Effects of unresolvable edges in grating-based X-ray differential phase imaging

Wataru Yashiro; Atsushi Momose

doi:10.1364/OE.23.009233

1. Introduction

X-ray grating interferometry (X-ray Talbot or Talbot-Lau interferometry) has been increasingly attracting attention for the last decade [1–7] as a quantitative X-ray phase imaging techniques because it attains higher sensitivity than that conventional absorption-contrast X-ray imaging and works with a low-brilliance laboratory X-ray source. Another feature of this interferometry is that it is a multi-modal X-ray imaging technique: it provides three images, i.e., absorption, differential-phase, and visibility-contrast images. The third image was first reported by F. Pfeiffer et al. and called ‘dark-field image’ [6]. It was later shown that the third image’s contrast (visibility contrast) can be quantitatively related to spatially fluctuating wavefront due to randomly distributed unresolvable microstructures (typically of the order of μm) [8, 9], from which ultra-small-angle X-ray scattering (USAXS) arises.

In this paper, we will show from numerical calculations and experiments for monochromatic X-rays that even an unresolvable sharp edge generates the visibility contrast. This means that the contrast has another major origin other than USAXS from randomly distributed unresolvable microstructures, which has been considered the main origin of the contrast.

Much research has thus far been reported on quantitative analysis of the visibility contrast [8–40]. However, how an unresolvable sharp edge affects the visibility-contrast image has never been reported. In the interferometry, slow variation in phase shift caused by a sample is assumed. In this case, the gradient of the phase shift, i.e., change in the direction of X-ray propagation, due to refraction of the sample is determined from distortion of moiré fringes, which are generated downstream of two gratings in this interferometer (see Fig. 1). However, for an unresolvable sharp edge, which can take an infinite value of differential-phase shift, this assumption fails, so that a more general description of the intensity of moiré fringes is required. In this paper, we provide general expressions to describe the intensity of the moiré fringes for an unresolvable sharp edge and show the results of numerical calculations from the expressions. The results showed that an unresolvable sharp edge generates contrast in not only a differential-phase image but also a visibility-contrast image for monochromatic X-rays. We demonstrated that these effects indeed occur in experiments using a synchrotron X-ray source.

Fig. 1 Setup of a spherical-wave X-ray Talbot interferometer.

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In Section 2.1, we will review the conventional simplified theoretical description of moiré fringes for the slow variation in phase shift. We will start by theoretically describing the moiré fringes for a point X-ray source, which will then be extended for a finite-size X-ray source. In Section 2.2, we will show more general expressions that can describe how an unresolvable edge appears in the three images. The results of numerical calculations for unresolvable edges done on the basis of the expressions given in Section 2.2 will be shown in Section 3. In Section 4, we will demonstrate experimental results confirming that the visibility contrast occurs even with an unresolvable edge.

2. Theoretical background

2.1. Conventional simplified theoretical description for X-ray grating interferometry

2.1.1. For a point X-ray source

First we consider a spherical-wave X-ray Talbot interferometer with a monochromatic (quasi-monochromatic [41]; wavelength: λ) point X-ray source as shown in Fig. 1, where a sample, two sufficiently large gratings (G1 and G2), and an X-ray image detector are located at distances of R_s, R₁, R₂, and R_D from the X-ray source along the optical axis (z-axis). Here we define the origin of the z-axis at the position of the X-ray source and xy-coordinate systems perpendicular to the z-axis, (x_s, y_s), (x₁, y₁), (x₂, y₂), and (x_D, y_D), at the positions of the sample, G1, G2, and the detector. On the xy-coordinate systems, four points (x_s0, y_s0), (x₁₀, y₁₀), (x₂₀, y₂₀), and (x_D0, y_D0) are given, which are related by (x_i0, y_i0) = (M_ijx_j0, M_ijy_j0), where M_ij ≡ R_i/R_j (i, j = s, 1, 2, D). We also define an xy-coordinate system (X₀, Y₀) at z = 0 to discuss the effect of a finite size of the X-ray source later. We assume that the gratings are set to be perpendicular to the z-axis, and, without loss of generality, their lines are parallel to the y-axes.

Effective pitches d_i for i = s, 2, D are conveniently defined by

d_{i} \equiv M_{i 1} d_{1},

where d₁ is the pitch of G1. We can also define effective distances z_eff,ij and effective Talbot orders p_ij for i, j = s, 1, 2, D by

z_{eff, i j} \equiv {(\frac{1}{R_{i}} + \frac{1}{R_{j} - R_{i}})}^{- 1}

and

p_{i j} \equiv z_{eff, i j} \cdot \frac{λ}{{d_{i}}^{2}}

= \frac{λ}{d_{i} d_{j}} (R_{j} - R_{i}) .

In this paper, we consider electric fields and intensities only near the optical axis (i.e., in the paraxial approximation), but we can extend the following results even for those far from the optical axis by appropriately introducing effective Talbot orders and effective amplitude transmission functions of the sample, G1, and G2.

In the paraxial approximation, the electric field E₂₋(x₂₀, y₂₀) just upstream of G2 generated by a point X-ray source at the origin O, with an intensity distribution I₀δ(X₀, Y₀) [cps/sr/m²] (∝ |E₀|² δ(X₀, Y₀)), is given by

E_{2 -} (x_{20}, y_{20}) \approx E_{0} \frac{exp [\frac{2 π i}{λ} R_{2}]}{R_{2}} \sum_{n = - \infty}^{\infty} {a^{'}}_{n} T_{s 2, n}^{F} (x_{s 0}, y_{s 0}) exp [2 π i \frac{n x_{10}}{d_{1}}],

where

{a^{'}}_{n} \equiv a_{n} exp [- π i p_{12} n^{2}],

T_{s 2, n}^{F} (x_{s 0}, y_{s 0}) \equiv T_{s 2}^{F} (x_{s 0} - n p_{s} d_{s}, y_{s 0}),

T_{s 2}^{F} (x_{s 0}, y_{s 0}) \equiv T_{s} * P_{s 2} (x_{s 0}, y_{s 0}),

P_{s 2} (x_{s 0}, y_{s 0}) \equiv \frac{- i}{λ z_{eff, s 2}} exp [\frac{π i}{λ z_{eff, s 2}} ({x_{s 0}}^{2} + {y_{s 0}}^{2})] .

