Effect of insufficient temporal coherence on visibility contrast in X-ray grating interferometry

Wataru Yashiro; Daiji Noda; Kentaro Kajiwara

doi:10.1364/OE.26.001012

1. Introduction

For the last decade, X-ray grating interferometry [1 62], which is an X-ray phase-contrast imaging technique [63–65], has attracted increasing attention because it works even with continuous-spectrum X-rays from a low-brilliance laboratory source. One of the advantages of this sort of interferometry is its multi-modality: it can provide three independent images i.e., transmittance, differential-phase, and visibility-contrast [7, 12, 14–17, 19–21, 26–34, 36–44, 46–60] images. The third modality was first reported by Pfeiffer et al. [7] as a dark-field contrast, and the visibility contrast due to unresolvable random structures (typically of the order of µm), arising from ultra-small-angle X-ray scattering, was quantitatively explained [19].

So far, several other causes of visibility contrast have been reported for monochromatic [36, 55, 57, 58] and continuous-spectrum [56, 59] X-rays. For monochromatic X-rays, it was found that the effects of more than first-order differential phase [36, 58] and unresolvable sharp edges [55] can cause visibility contrast. For continuous-spectrum X-rays, even a uniform sample can cause the visibility contrast due to the so-called beam-hardening effect [56, 59].

In this paper, we report another cause of the visibility contrast, i.e., the effect of insufficient temporal coherence. This effect occurs when continuous-spectrum X-rays are largely refracted. In the next section, the effect of insufficient temporal coherence on the visibility contrast is theoretically described. In Section 3, we show the results of an experiment using a low-temporal-coherence X-ray beam that demonstrates the effect occurs. In Section 4, we will discuss the potential applications of the effect.

2. Effect of temporal coherence

A typical experimental setup of X-ray grating interferometry is shown in Fig. 1. X-ray grating interferometry uses a self-imaging phenomenon called the Talbot effect [66, 67]. In the figure, a grating (G1) is illuminated by partially spatially coherent X-rays, so that self-images of G1 are generated downstream of it. When a sample is located in front of or behind the grating, it refracts the X-rays, so that the self-image becomes deformed. From this deformation, an image called a differential-phase image can be obtained, and the deformation is often detected as a deformation of moiré fringes arising from overlaying the second grating (G2; an absorption grating). Besides the differential phase image, two other images, called transmittance and visibility-contrast images, can be formed from the average intensity and the visibility of the moiré fringes.

Fig. 1 Typical experimental setup of millisecond-order X-ray tomography (side view).

Download Full Size | PDF

To discuss the effect of insufficient temporal coherence on the visibility of the self-image, which determines the visibility of the moiré fringes, we briefly review how the self-image is described mathematically. We consider a case where an X-ray grating interferometer with two gratings G1 and G2 (an X-ray Talbot interferometer) is constructed around the optical axis (the z-axis), as shown in Fig. 1. Here, G1 is a one-dimensional grating with a pitch of d₁, the lines of which are aligned parallel and perpendicular to the y and z axes, respectively. First, we assume that G1 is illuminated by perfectly spatially coherent X-rays from an infinitely small source at a distance of R₁ upstream from G1. The electric field E_G1 for monochromatic X-rays immediately behind G1 is expressed as

E_{G 1} (x_{1}, y_{1}) = E_{1} \sum_{n} T_{n} (x_{1})

= E_{1} \sum_{n} a_{n} \exp (2 π i \frac{n x_{1}}{d_{1}}),

where the x₁−y₁ plane is defined on G1 perpendicular to the z-axis, E₁ is the electric field just in front of G1, T_n is the amplitude (complex) transmission function of G1 for the nth diffraction order, given by

T_{n} (x_{1}) \equiv a_{n} \exp (2 π i \frac{n x_{1}}{d_{1}})

and a_n is the nth Fourier coefficient of the complex transmission function of G1. The electric field E_self of the self-image generated at a distance of z₁₂ downstream of G1 for monochromatic X-rays with a wavelength λ can be expressed in the paraxial approximation as

E_{self} (x_{2}, y_{2}) \approx E_{2} \sum_{n} {a^{'}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}),

where

E_{2} \equiv E_{1} \exp [2 π i \frac{z_{12}}{λ}] \cdot \frac{R_{1}}{R_{2}},

{a^{'}}_{n} \equiv a_{n} \exp (- n^{2} π i \frac{λ z_{12}}{d_{1} d_{2}}),

R₂ ≡ R₁ + z₁₂, and d₂ = d₁(R₂/R₁), corresponding to the pitch of the self-image of G1 with a compression ratio of 1. Here, the x₂−y₂ plane is defined on the self-image parallel to the x₁−y₁ plane such that it satisfies

