray intensity incident on
the sample feature that is being imaged. When the relevant Fresnel number is large, the
expressions for the image contrast, spatial resolution and signaltonoise in the case of
polychromatic radiation are virtually the same as in the monochromatic case with the suitable
replacement of the conventional phase by the generalized eikonal of the polychromatic beam.
The behaviour is much more complicated in the case of small Fresnel numbers (large
propagation distances) and polychromatic radiation, where we could not obtain simple and
general analytical expressions for the image characteristics.
λ
R′ , of the first Fresnel zone. We showed that the contrast and the
Acknowledgement
The authors are grateful to XRT Ltd for encouragement of this work.
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