Here the factor a_n in Eq. (6) is the nth Fourier coefficient of the amplitude transmission function T₁(x₁₀, y₁₀) of G1, the parameter p_s in Eq. (7) is defined by

p_{s} \equiv {\begin{array}{l} p_{12} & (R_{s} \leq R_{1}) \\ p_{s 2} & (R_{s} \geq R_{1}) \end{array},

and the function

T_{s 2}^{F} (x_{s 0}, y_{s 0})

in Eq. (8) corresponds to the effective amplitude transmission function of the sample (the superscript ‘F’ meaning that Fresnel diffraction by the sample is taken into account), which is expressed by a convolution of the sample’s amplitude transmission function T_s (x_s0, y_s0) and the propagator P_s2 (x_s0, y_s0) defined by Eq. (9) (the operator * denoting convolution (f * g(x, y) ≡ ∬ f (x − x′, y − y′)g(x′, y′)dx′dy′)).

The sample’s amplitude transmission function T_s (x_s0, y_s0) in Eq. (8) can be expressed by

T_{s} (x_{s 0}, y_{s 0}) = exp [i Φ_{s} (x_{s 0}, y_{s 0})],

where Φ_s (x_s0, y_s0) is generally a complex number whose real and imaginary parts (

Φ_{s}^{r}

and

Φ_{s}^{i}

) correspond to the phase shift and the amplitude attenuation caused by the sample, and related to the electric field E_s+ (x_s0, y_s0) just downstream of the sample by

E_{s +} (x_{s 0}, y_{s 0}) = T_{s} (x_{s 0}, y_{s 0}) \frac{E_{0} exp [\frac{2 π i}{λ} R_{s}]}{R_{s}} exp [2 π i ({x_{s 0}}^{2} + {y_{s 0}}^{2})] .

In a conventional simplified description, the effect of the Fresnel diffraction by the sample is neglected:

T_{s 2, n}^{F} (x_{s 0}, y_{s 0}) \approx T_{s, n} (x_{s 0}, y_{s 0}),

where T_s,n (x_s0, y_s0) ≡ T_s (x_s0 − np_sd_s, y_s0). We call this approximation ‘projection approximation’ in a broad sense. It is also assumed that Φ_s (x_s0, y_s0) is a slowly varying function of x_s0, so that the exponent of T_s,n (x_s0, y_s0) can be approximated by the Taylor series up to the first order as

T_{s, n} (x_{s 0}, y_{s 0}) \approx exp [- Φ_{s}^{i} (x_{s 0}, y_{s 0})] exp [i (Φ_{s}^{r} (x_{s 0}, y_{s 0}) - n p_{s} d_{s} \frac{\partial Φ_{s}^{r}}{\partial x_{s 0}} (x_{s 0}, y_{s 0}))] .

Therefore, from Eqs. (5), (13), and (14), the intensity I₂₋(x₂₀, y₂₀) just upstream of G2 is given by

I_{2 -} (x_{20}, y_{20}) \approx \frac{I_{0}}{{R_{2}}^{2}} \sum_{n, n^{'}} {a^{'}}_{n} \bar{{a^{'}}_{n - n_{1}}} T_{s 2, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{s 2, n - n_{1}}^{F} (x_{s 0}, y_{s 0})} exp [2 π i \frac{n_{1} x_{10}}{d_{1}}],

\approx \frac{I_{0} exp [- 2 Φ_{s}^{i}]}{{R_{2}}^{2}} \sum_{n_{1}} b_{n_{1}} exp [- i n_{1} p_{s} d_{s} \frac{\partial Φ_{s}^{r}}{\partial x_{s 0}}] exp [2 π i \frac{n_{1} x_{10}}{d_{1}}],

where overline means complex conjugate, and

b_{n_{1}} \equiv \sum_{n} {a^{'}}_{n} \bar{{a^{'}}_{n - n_{1}}}

. The factor

exp [- 2 Φ_{s}^{i}]

represents the X-ray intensity attenuation by the sample, and for energy conservation,

Φ_{s}^{i}

has to be a positive number.

The intensity I₂₊(x₂₀, y₂₀) just downstream of G2 is written by

I_{2 +} (x_{20}, y_{20}) = I_{2 -} (x_{20}, y_{20}) {| T_{2} (x_{20}, y_{20}) |}^{2},

and from Eq. (16),

\begin{array}{l} I_{2 +} (x_{20}, y_{20}) \approx & \frac{I_{0} exp [- 2 Φ_{s}^{i}]}{{R_{2}}^{2}} \\ \times \sum_{n_{1}, n_{2}} b_{n_{1}} c_{n_{2}} exp [- i n_{1} p_{s} d_{s} \frac{\partial Φ_{s}^{r}}{\partial x_{s 0}}] exp [2 π i \frac{(n_{1} + n_{2}) x_{20} + n_{2} χ_{2}}{d_{2}}], \end{array}

where |T₂(x₂₀, y₂₀)|² is the intensity transmission function of G2, which was expanded into a Fourier series ∑_n₂ c_n₂ exp [2πin₂x₂₀/d₂] and χ₂ is the displacement of G2 along the x₂ direction.

Noted that we can include the cases where the image of G1 generated just upstream of G2 is compressed. For a compressed image of G1, b_n₁ is zero when n₁ ≠ n′₁α, where n′₁ is an integer and α is the compression ratio [42] determining the pitch of the image of G1. In such a case, the pitch of G2 has to be d₂/α.

Finally, assuming that the detector is located sufficiently close to G2 (p_2D ≪ 1) and that the width w_D of the point spread function (PSF) of the X-ray image detector is sufficiently larger than d_D (w_D ≫ d_D), we can write the intensity detected at (x_D0, y_D0) on the detector as a magnified image of I₂₊(x₂₀, y₂₀):

I_{D} (x_{D 0}, y_{D 0}) = {(M_{2 D})}^{2} \cdot I_{2 +} (x_{20}, y_{20}) .

A shift-invariant PSF Ŵ_D(x_D0, y_D0) of the detector assumed, and its pixel size ΔS [m²] taken into account,

I_{D} (x_{D 0}, y_{D 0} Δ S) [cps] = Δ S {(M_{2 D})}^{2} {I_{2 +} * {\hat{W}}_{D} (x_{D 0}, y_{D 0})} .