(x_{2}, y_{2}) = \frac{R_{2}}{R_{1}} (x_{1}, y_{1})

, and we have assumed that d₁ ≫ λ for the Talbot effect to occur. The intensity I_self,0 of the self-image for the infinitely small X-ray source is thus expressed as

I_{self, 0} (x_{2}, y_{2}) \approx I_{2} \sum_{n} b_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}),

where

I_{2} \equiv {| E_{1} |}^{2} \cdot \frac{{R_{2}}^{2}}{{R_{1}}^{2}},

and

b_{n} = \sum_{n^{'}} a_{n^{'} + n} a_{n^{'}}^{*} \exp [\frac{- π i ({(n^{'} + n)}^{2} - {n^{'}}^{2}) z_{12}}{d_{1} d_{2}}] .

For considering the effect of partial coherence of X-rays, we interpret Eq. (4) in geometrical wave optics as follows. We rewrite Eq. (4) into

E_{self} (x_{2}, y_{2}) = E_{2} \sum_{n} a_{n} \exp (n^{2} π i \frac{λ z_{12}}{d_{1} d_{2}}) \exp (2 π i \frac{n x_{1}}{d_{1}}) \exp (- n^{2} 2 π i \frac{λ z_{12}}{d_{1} d_{2}}),

= E_{2} \sum_{n} a_{n} \exp (n^{2} π i \frac{λ z_{12}}{d_{1} d_{2}}) \exp (2 π i \frac{n (x_{1} - n \frac{λ z_{12}}{d_{2}})}{d_{1}}),

= E_{2} \sum_{n} a_{n} \exp (n π i \frac{Δ θ_{n} z_{12}}{d_{1}}) \exp (2 π i \frac{n (x_{1} - Δ θ_{n} z_{12})}{d_{1}}),

= E_{2} \sum_{n} a_{n} \exp (2 π i \frac{n Δ x_{n}}{2 d_{1}}) \exp (2 π i \frac{n (x_{1} - Δ x_{n})}{d_{1}})

Here, we used x₂/d₂ = x₁/d₁, and defined Δθ_n and Δx_n by

Δ θ_{n} \equiv \frac{n λ}{d_{2}},

Δ x_{n} \equiv z_{12} Δ θ_{n} .

Equation (13) can be more simply rewritten by

E_{self} (x_{2}, y_{2}) = E_{2} \sum_{n} T_{n} (x_{1} - Δ x_{n}) \exp (2 π i \frac{Δ l_{2}}{λ}),

where

Δ l_{n} \equiv \frac{n Δ x_{n} λ}{2 d_{1}} .

Noted that Δθ_n corresponds to an angle geometrically given by z₁₂ and Δx_n, as shown in Fig. 2(a).

Fig. 2 Schematic illustration of geometrical-optics interpretation of the Talbot effect for monochromatic X-rays (a) without and (b) with a sample. Without a sample, the electric field at a distance z₁₂ downstream of G1 is formed from the interference of the waves passing through the points P_n on G1 (n = 0, ±1, ±2, ⋯), where P_n is separated at a distance of Δx_n from P₀. Here, Δl_n is the optical path difference between the nth and 0th order, and Δθ_n is geometrically defined by Δx_n and z₁₂. With a sample, an additional optical path cτ_s,_n is introduced into the nth order by the refraction of the sample.

Download Full Size | PDF

Equation (16) allows for the Talbot effect to be incorporated in the interpretation of geometrical wave optics: the nth-order diffracted wave propagates along the path passing through the point P_n shown in Fig. 2(a) and E_self is formed from the interference of all the diffraction orders. In fact, T_n (x₁ − Δx_n) corresponds to the complex transmission function of the nth order at P_n, and the optical path difference between the nth and the 0th orders, passing through the points P_n and P₀, respectively, is approximately given by

(\sqrt{{R_{1}}^{2} + Δ {x_{n}}^{2}} + \sqrt{{z_{12}}^{2} + Δ {x_{n}}^{2}}) - (R_{1} + z_{12}) \approx \frac{Δ {x_{n}}^{2}}{2} (\frac{1}{R_{1}} + \frac{1}{z_{12}}),

= \frac{Δ {x_{n}}^{2}}{2} \cdot \frac{R_{1} + z_{12}}{R_{1}} \cdot \frac{1}{z_{12}},

= \frac{Δ x_{n}}{2 d_{1}} \cdot \frac{d_{2} Δ x_{n}}{z_{12}},

= \frac{Δ x_{n}}{2 d_{1}} \cdot (n λ),

which corresponds to Δl_n in Eq. (16). We used R₁, z₁₂ ≫ Δx_n in Eqs. (18) and R₂/R₁ = d₂/d₁ from Eq. (19) to Eq. (20).