If it can be assumed that

\partial Φ_{s}^{r} / \partial x_{s 0}

is a slowly varying function of x_s0 (and the condition d_D ≪ w_D is used again), only the terms that satisfy n₁ + n₂ = 0 are dominant in Eq. (18), so that

I_{D} (x_{D 0}, y_{D 0}) \approx \frac{I_{0} exp [- 2 Φ_{s}^{i}]}{{R_{D}}^{2}} \sum_{n} b_{- n} c_{n} exp [- \frac{2 π i n p_{s} d_{s}}{λ} φ_{x_{s 0}}] exp [2 π i \frac{n χ_{2}}{d_{2}}],

where

φ_{x_{s 0}} \equiv (λ / 2 π) \partial Φ_{s}^{r} / \partial x_{s 0}

, corresponding to the angle of the beam deflection by the refraction of the sample.

Absorption by the air and the substrates of the gratings and the detection efficiency of the detector can also be taken into account; hereinafter we define I₀ [cps/sr] as the intensity including the effects of the absorption and the detection efficiency.

2.1.2. For a finite size X-ray source

Next we consider the effect of a finite size of X-ray source. The intensity distribution of the X-ray source is given by I₀ · Î₀(X₀, Y₀), where Î₀(X₀, Y₀) is the normalized intensity distribution satisfying ∫ ∫ Î₀(X₀, Y₀)dX₀dY₀ = 1. We can regard I_D(x_D0, y_D0) in Eq. (20) for a point X-ray source given as a functional of T_s(x_s0, y_s0), T₁(x₁₀, y₁₀), and T₂(x₂₀, y₂₀):

I_{D} [x_{D 0}, y_{D 0}; T_{s} (x_{s 0}, y_{s 0}), T_{1} (x_{10}, y_{10}), T_{2} (x_{20}, y_{20}); {\hat{W}}_{D} (x_{D 0}, y_{D 0})] .

As shown in Fig. 2, the contribution of a point source at (X₀, Y₀) to the intensity at (x_D0, y_D0) on the detector is given by

\begin{array}{l} I_{D} [x_{D 0}, y_{D 0}; T_{s} (x_{s 0} + Δ x_{s} + y_{s 0} + Δ y_{s}), T_{1} (x_{10} + Δ x_{1}, y_{10} + Δ_{y 1}), \\ T_{2} (x_{20} + Δ x_{2}, y_{20} + Δ y_{2}); {\hat{W}}_{D} (x_{D 0}, y_{D 0})], \end{array}

where (Δx_i, Δy_i) (i = s, 1, 2) are functions of (X₀, Y₀):

(Δ x_{i}, Δ y_{i}) = \frac{R_{D} - R_{i}}{R_{D}} (X_{0}, Y_{0}) .

Fig. 2 Effect of finite size of X-ray source in X-ray Talbot interferometer.

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Noted that, in Eq. (23), we assumed that the amplitude transmission function of the sample for the point source at (X₀, Y₀) can be approximately given by a translation of T_s(x_s, y_s). This approximation can be applied for a sufficiently thin sample. In this paper, we consider such a thin sample; a thick sample needs a more rigorous description taking into account optical path integral [43] instead of simple projection, but this is beyond the scope of this paper. We also neglect the effect of total external reflection and assume that R_D − R_s is sufficiently larger than the thickness of the sample for the simple projection to be applied [44].

Assuming the finite-size X-ray source is chaotic, we can obtain the intensity I_D,σ(x_D0, y_D0) at (x_D0, y_D0) on the detector by simply integrating over X₀ and Y₀:

I_{D, σ} (x_{D 0}, y_{D 0}) \approx I_{0} \iint {\hat{I}}_{0} (X_{0}, Y_{0}) I_{D} (x_{D 0}, y_{D 0}; X_{0}, Y_{0}) d X_{0} d Y_{0} .

For example, for I₂₊(x₂₀, y₂₀) in Eq. (18),

\begin{array}{l} I_{D, σ} (x_{D 0}, y_{D 0}) \approx \frac{I_{0} exp [- 2 Φ_{s}^{i}]}{{R_{D}}^{2}} \\ \times \iint {\hat{I}}_{0} (X_{0}, Y_{0}) \sum_{n_{1}, n_{2}} b_{n_{1}} c_{n_{2}} exp [- i n_{1} p_{s} d_{s} \frac{\partial Φ_{s}^{r}}{\partial x_{s 0}} (x_{s 0} + Δ x_{s}, y_{s 0} + Δ y_{s})] \\ \times exp [2 π i \frac{(n_{1} + n_{2}) x_{20} + n_{2} χ_{2}}{d_{2}}] exp [2 π i (\frac{n_{1} Δ x_{1}}{d_{1}} + \frac{n_{2} Δ x_{2}}{d_{2}})] d X_{0} d Y_{0} . \end{array}

If we assume that

\partial Φ_{s}^{r} / \partial x_{s 0}

is almost constant in the integral over X₀ and Y₀ (i.e., the penumbra effect by the finite size of the X-ray source is negligible for the sample) and that the width w_D of the PSF of the detector is sufficiently larger than d_D, only the terms that satisfy n₁ + n₂ = 0 in Eq. (26) are dominant. As a result,

\begin{array}{l} I_{D, σ} (x_{D 0}, y_{D 0}) \approx \frac{I_{0} exp [- 2 Φ_{s}^{i}]}{{R_{D}}^{2}} \sum_{n} μ (- n \frac{p_{12} d_{1}}{λ R_{1}}, 0) b_{- n} c_{n} \\ \times {〈 exp [- \frac{2 π i n p_{s} d_{s}}{λ} φ_{x_{s 0}}] 〉}_{D} exp [2 π i \frac{n χ_{2}}{d_{2}}], \end{array}

where 〈〉_D expresses the convolution with the PSF of the detector (Ŵ_D). The factor μ is the complex coherence factor given by the Fourier transform of the normalized intensity distribution Î₀ [41]:

μ (k_{x}, k_{y}) = \iint {\hat{I}}_{0} (X_{0}, Y_{0}) exp [2 π i (k_{x} X_{0} + k_{y} Y_{0})] d X_{0} d Y_{0},

and in the case of Eq. (27),

(k_{x}, k_{y}) = (\frac{- n}{d_{2}} \frac{R_{2} - R_{1}}{R_{1}}, 0),

= (- n \frac{p_{12} d_{1}}{λ R_{1}}, 0) .

Equation (28) corresponds to the well-known van Cittert-Zernike theorem. Thus, we can obtain a conventional simplified formula for the intensity on the detector in an X-ray Talbot interferometer [7, 45].

2.1.3. Image acquisition

In X-ray grating interferometry, we can obtain the following three independent images, which we call absorption (or transmittance), differential-phase (or moiré-phase), and visibility-contrast (or normalized-visibility) images regardless of their physical meanings. Here we briefly review how these images are acquired and described theoretically.

To acquire the images, a fringe-scanning method [46–48] or the Fourier transform method [49] is often used. Here we describe only a fringe-scanning method where spatial resolution is not degraded in the process of calculating the three images from experimentally obtained images (moiré images) that are generated downstream of G2. In a fringe-scanning method, we introduce phase shifts to the moiré images by moving G1 or G2. In the equal sampling algorithm [46–48], which is the most typical fringe-scanning algorithm, we capture moiré images at χ₂ = k/M (k = 0, 1, ··· , M − 1) for an integer M, and they are added with a weight of exp(−2πiNk/M). As a result, we can obtain the Nth order Fourier coefficient q_N of I_D,σ(x_D0, y_D0) for each pixel on the detector. Noted that here we assumed the effect of aliasing (contributions from higher order harmonics) to be negligible.

For example, when the fringe-scanning algorithm for N = 0 is applied to Eq. (27),

q_{0} \approx \frac{I_{0} exp [- 2 Φ_{s}^{i}]}{{R_{D}}^{2}} b_{0} c_{0},

where we used μ(0, 0) = 1. The absorption (transmittance) image 𝒯(x_D0, y_D0) is defined by

q_{0} / q_{0}^{ref}

, where

q_{0}^{ref}

is the 0th Fourier component of the intensity on the detector without the sample, that is,

𝒯 (x_{D 0}, y_{D 0}) \equiv \frac{q_{0}}{q_{0}^{ref}},

\approx exp [- 2 Φ_{s}^{i} (x_{s 0}, y_{s 0})] .

Similarly, the moiré-phase image 𝒫(x_D0, y_D0) is defined by

arg [q_{1} / q_{1}^{ref}]

, where q₁ and

q_{1}^{ref}

are the 1st Fourier component of the intensity on the detector with and without the sample:

𝒫 (x_{D 0}, y_{D 0}) \equiv arg [\frac{q_{1}}{q_{1}^{ref}}],

\approx arg {〈 exp [- i p_{s} d_{s} \frac{\partial Φ_{s}^{r}}{\partial x_{s 0}} (x_{s 0}, y_{s 0})] 〉}_{D} .

When

\partial Φ_{s}^{r} / \partial x_{s 0}

can be regarded as a constant in 〈〉_D,

\frac{1}{p_{s} d_{s}} 𝒫 (x_{D 0}, y_{D 0}) \approx \frac{\partial Φ_{s}^{r}}{\partial x_{s 0}} (x_{s 0}, y_{s 0}),

which explains why 𝒫(x_D0, y_D0) is called a differential-phase image. The visibility-contrast (normalized-visibility) image 𝒱(x_D0, y_D0) is defined by

𝒱 (x_{D 0}, y_{D 0}) \equiv \frac{| q_{1} | / q_{0}}{| q_{1}^{ref} | / q_{0}^{ref}} .

2.2. More rigorous description

2.2.1. General expression

In Section 2.1, we obtained simplified but practically useful formulae. We used the projection approximation in Eq. (13) and assumed that Φ_s(x_s0, y_s0) is a slowly varying function of x_s0 in Eq. (14). Here we obtain stricter formulae for more general cases. Without the above two assumptions, the intensity I_D(x_D0, y_D0) at (x_D0, y_D0) on the detector for a point X-ray source is given from Eqs. (5), (15), (17), and (20), and, similar to Eqs. (23) and (25), the intensity I_D,σ(x_D0, y_D0) for a finite size X-ray source with a normalized intensity distribution Î₀(X₀, Y₀) can be written by

\begin{array}{l} I_{D, σ} (x_{D 0}, y_{D 0}) \approx \iint {\hat{I}}_{0} (X_{0}, Y_{0}) \\ \times I_{D} [x_{D 0}, y_{D 0}; T_{sD}^{F} (x_{S 0} + Δ x_{s}, y_{s 0} + Δ y_{s}), T_{1} (x_{10} + Δ x_{1}, y_{10} + Δ y_{1}), \\ T_{2} (x_{20} + Δ x_{2}, y_{20} + Δ y_{2}); {\hat{W}}_{D} (x_{D 0}, y_{D 0})] d X_{0} d Y_{0}, \end{array}

where

(x_{i 0}, y_{i 0}) = \frac{R_{i}}{R_{D}} (x_{D 0}, y_{D 0}),

(Δ x_{i}, Δ y_{i}) \equiv \frac{R_{D} - R_{i}}{R_{D}} (X_{0}, Y_{0})

(i = s, 1, 2) and

T_{sD}^{F} (x_{s 0}, y_{s 0})

is defined in the same manner as Eqs. (8) and (9) but suffixes ‘2’ are replaced by ‘D’. Noted that this formula can include even the expression of the intensity for an X-ray Talbot-Lau interferometer with a negligible gap between the X-ray source and G0.

More explicitly given,

I_{D, σ} (x_{D 0}, y_{D 0}) Δ S [cps] \approx Δ S 𝒫 {\hat{I}}_{0} (X_{0}, Y_{0}) [I_{D -} * {\hat{W}}_{D} (x_{D 0}, y_{D 0})] d X_{0} d Y_{0},

where

\begin{array}{l} I_{D -} [x_{D 0}, y_{D 0}; T_{sD}^{F} (x_{s 0} + Δ x_{s}, y_{s 0} + Δ y_{s}), T_{1} (x_{10} + Δ x_{1}, y_{10} + Δ y_{1}), \\ T_{2} (x_{20} + Δ x_{2}, y_{20} + Δ y_{2})] \\ \equiv \frac{I_{0}}{{R_{D}}^{2}} \sum_{n, n_{1}, n_{2}} {a^{'}}_{n} \bar{{a^{'}}_{n - n_{1}}} T_{sD, n}^{F} (x_{s 0} + Δ x_{s}, y_{s 0} + Δ y_{s}) \bar{T_{sD, n - n_{1}}^{F} (x_{s 0} + Δ x_{s}, y_{s 0} + Δ y_{s})} \\ \times c_{n_{2}} exp [2 π i \frac{(n_{1} + n_{2}) x_{20}}{d_{2}}] exp [2 π i (\frac{n_{1} Δ x_{1}}{d_{1}} + \frac{n_{2} Δ x_{2}}{d_{2}})] exp [2 π i \frac{n_{2} χ_{2}}{d_{2}}], \end{array}

where

T_{sD, n}^{F} (x_{s 0}, y_{s 0})

is defined by Eq. (7) with suffixes ‘2’ replaced by ‘D’. In the following, we will show three approximations that are more rigorous than those in Section 2.1 but useful for practical purposes.