Note that, using the Talbot order p defined such that it satisfies

z_{12} = \frac{p d_{1} d_{2}}{λ},

Δx_n is simply given by

Δ x_{n} = n p d_{1} .

For convenience in the following description, we define a″_n as

{a^{″}}_{n} \equiv a_{n} \exp (- 2 π i \frac{n Δ x_{n}}{d_{1}}),

by which E_self can be expressed as

E_{self} (x_{2}, y_{2}) = E_{2} \sum_{n} {a^{″}}_{n} \exp (2 π i \frac{Δ l_{n}}{λ}) \exp (2 π i \frac{n x_{n}}{d_{1}}) .

Next, we consider a case where G1 is illuminated by X-rays with a continuous spectrum from a chaotic X-ray source with a finite size. By defining the X₀−Y₀ coordinate on a plane including the X-ray source such that it satisfies (X₀, Y₀) = (x₂, y₂), we can express the intensity I_self of the self-image as

I_{self} (x_{2}, y_{2}) \approx \int I_{self, 0} (x_{2}, y_{2}; X_{0}, Y_{0}; ν) d X_{0} d Y_{0} d ν,

where ν ≡ c/λ (c : the velocity of light) and I_self,0 (x₂, y₂; X₀,Y₀; ν) is the intensity of the self-image at (x₂, y₂) generated by a point source at (X₀, Y₀). From Eq. (7), the intensity can be written in the following form:

I_{self} (x_{2}, y_{2}) \approx \sum_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}) \int μ_{n} (ν) b_{n} (ν) d ν,

where

μ_{n} (ν) \equiv \int I_{2} (x_{2}, y_{2}; X_{0}, Y_{0}; ν) \exp (2 π i \frac{X_{0} Δ x_{n}}{R_{1} λ}) d X_{0} d Y_{0},

Here, I₂(x₂, y₂; X₀, Y₀; ν) is the contribution from a point source at (X₀, Y₀) to the intensity at (x₂, y₂) without G1. For simplicity, we assume that the spectrum of X-rays is independent of (X₀, Y₀) (the condition of the so-called cross-spectral purity [68] is satisfied). In this case, I₂(x₂, y₂; X₀, Y₀; ν) can be expressed by

I_{2} (x_{2}, y_{2}; X_{0}, Y_{0}, ν) = S (ν) {\hat{I}}_{2} (x_{2}, y_{2}; X_{0}, Y_{0}),

where S(ν) is the spectrum of the X-ray source and

{\hat{I}}_{2} (x_{2}, y_{2}; X_{0}, Y_{0})

is the normalized intensity that satisfies

\int {\hat{I}}_{2} (x_{2}, y_{2}; X_{0}, Y_{0}) d X_{0} d Y_{0} = 1

. Substituting Eq. (29) into Eq. (28)I_self(x₂, y₂) can be expressed as

I_{self} (x_{2}, y_{2}) \approx \sum_{n} {\hat{μ}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}) \int S (ν) b_{n} (ν) d ν,

where

{\hat{μ}}_{n} \equiv \int {\hat{I}}_{2} (x_{2}, y_{2}; X_{0}, Y_{0}) \exp (2 π i \frac{X_{0} Δ x_{n}}{R_{1} λ}) d X_{0} d Y_{0},

corresponding to a complex coherence factor [68, 69] at two point separated by Δx_n on G1. Equation (31) is equivalent to the van-Cittert-Zernike theorem for the X-ray source with a finite size. Note that, because Δx_n is proportional to λ,

{\hat{μ}}_{n}

is independent of ν. In addition, in the paraxial approximation,

{\hat{μ}}_{n}

is independent of (x₂, y₂) when the size of the X-ray source is sufficiently smaller than R₁.

Equation (30) can be related to the Wiener-Khintchine’s theorem as follows. From Eqs. (7) and (25), b_n (ν) can be rewritten as

b_{n} (ν) = \sum_{n^{'}} {a^{″}}_{n^{'} + n} (ν) {a^{″ *}}_{n^{'}} (ν) \exp [2 π i ν (τ_{n^{'} + n} - τ_{n^{'}})],

where

τ_{n} \equiv Δ l_{n} / c .