2.2.2. Small penumbra approximation

The convolution in Eq. (41) is explicitly given by

I_{D -} * {\hat{W}}_{D} (x_{D 0}, y_{D 0}) = \int I_{D -} (x_{D 0} - x^{'}, y_{D 0} - y^{'}) {\hat{W}}_{D} (x^{'}, y^{'}) d x^{'} d y^{'} .

Here we introduce x″ and y″ defined by

(x^{″}, y^{″}) \equiv (x^{'}, y^{'}) - (Δ {X^{'}}_{0}, {Y^{'}}_{0}),

where

(Δ {X^{'}}_{0}, {Y^{'}}_{0}) \equiv \frac{R_{D} - R_{s}}{R_{s}} (X_{0}, Y_{0}) .

By using x″ and y″ instead of x′ and y′ in Eq. (43),

I_{D -} * {\hat{W}}_{D} (x_{D 0}, y_{D 0}) = \int {I^{'}}_{D -} (x_{D 0} - x^{″}, y_{D 0} - y^{″}) {\hat{W}}_{D} (x^{″} + Δ {X^{'}}_{0}, y^{″} + Δ {Y^{'}}_{0}) d x^{″} d y^{″},

where

\begin{array}{l} {I^{'}}_{D -} (x_{D 0}, y_{D 0}) \equiv I_{D -} [x_{D 0}, y_{D 0}; T_{s} (x_{s 0}, y_{s 0}), T_{1} (x_{10} + Δ {x^{'}}_{1}, y_{10} + Δ {y^{'}}_{1}) \\ T_{2} (x_{20} + Δ {x^{'}}_{2}, y_{20} + Δ {y^{'}}_{2})], \end{array}

and

(Δ {x^{'}}_{i}, Δ {y^{'}}_{i}) \equiv - \frac{R_{i} - R_{s}}{R_{s}} (X_{0}, Y_{0})

for i = 1, 2.

When penumbra on the detector is much narrower than the width w_D of the point spread function (i.e., |ΔX′₀|, |ΔY′₀| ≪ w_D), Ŵ_D(x″ + ΔX′₀, y″ + ΔY′₀) can be approximated by Ŵ_D(x″, y″) in Eq. (46). As a result, only (Δx′₁, Δy′₁) and (Δx′₂, Δy′₂) in Eq. (47) depend on (X₀, Y₀). The intensity I_D,σ(x_D0, y_D0) in Eq. (41) is therefore rewritten into

I_{D, σ} (x_{D 0}, y_{D 0}) \approx I_{D -, σ} * {\hat{W}}_{D} (x_{D 0}, y_{D 0}),

where

\begin{array}{l} I_{D -, σ} (x_{D 0}, y_{D 0}) \equiv \frac{I_{0}}{{R_{D}}^{2}} \sum_{n, n_{1}, n_{2}} μ (- (n_{1} p_{s 1} + n_{2} p_{s 2}) \frac{d_{s}}{λ R_{s}}, 0) {a^{'}}_{n} \bar{{a^{'}}_{n - n_{1}}} \\ \times T_{sD, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, n - n_{1}}^{F} (x_{s 0}, y_{s 0})} \\ \times c_{n_{2}} exp [2 π i \frac{(n_{1} + n_{2}) x_{20}}{d_{2}}] exp [2 π i \frac{n_{2} χ_{2}}{d_{2}}] . \end{array}

Here we used the definition of μ given in Eq. (28). The notation 〈〉_D in Eq. (27) used,

\begin{matrix} I_{D, σ} (x_{D 0}, y_{D 0}) \approx \frac{I_{0}}{{R_{D}}^{2}} \sum_{n, n_{1}, n_{2}} μ (- (n_{1} p_{s 1} + n_{2} p_{s 2}) \frac{d_{s}}{λ R_{s}}, 0) {a^{'}}_{n} \bar{{a^{'}}_{n - n_{1}}} c_{n_{2}} exp [2 π i \frac{n_{2} χ_{2}}{d_{2}}] \\ \times {〈 T_{sD, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, n - n_{1}}^{F} (x_{s 0}, y_{s 0})} exp [2 π i \frac{(n_{1} + n_{2}) x_{20}}{d_{2}}] 〉}_{D} . \end{matrix}

2.2.3. No resonance approximation

When the sample has no periodic structures (or even periodic structures that can be regarded to be incommensurate with d_s), Eq. (51) is further simplified; in this case, only the terms satisfying n₁ + n₂ = 0 are dominant:

\begin{array}{l} I_{D, σ} (x_{D 0}, y_{D 0}) \approx \frac{I_{0}}{{R_{D}}^{2}} \sum_{n_{2}} μ (- n_{2} \frac{p_{12} d_{1}}{λ R_{1}}, 0) c_{n_{2}} exp [2 π i \frac{n_{2} χ_{2}}{d_{2}}] \\ \times \sum_{n} {a^{'}}_{n} \bar{{a^{'}}_{n + n_{2}}} {〈 T_{sD, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, n + n_{2}}^{F} (x_{s 0}, y_{s 0})} 〉}_{D} . \end{array}

As a result, the transmittance, moiré-phase, and normalized-visibility images are given by

𝒯 (x_{D 0}, y_{D 0}) \approx \frac{\sum_{n} {| {a^{'}}_{n} |}^{2} {〈 {| T_{sD, n}^{F} (x_{s 0}, y_{s 0}) |}^{2} 〉}_{D}}{\sum_{n} {| {a^{'}}_{n} |}^{2}},

𝒫 (x_{D 0}, y_{D 0}) \approx arg [\frac{\sum_{n} {a^{'}}_{n} \bar{{a^{'}}_{n + 1}} {〈 T_{sD, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, n + 1}^{F} (x_{s 0}, y_{s 0}} 〉}_{D}}{\sum_{n} {a^{'}}_{n} \bar{{a^{'}}_{n + 1}}}],

𝒱 (x_{D 0}, y_{D 0}) \approx \frac{| \sum_{n} {a^{'}}_{n} \bar{{a^{'}}_{n + 1}} {〈 T_{sD, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, n + 1}^{F} (x_{s 0}, y_{s 0}} 〉}_{D} | / | \sum_{n} {a^{'}}_{n} \bar{{a^{'}}_{n + 1}} |}{{| \sum_{n} {a^{'}}_{n} |}^{2} {〈 {| T_{sD, n}^{F} (x_{s 0}, y_{s 0}) |}^{2} 〉}_{D} / {| \sum_{n} {a^{'}}_{n} |}^{2}} .

These are independent of the complex coherence factor μ. Later these expressions will be used to explain experimental results for unresolvable edges.

2.2.4. Homogeneous approximation

When $T_{sD}^{F} (x_{D 0}, y_{D 0})$ is regarded to be homogeneous at a pixel within a range that is comparable to the width w_D of PSF, we can consider that ${| T_{sD, n}^{F} (x_{s 0}, y_{s 0}) |}^{2}$ and $T_{sD, n}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, n + 1}^{F} (x_{s 0}, y_{s 0})}$ in Eqs. (53), (54), and (55) do not depend on n. As a result, these expressions are further approximated by

𝒯 (x_{D 0}, y_{D 0}) \approx {〈 {| T_{sD, 0}^{F} (x_{s 0}, y_{s 0}) |}^{2} 〉}_{D},

𝒫 (x_{D 0}, y_{D 0}) \approx arg [{〈 T_{sD, 0}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, 1}^{F} (x_{s 0}, y_{s 0})} 〉}_{D}],

𝒱 (x_{D 0}, y_{D 0}) \approx \frac{| {〈 T_{sD, 0}^{F} (x_{s 0}, y_{s 0}) \bar{T_{sD, 1}^{F} (x_{s 0}, y_{s 0})} 〉}_{D} |}{{〈 {| T_{sD, 0}^{F} (x_{s 0}, y_{s 0}) |}^{2} 〉}_{D}} .

Note that the small penumbra approximation in the Section 2.2.2 is not necessary to derive these expressions.

This homogeneous approximation includes the random phase approximation [8,9,19,50] for the wavefront arising from unresolvable random microstructures. In this case, $T_{sD}^{F} (x_{s 0}, y_{s 0})$ is expressed by

T_{sD}^{F} (x_{s 0}, y_{s 0}) = exp [i Φ_{sD}^{F} (x_{s 0}, y_{s 0})],

= exp [i {Φ_{sD, sm}^{F} (x_{s 0}, y_{s 0}) + Δ Φ_{sD}^{F} (x_{s 0}, y_{s 0})}],

where

Φ_{sD}^{F}

is a complex number and

Φ_{sD, sm}^{F}

and

Δ Φ_{sD}^{F}

represent smooth and fine features of the wavefront. The fine feature

Δ Φ_{sD}^{F}

is attributed to the wavefront due to the unresolvable random microstructures in a sample. Here we define

Δ Φ_{sD}^{F}

as it satisfies

〈 exp {[i Δ Φ_{sD}^{F}]}_{D} 〉 = 1

. On the other hand,

Φ_{sD, sm}^{F}

expresses a resolvable, slowly varying component of the wavefront due to averaged refractive index of the sample. In the homogeneous approximation,

exp [i Φ_{sD, sm}^{F} (x_{s 0}, y_{s 0})] \bar{exp [i Φ_{sD, sm}^{F} (x_{s 0} + p_{s} d_{s}, y_{s 0})]}

in Eqs. (57) and (58) should be only a function of p_sd_s and independent of x_s0 within the range comparable to w_D. As a result,

Φ_{sD, sm}^{F}

has to be expressed in the form of

Φ_{sD, sm}^{F} (x_{s 0}, y_{s 0}) = C^{r} \cdot x_{s 0} + i C^{i},

where C^r and Cⁱ are real numbers that are locally constant around the point (x_s0, y_s0). In fact, Eq. (61) is necessary and sufficient for the condition

\frac{\partial (exp [i Φ_{sD, sm}^{F} (x_{s 0}, y_{s 0})] \bar{exp [i Φ_{sD, sm}^{F} (x_{s 0} + p_{s} d_{s}, y_{s 0})]})}{\partial x_{s 0}} = 0

to be satisfied. We can finally obtain the following forms:

𝒯 (x_{D 0}, y_{D 0}) \approx exp [- 2 {(Φ_{sD, sm}^{F} (x_{s 0}, y_{s 0}))}^{i}],

𝒫 (x_{D 0}, y_{D 0}) \approx p_{s} d_{s} [{(\frac{\partial Φ_{sD, sm}^{F} (x_{s 0}, y_{s 0})}{x_{s 0}})}^{r}],

𝒱 (x_{D 0}, y_{D 0}) \approx | {〈 exp [i Δ Φ_{sD}^{F} (x_{s 0}, y_{s 0})] \bar{exp [i Δ Φ_{sD}^{F} (x_{s 0} + p_{s} d_{s}, y_{s 0})]} 〉}_{D} | .

Here we rewrote C^r and Cⁱ by

{(\partial Φ_{sD, sm}^{F} / x_{s 0})}^{r}

and

{(Φ_{sD, sm}^{F})}^{i}

, where the suffixes r and i denotes real and imaginary parts. Equation (65) is a general form of the normalized visibility for a sample with unresolvable random microstructures with absorption taken into account. As mentioned in previous papers [8, 19], the Fourier transform of Eq. (65) with respect to p_sd_s corresponds to angular distribution of USXAS that is integrated in the y-direction. Noted that a small correction term is necessary in Eq. (64) for a large absolute value of the imaginary part of

Δ Φ_{sD}^{F}

.

3. Numerical calculations

In this section, we show results of numerical calculations for unresolvable edges done on the basis of Eqs. (53), (54), and (55). Similar to in Section 2.2.2, we assume here that the penumbra is sufficiently narrower than the PSF of the detector used.