Substituting Eq. (32) into Eq. (30), we obtain

I_{self} (x_{2}, y_{2}) \approx \sum_{n} {\hat{μ}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}})

\times \sum_{n^{'}} \int S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν) \exp [2 π i ν (τ_{n^{'} + n} - τ_{n^{'}})] d ν,

and, we finally obtain

I_{self} (x_{2}, y_{2}) \approx \sum_{n} {\hat{μ}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}) [\sum_{n^{'}} {\hat{γ}}_{n^{'} + n, n^{'}} (Δ τ_{n^{'} + n, n^{'}}) S_{n^{'} + n, n^{'}}],

where

Δ τ_{n^{'} + n, n^{'}} \equiv τ_{n^{'} + n} - τ_{n^{'}},

{\hat{γ}}_{n^{'} + n, n^{'}} (τ) \equiv \frac{ℱ T_{ν} [S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν)]}{\int S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν) d ν},

and

S_{n^{'} + n, n^{'}} \equiv \int S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν) d ν .

Here,

ℱ T_{ν}

express the Fourier transform with respect to ν.

The factor ${\hat{γ}}_{n^{'} + n, n^{'}} (Δ τ_{n^{'} + n, n^{'}})$ in Eq. (36) corresponds to the Wiener-Khintchine’s theorem, i.e., it represents the effect of temporal coherence for the n′ + n and n′ orders, and ${\hat{μ}}_{n} {\hat{γ}}_{n^{'} + n, n^{'}} (Δ τ_{n^{'} + n, n^{'}})$ corresponds to the complex coherence factor for the n′ + n and n′ orders. It should be noted that, for monochromatic (more precisely, quasi-monochromatic [69]) X-rays with a frequency of ν₀ illuminating G1 with a perfect temporal coherence, S(ν) = S(ν₀)δ(ν − ν₀) and

{\hat{γ}}_{n^{'} + n, n^{'}} (Δ τ_{n^{'} + n, n^{'}}) = \exp [2 π i ν_{0} (Δ τ_{n^{'} + n, n^{'}})] .

As a result, we only have to take the effect of the spatial coherence of the X-rays on G1 into account:

I_{self} (x_{2}, y_{2}) \approx S (ν_{0}) \sum_{n} {\hat{μ}}_{n} b_{n} (ν_{0}) \exp (2 π i \frac{n x_{2}}{d_{2}}) .

When the width of the spectrum of

S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν)

increases,

| {\hat{γ}}_{n^{'} + n, n^{'}} |

decreases, and the visibility of the self-image therefore decreases. However, this reduction in visibility is not large even for a broad spectrum because, for the Talbot effect to occur, d₁ ≫ λ, and

τ_{n^{'} + n} - τ_{n^{'}}

for a small n is very small. This means that an X-ray beam with a wide energy bandwidth is available for the Talbot effect to be observed.

The visibility $V$ of the self-image, which is normally obtained with a fringe scanning method [70–72] or a Fourier transform method [73], is given by the 0th- and 1st-order Fourier components of I_self(x₂, y₂):

V = \frac{{\hat{μ}}_{1} [\sum_{n^{'}} {\hat{γ}}_{n^{'} + 1, n^{'}} (Δ τ_{n^{'} + 1, n^{'}}) S_{n^{'} + 1, n^{'}}]}{{\hat{μ}}_{0} [\sum_{n^{'}} {\hat{γ}}_{n^{'}, n^{'}} (0) S_{n^{'}, n^{'}}]},

= \frac{{\hat{μ}}_{1} [\sum_{n^{'}} {\hat{γ}}_{n^{'} + 1, n^{'}} (Δ τ_{n^{'} + 1, n^{'}}) S_{n^{'} + 1, n^{'}}]}{\sum_{n^{'}} S_{n^{'}, n^{'}}} .

Here, we used

{\hat{μ}}_{0} = 1

and

{\hat{γ}}_{n^{'}, n^{'}} (0) = 1

. Especially when a π/2 phase grating with a duty cycle of 0.5 is used, Eq. (44) can be further simplified into

V = \frac{{\hat{μ}}_{1} ℜ [{\hat{γ}}_{1, 0} (Δ τ_{1, 0}) S_{1, 0}]}{\sum_{n^{'}} S_{n^{'}, n^{'}}},

where ℜ[z] represents the real part of a complex number z.