Figures 3, 4, 5, and 6 show examples of unresolvable edges appearing in the transmittance (𝒯), moiré-phase (𝒫), and normalized-visibility (𝒱) images for λ = 1 Å. The samples used for the numerical calculations are uniform polymethyl methacrylate (PMMA; complex refractive index n_PMMA = 1 − δ_PMMA + iβ_PMMA, where δ_PMMA = 1.722 × 10⁻⁶ and β_PMMA = 1.559 × 10⁻⁹ for λ = 1 Å) with shapes of a cuboid plate (Figs. 3 and 5) and a cylinder (Figs. 4 and 6). We assume that the samples are sufficiently long in the direction parallel to the lines of G1, in which the edges of the samples are aligned as shown in the insets of Figs. 3 (a) and 4 (a), so that the problem is reduced to one dimension. The samples are assumed to be located just upstream of G1, which is a π/2-phase grating made of Au with a pitch (d₁) of 5.3 μm and a duty cycle of 0.5. The source-to-G1 and G1-to-G2 distances are set to be 30 m and 0.1411 m, which correspond to a Talbot order p₁₂ of 0.5, and the distance between G2 and the detector is negligible. The pixel size of the detector was set as 0.1 μm and its PSF was approximated by a Gaussian function.

Fig. 3 Results of numerical calculations for line profiles across the edges of PMMA cuboid plates with several thicknesses (T) in (a) transmittance (𝒯), (b) moiré-phase (𝒫), and (c) normalized-visibility (𝒱) images, and dependences of (d) the maximum value of |𝒫| and (e) the minimum value of 𝒱 on the thickness of the plates. Solid and broken lines are the results with and without the Fresnel diffraction by the samples taken into account.

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Fig. 4 Results of numerical calculations for line profiles across the edges of PMMA cylinders with several diameters (D) in transmittance (𝒯; (a) and (d)), moiré-phase (𝒫; (b) and (e)), and normalized-visibility (𝒱; (c) and (f)) images. Solid and broken lines are the results with and without the Fresnel diffraction by the samples taken into account.

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Fig. 5 Minimum value of the normalized visibility (𝒱) for PMMA cuboid plates is plotted with respect to a universal parameter (U_plate). Here $w_{D} \equiv 2 \sqrt{2 ln 2} σ_{D}$ .

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Fig. 6 Minimum value of the normalized visibility (𝒱) for PMMA cylinders is plotted with v respect to a universal parameter (U_cylinder). Here $w_{D} \equiv 2 \sqrt{2 ln 2} σ_{D}$ . Red lines: w_D changed; blue lines: distance between G1 and G2 (corresponding to p₁₂) changed; green lines: mass density of PMMA changed (but kept uniform). Solid, broken, and dotted lines correspond to diameters (D) of 1, 2, and 5 mm.

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3.1. Cuboid plates

Figure 3 (a), (b), and (c) show line profiles across the edges of PMMA cuboid plates with several thicknesses (T) in the transmittance, moiré-phase, and normalized-visibility images. Here the width w_D of the PSF of the detector, which is represented by the full width at half maximum (FWHM), is fixed at 100 μm. The origin of the horizontal axes corresponds to the position of the edges, and positive and negative sides are the inside and outside of the plates. The broken lines in the figures are the results under the projection approximation of Eq. (13), i.e., for the case where the Fresnel diffraction by the samples is neglected, while the solid lines are the results with the Fresnel diffraction taken into account.

It can be seen that the unresolvable edges generate contrast not only in the moiré-phase image but also the normalized-visibility image. In addition, the effect of the Fresnel diffraction by the samples is negligible except for the transmittance image; this fact shows that the projection approximation is a good approximation to describe the moiré-phase and normalized-visibility images. Figure 3 (d) and (e) show dependences of the extremum value of the moiré phase (|𝒫|_max) and the minimum value of the normalized visibility (𝒱_min) on the thickness T. The periodic behaviors in the figures are due to the periodic change in the phase shift as T increases. Noted that the positions of |𝒫|_max and 𝒱_min are almost the same as the edge position when the absorptions by the samples are negligible (see Fig. 3 (b) and (c)). When absorption becomes large as T increases, the positions of |𝒫|_max and 𝒱_min move inward. This fact means that we cannot determine the exact edge position only from the moiré-phase and the normalized-visibility images.

The reduced normalized visibility shown in Fig. 3 (c) is not straightforwardly explained by the dark-field model, i.e., the model where the intensity of the image generated downstream of G1 is convoluted with the angular distribution of USAXS. When absorption by a plate with a thickness T is negligible, the angular distribution of USAXS for its edge should be proportional to

\frac{{| exp [i δ_{PMMA} T] - 1 |}^{2}}{{(2 π k_{x})}^{2}} .

That is, the angular distribution should be localized very close to k_x = 0. As a result, the visibility of the image of G1 appears to be unchanged after convolution with the angular distribution.

3.2. Cylinders

Figure 4 (a)–(f) show simulated line profiles across the edges of PMMA cylinders with several diameters (D) ((a)–(c): for small diameters; (d)–(f): for large diameters). Here w_D is fixed at 100 μm as well as in Fig. 3. The cylinders also generated contrast at their edges in not only the moiré-phase images ((Fig. 4 (b) and (e)) but also the normalized-visibility images (Fig. 4 (c) and (f)). In addition, the effect of the Fresnel diffraction was seen only on the line profiles in the transmittance images (see solid and broken lines in the figures). However, in contrast to the case of the cuboid plates, |𝒫|_max and 𝒱_min do not behave periodically but change monotonically as D increases.

Figure 4 (e) and (f) show that the positions of |𝒫|_max and 𝒱_min shift inward as D increases. Noted that, for D = 100 μm in Fig. 4 (e) and (f), both the left and right edges contribute to the contrasts; the inward shifting for D from 200 μm on up is an effect of an unresolvable edge.

3.3. Universal parameters

There are parameters to roughly describe the behavior of the minimum normalized visibility 𝒱_min. From a simple consideration, we can find that

U_{plate} = \frac{p_{s} d_{s}}{σ_{D}} (1 - cos \frac{2 π δ_{PMMA} T}{λ})

is a universal parameter for cuboid plates (σ_D: the standard deviation of the Gaussian PSF). Figure 5 shows dependence of 𝒱_min on U_plate for PMMA cuboid plates. Red, black, and green lines in the figure, corresponding to the cases where absorption is negligible, are all on a monotonically decreasing universal line. As absorption increases (see blue lines), the 𝒱_min line shifts downward, but deviation from the universal line is small.

For cylinders

U_{cylinder} = \frac{δ_{PMMA}}{λ} p_{s} d_{s} \sqrt{\frac{D}{σ_{D}}}

is a universal parameter when D ≫ σ_D ≫ p_sd_s. Figure 6 shows dependence of 𝒱_min on U_cylinder for PMMA cylinders. As seen in the figure, all simulated lines were mostly on the same monotonically decreasing line for U_cylinder less than 0.55. Around U_cylinder ≈ 0.55, 𝒱_min takes 0. Noted that, after this turning point, abnormal wrapping occurs in the moiré-phase image, which will be discussed in a separate paper.