Finally, we consider the effect of a sample. We define R_s to be the distance of the sample from the X-ray source and (x_s, y_s) to be a coordinate on the sample such that $(x_{s}, y_{s}) = \frac{R_{s}}{R_{1}} (x_{1}, y_{1})$ , as shown in Fig. 1. For X-rays passing through the sample, T_n (x₁ − Δx_n) in Eq. (16) is replaced with

T_{n} (x_{1} - Δ x_{n}) \exp (2 π i {\hat{Φ}}_{n} (x_{1}, y_{1}, ν),

where

{\hat{Φ}}_{n} (x_{1}, y_{1}, ν) \equiv \hat{Φ} (x_{s} - n p_{s} d_{1}, y_{s}, ν),

= {\begin{array}{l} \hat{Φ} (\frac{R_{s}}{R_{1}} (x_{1} - Δ x_{n}), \frac{R_{s}}{R_{1}} y_{1}, ν) & (0 < R_{s} \leq R_{1}) \\ \hat{Φ} (\frac{R_{s}}{R_{1}} x_{1} - \frac{R_{2} - R_{s}}{R_{2} - R_{1}} Δ x_{n}, \frac{R_{s}}{R_{1}} y_{1}, ν) & (R_{1} \leq R_{s} \leq R_{2}) \end{array}

Here, p_s is the effective Talbot order for the sample, defined by

p_{s} \equiv {\begin{array}{l} p \frac{R_{s}}{R_{1}} & (0 < R_{s} \leq R_{1}) \\ p \frac{R_{2} - R_{s}}{R_{2} - R_{1}} & (R_{1} \leq R_{s} \leq R_{2}) \end{array},

\exp [2 π i \hat{Φ} (x_{s}, y_{s}, ν)]

represents the transmittance of the sample, and the real and imaginary parts of

\hat{Φ} (x_{s}, y_{s}, ν)

represent the effects of the phase shift and absorption caused by the sample, respectively. From Eq. (46)a″_n in Eq. (24) is replaced by

{a^{″}}_{n} \exp (2 π i {\hat{Φ}}_{n} (x_{1}, y_{1}, ν)) .

For simplicity, we neglect the imaginary part of

\hat{Φ} (x_{s}, y_{s}, ν)

for a while. Expressing

\hat{Φ} (x_{s}, y_{s}, ν)

by

{\hat{Φ}}_{n} (x_{s}, y_{s}, ν) = {\hat{Φ}}_{n} (x_{s}, y_{s}, ν_{0}) (\frac{ν}{ν_{0}} + C_{n} (x_{s}, y_{s}, ν)),

where ν₀ is a frequency around the center of the spectrum of

\sum_{n} {\hat{μ}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}) S (ν) b_{n} (ν)

in Eq. (30), Eq. (35) can be rewritten in the following form:

I_{self} (x_{2}, y_{2}) \approx \sum_{n} {\hat{μ}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}})

\times \sum_{n^{'}} \int S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*}

\times \exp [2 π i ({\hat{Φ}}_{n^{'} + n} (ν_{0}) C_{n^{'} + n} (ν) - {\hat{Φ}}_{n^{'}} (ν_{0}) C_{n^{'}} (ν))]

\times \exp [2 π i ν (Δ τ_{n^{'} + n, n^{'}} + Δ τ_{s, n^{'} + n, n^{'}})] d ν,

where

τ_{s, n} = \frac{{\hat{Φ}}_{n} (ν_{0})}{ν_{0}},

Δ τ_{s, n^{'} + n, n^{'}} \equiv τ_{s, n^{'} + n} - τ_{s, n^{'}} .

Here, we omitted the dependence of

{\hat{Φ}}_{n} (ν)

and C_n(ν) on the coordinate (x_s, y_s). Hence, we obtain the intensity of the self-image with the sample:

I_{self} (x_{2}, y_{2}) \approx \sum_{n} {\hat{μ}}_{n} \exp (2 π i \frac{n x_{2}}{d_{2}}) [\sum_{n^{'}} {\hat{γ}}^{'}_{n^{'} + n, n^{'}} (Δ τ_{n^{'} + n, n^{'}} + Δ τ_{s, n^{'} + n, n^{'}}) {S^{'}}_{n^{'} + n, n^{'}}],

where

{\hat{γ}}^{'}_{n^{'} + n, n^{'}} (τ) \equiv \frac{ℱ T_{ν} [S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν) \exp [2 π i ({\hat{Φ}}_{n^{'} + n} (ν_{0}) C_{n^{'} + n} (ν) - {\hat{Φ}}_{n^{'}} (ν_{0}) C_{n^{'}} (ν))]]}{\int S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν) \exp [2 π i ({\hat{Φ}}_{n^{'} + n} (ν_{0}) C_{n^{'} + n} (ν) - {\hat{Φ}}_{n^{'}} (ν_{0}) C_{n^{'}} (ν))] d ν},

and

{S^{'}}_{n^{'} + n, n^{'}} \equiv \int S (ν) {a^{″}}_{n^{'} + n} (ν) {a^{″}}_{n^{'}}^{*} (ν) \exp [2 π i ({\hat{Φ}}_{n^{'} + n} (ν_{0}) C_{n^{'} + n} (ν) - {\hat{Φ}}_{n^{'}} (ν_{0}) C_{n^{'}} (ν))] d ν .

The visibility

V_{s}

with the sample is thus expressed as

V_{s} = \frac{{\hat{μ}}_{1} [\sum_{n^{'}} {\hat{γ}}^{'}_{n^{'} + 1, n^{'}} (Δ τ_{n^{'} + 1, n^{'}} + Δ τ_{s, n^{'} + 1, n^{'}}) {S^{'}}_{n^{'} + 1, n^{'}}]}{\sum_{n^{'}} {S^{'}}_{n^{'}, n^{'}}} .

The normalized visibility

V_{s} / V

, which is the visibility with the sample divided by that without the sample, is given by

\frac{V_{s}}{V} = \frac{\sum_{n^{'}} S_{n^{'}, n^{'}}}{\sum_{n^{'}} {S^{'}}_{n^{'}, n^{'}}} \cdot \frac{\sum_{n^{'}} {\hat{γ}}^{'}_{n^{'} + 1, n^{'}} (Δ τ_{n^{'} + 1, n^{'}} + Δ τ_{s, n^{'} + 1, n^{'}}) {S^{'}}_{n^{'} + 1, n^{'}}}{\sum_{n^{'}} {\hat{γ}}^{'}_{n^{'} + 1, n^{'}} (Δ τ_{n^{'} + 1, n^{'}}) S_{n^{'} + 1, n^{'}}}

For a π/2 phase grating, we have

V_{s} = \frac{{\hat{μ}}_{1} [{S^{'}}_{1, 0} {\hat{γ}}^{'}_{1, 0} (Δ τ_{1, 0} + Δ τ_{s, 1, 0}) + {S^{'}}_{0, - 1} {\hat{γ}}^{'}_{0, - 1} (Δ τ_{0, - 1} + Δ τ_{s, 0, - 1})]}{\sum_{n^{'}} {S^{'}}_{n^{'}, n^{'}}},

\frac{V_{s}}{V} = \frac{\sum_{n^{'}} S_{n^{'}, n^{'}}}{\sum_{n^{'}} {S^{'}}_{n^{'}, n^{'}}} \cdot \frac{{S^{'}}_{1, 0} {\hat{γ}}^{'}_{1, 0} (Δ τ_{0, 1} + Δ τ_{s, 0, 1}) + {S^{'}}_{0, - 1} {\hat{γ}}^{'}_{0, - 1} (Δ τ_{- 1, 1} + Δ τ_{s, - 1, 1})}{S_{1, 0} {\hat{γ}}_{1, 0} (Δ τ_{0, 1}) + S_{0, - 1} {\hat{γ}}_{0, - 1} (Δ τ_{- 1, 1})} .