Especially when X-ray energy is far from absorption edges of all the elements in a sample, refractive index decrement (δ_PMMA in Eq. (68) for the case of PMMA) is almost proportional to λ². In this case

U_{cylinder} \propto ρ_{electron} λ p_{s} d_{s} \sqrt{\frac{D}{σ_{D}}},

where ρ_electron is the number density of electrons in the sample. Hence, the behavior of 𝒱_min for λ is the same as that for p_sd_s (see solid lines in Fig. 8).

Fig. 7 Experimental results (open circles) of line profiles across edges of three PMMA cylinders with diameters of 3, 6, and 10 mm in (a) transmittance, (b) moiré-phase, and (c) normalized-visibility images. Solid lines were results of numerical calculations based on Eqs. (53), (54), and (55) with the Fresnel diffraction by the samples taken into account.

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Fig. 8 Experimental results (open circles) of |𝒫|_max and 𝒱_min near the edges of three PMMA cylinders with diameters of 3, 6, and 10 mm. Solid lines were results of numerical calculations based on Eqs. (53), (54), and (55) with the Fresnel diffraction by the samples taken into account. (a) and (b): Talbot-order (p_s)-dependence at 25 keV; (c) and (d): X-ray-wavelength-dependence for p_s = 0.5

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4. Experiments

The prediction of the numerical calculation in the previous section was confirmed by experiments using commercially available PMMA cylinder samples. The experiments were performed at the beamline 14C, Photon Factory (PF), KEK, Japan, where a 8 mm (horizontal) × 38 mm (vertical) X-ray beam from a vertical wiggler (source size: 532 μm (horizontal) × 45 μm (vertical)) can be used [51]. We used an X-ray beam monochromatized by a Si(220) double crystal monochromator before going into the experimental hutch of the beamline, which is located about 30 m downstream from the X-ray source. In the experimental hutch, an X-ray Talbot interferometer with two gratings (G1 and G2) was constructed. The lines of the two gratings were aligned in the horizontal direction. Three 5.3-μm-pitch π/2-phase Au gratings designed for 17.5, 25.0, and 40.0 keV were used as G1, while a 5.3-μm-pitch Au absorption grating about 40 μm thick was used as G2.

We used an X-ray image detector (Spectral Instruments), in which a charge coupled device (CCD) is connected to a 40-μm-thick Gd₂O₂S (GOS) scintillator screen with a 2:1 fiber coupling. The effective pixel size of the detector was 18 μm.

PMMA cylinders with diameters of 3, 6, and 10 mm were used as the samples and located at 165 mm upstream of G1 through the experiments. The axes of the cylinders were aligned in the horizontal direction. The transmittance, moiré-phase, and normalized-visibility images were obtained by the equal-sampling five-step fringe scanning (G2 was scanned with respect to G1). Open circles in Fig. 7 show experimentally obtained line profiles across the edges of the samples in the three images at 25 keV. Solid lines are results of numerical calculations based on Eqs. (53), (54), and (55) with the Fresnel diffraction by the samples taken into account. The point spread function (PSF) of the detector that was used for the numerical calculations was determined from an experimentally obtained modulation transfer function (MTF) with Coltman correction. Error bars were given in the same manner as Ref. [52]. In all the three images, the results of the numerical calculations agree well with the experimental results.

Figure 8 shows change in |𝒫|_max and 𝒱_min around the edges of the PMMA cylinders. Figure 8 (a) and (b) show the dependences of |𝒫|_max and 𝒱_min on p_s (defined in Eq. (10)) at 25 keV. We changed p_s by changing the distance between G1 and G2. Figure 8 (c) and (d) show the dependences of |𝒫|_max and 𝒱_min on X-ray wavelength λ for p_s = 0.5. All the experimental results (open circles) were well explained by the results of the numerical calculations (solid lines). Slight deviations of open circles from the solid lines might be attributed to digitized error due to a finite pixel size (18 μm) and small error in the experimentally obtained PSF.

5. Summary

We obtained general formulae for the three images (transmittance, moiré-phase, and normalized-visibility images) obtained in X-ray Talbot interferometry as described in Section 2.2. On the basis of the formulae, we performed numerical calculations and showed that an unresolvable sharp edge generates contrast in not only a differential-phase image but also a normalized-visibility image. The reduction in the normalized visibility is not straightforwardly explained by the ‘dark-field’ model, i.e., not attributed to USAXS.

The results presented in this paper can also be applied to neutron grating interferometry [53–62]. Noted that it might be possible to discriminate the reduction in visibility caused by an unresolvable edge from that by USAXS. One way to discriminate between them is to move the sample slightly in the direction perpendicular to the grating lines. As shown in Figs. 3, 4, and 7, the normalized visibility around an unresolvable edge changes with sensitivity to the position of the edge: even if the displacement of the sample is smaller than the width of the PSF of the detector, it can be detected as a change in normalized visibility. On the other hand, when the visibility is reduced by USAXS, no change in normalized visibility should be expected with the displacement of the sample. Another way is to change experimental parameters (p_sd_s, λ, and w_D). As shown in Figs. 5 and 6, normalized visibility can be changed by these parameters differently from the case of USAXS (see Ref. [8]).

Acknowledgments

The experiments were performed at Photon Factory (PF), Japan, under a proposal approved ( 2013G077). We thank Dr. K. Hyodo in KEK, Dr. M.P. Olbinado, and Mr. G. Murakami in Tohoku University for their help before the experiments.

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Effects of unresolvable edges in grating-based X-ray differential phase imaging

Abstract

1. Introduction

2. Theoretical background

2.1. Conventional simplified theoretical description for X-ray grating interferometry

2.1.1. For a point X-ray source

2.1.2. For a finite size X-ray source

2.1.3. Image acquisition

2.2. More rigorous description

2.2.1. General expression

2.2.2. Small penumbra approximation

2.2.3. No resonance approximation

2.2.4. Homogeneous approximation

3. Numerical calculations

3.1. Cuboid plates

3.2. Cylinders

3.3. Universal parameters

4. Experiments

5. Summary

Acknowledgments

References and links

Cited By

Figures (8)

Equations (69)

Optics Express