Comparing Eqs. (61) and (63) with Eqs. (44) and (45), we find that $S_{n^{'} + n, n^{'}}$ and ${\hat{γ}}_{n^{'} + n, n^{'}}$ in Eqs. (44) and (45) are replaced by ${S^{'}}_{n^{'} + n, n^{'}}$ and ${\hat{γ}}^{'}_{n^{'} + n, n^{'}}$ in Eqs. (61) and (63) and that an additional term $Δ τ_{s, n^{'} + 1, n^{'}}$ is added to $Δ τ_{n^{'} + 1, n^{'}}$ in the argument of ${\hat{γ}}^{'}_{n^{'} + n, n^{'}}$ . The replacements of $S_{n^{'} + n, n^{'}}$ and ${\hat{γ}}_{n^{'} + n, n^{'}}$ into ${S^{'}}_{n^{'} + n, n^{'}}$ and ${\hat{γ}}^{'}_{n^{'} + n, n^{'}}$ are due to the change in spectral phase when the sample is inserted, while the additional term $Δ τ_{s, n^{'} + n, n^{'}}$ comes from the additional path difference cτ_s,_n introduced when X-rays are refracted, as shown in Fig. 2(b). The change in the spectrum only slightly affects the temporal coherence, or rather makes the temporal coherence length longer for a large ${\hat{Φ}}_{n^{'} + n} (ν_{0}) C_{n^{'} + n} (ν) - {\hat{Φ}}_{n^{'}} (ν_{0}) C_{n^{'}}$ in Eq. (59), so that it may not reduce the visibility of the self-image. On the other hand, the term $Δ τ_{s, n^{'}, 1}$ can reduce the visibility because it makes the temporal coherence length that is necessary for the Talbot effect to occur longer. Thus, a large refraction by a sample can cause a reduction in the visibility of the self-image due to insufficient temporal coherence.

Note that, when G2 overlays the self-image to generate moiré fringes, the spectrum behind G2 changes. As a result, S(ν) in Eqs. (62) and (64) should be replaced with

S (ν) c_{- 1} (ν),

where c_n(ν) is the nth-order Fourier component of the intensity transmission function of G2. Furthermore, when we further take into consideration the absorptions by the air, the sample, the substrates of G1 and G2, and other optical components such as filters, together with the detection efficiency of the detector,

{\hat{Φ}}_{n} (ν)

is replaced with

ℜ [{\hat{Φ}}_{n} (ν)]

and S(ν) is replaced with

S (ν) c_{- 1} α (ν) ϵ (ν) \exp [- 4 π J [{\hat{Φ}}_{n} (ν)]],

where α(ν) represents the absorptions, ϵ(ν) is the detection efficiency, and

J [z]

represents the imaginary part of a complex number z.

3. Experiment

The experimental setup by which we observed the effect of insufficient temporal coherence on the visibility contrast is shown in Fig. 3. We constructed an X-ray grating interferometer with two gratings G1 and G2 (an X-ray Talbot interferometer) at BL28B2, SPring-8, Japan, where a white synchrotron X-ray beam from a bending magnet source is available. The size (standard deviation) of the X-ray source is 100 µm (horizontal) × 12.2 µm (vertical) [74]. A gold π/2-phase grating with a pitch of 5.3 µm for an X-ray energy of 25 keV was used as G1, and it was located 48.5 m downstream from the X-ray source. The lines of G1 were aligned in the horizontal direction in order to use the spatial coherence in the vertical direction, which is higher than that in the horizontal direction. A gold absorption grating with a pitch of 5.3 µm and a thickness of 70 µm was located at a distance of 283 mm downstream of G1, which corresponds to a Talbot order p of 0.5 for the energy of 25 keV. Both gratings were fabricated on 200-µm-thick Si substrates. An X-ray image detector consisting of a 40-µm-thick Ce:Gd₃Al₂Ga₃O12 (GAGG) single crystal scintillator screen, lenses, and a CMOS camera (Photoron FASTCAM MiniAX100) with an effective pixel size of 5.2 µm was located just behind G2. The full width at half maximum (FWHM) of the line spread function of the detector, which mainly determined the spatial resolution of the interferometer, was estimated to be 20 µm from the experimentally obtained modulation transfer function.

Fig. 3 Experimental setup for observing the effect of insufficient temporal coherence (side view).

Download Full Size | PDF

An acrylonitrile butadiene styrene (ABS) cylinder with a diameter of 40 mm, height of 20 mm, and density of 1.18 g/cm³ was used as the sample, and the X-ray beam was incident on its top flat surface at a small absolute value of glancing angle θ_in, as shown in Fig. 3, so as to introduce a sufficiently large Δτ_s,1,0 and Δτ_s,0,−1 in Eq. (63) (a large differential phase $2 π \partial \hat{Φ} / \partial x_{s}$ ) and heavily reduce the visibility of the self-image. The sample was located just in front of or/146 mm downstream of G1, corresponding to effective Talbot orders p_s (defined in Eq. (49)) of 0.5 and 0.24 for the sample, and the glancing angle of the X-ray beam to the top flat surface of the cylinder was changed while the axis of the cylinder was kept in the vertical plane including the optical axis.

4. Results

The three images in Fig. 4 are examples of 4(b) the transmittance, 4(c) moiré-phase, and 4(d) normalized visibility images in an area consisting of 600 × 400 pixels (in the square in 4(a)) obtained for the moiré image with the sample located in front of G1 (p_s = 0.5). Here, θ_in was fixed at −0.95°. A Fourier transform method was used to obtain the three images from moiré images with and without the sample. Each of the moiré images was captured with an exposure time of 25 µs. The middle area in the three images corresponds to the top flat surface with a small |θ_in| (a large constant value of $| 2 π \partial \hat{Φ} / \partial x_{s} |$ ). It can be seen in Fig. 4(d) that the normalized visibility is reduced by a large $| 2 π \partial \hat{Φ} / \partial x_{s} |$ .

Fig. 4 Examples of (b) transmittance, (c) moiré-phase (proportional to $p_{s} d_{1} \partial \hat{Φ} / \partial x_{s}$ ), and (d) normalized visibility images at a glancing angle θ_in of −0.95° in an area of 600 × 400 pixels (the square in Fig. 4(a)). Gray scales: (b)0-1.2, (c)-5.78-0.5, (c)0-1.2.

Download Full Size | PDF

The filled circles in Fig. 5 show the experimentally obtained θ_in-dependences of 5(a) the unwrapped moiré-phase and 5(b) normalized visibility averaged along the horizontal middle lines passing through the center of the top surface (see the dashed lines in Figs. 4(c) and 4(d)) for p_s = 0.5. In Fig. 5(b), the normalized visibility drastically changes depending on θ_in. This behavior of the normalized visibility can be qualitatively explained by the insufficient temporal coherence: a smaller |θ_in| (a larger $| 2 π \partial \hat{Φ} / \partial x_{s} |$ ) causes a longer cτ_s,1 in Fig. 2 (b), which makes the temporal coherence length for the Talbot effect to occur longer, and hence, for a given temporal coherence, the visibility of the self-image decreases. It should be noted that, because the thickness of the sample (20 mm/cos(θ_in )) was almost constant in this range of θ_in, the reduction in visibility can not be explained by the beam hardening effect reported in a previous paper [56] or unresolvable random structures. In addition, the reduced visibility is not due to the total external reflection by the top flat surface, because the visibility decreases even for a negative θ_in. Furthermore, the observed reduced visibility is not due to the effect of unresolvable sharp edges, which occurs even for monochromatic X-rays as reported in a previous paper [55], since $| 2 π \partial \hat{Φ} / \partial x_{s} |$ is constant (resolvable) within the spatial resolution of the detector.

Fig. 5 θ_in-dependences of (a) the unwrapped moiré-phase and (b) normalized visibility averaged along the horizontal middle lines passing through the center of the top surface (the dashed lines in Fig. 4(c) and 4(d)) for p_s = 0.5.

Download Full Size | PDF

The solid lines in Figs. 5(a) and 5(b) are results of the simulation based on the theoretical formulation described in Section 2. In the simulation, we used the on-axis spectrum of the incident X-ray beam calculated by SPECTRA [75]. In addition, we took into account the absorptions by the sample, the two gratings and their substrates, and the air, and assumed that the number of scintillation photons detected by the CMOS camera is proportional to the energy of X-rays absorbed by the scintillator screen, but that the K-fluorescent X-rays from Gd in the screen escape. We also assumed a constant background level for the captured moiré images for parasitic X-ray scattering. The simulation results are in good agreement with the experimentally obtained ones, which that our formulation for the normalized visibility is correct even for a large indicates $| 2 π \partial \hat{Φ} / \partial x_{s} |$ .

Figure 6 shows the θ_in-dependences of 6(a) the unwrapped moiré-phase and 6(b) normalized visibility averaged along the middle lines for p_s = 0.24. Similar to the result for p_s = 0.5, the normalized visibility drastically changes depending on θ_in, but the width of the region where the reduced visibility is observed is about half of that for p_s = 0.5. This narrower width can be simply explained from Fig. 2(b) or Eqs. (47), (49), (56), (57), and (61): because the additional path difference cτ_s,_n due to the refraction by the sample is proportional to p_s, $| 2 π \partial \hat{Φ} / \partial x_{s} |$ at |θ_in| for p_s = 0.5 induces almost the same effect on the visibility as about twice of $| 2 π \partial \hat{Φ} / \partial x_{s} |$ (i.e., the differential phase at half of |θ_in|) for p_s = 0.24. The simulation results for p_s = 0./24 ((solid curves in Figs. 6(a) and 6(b))) also matched the experimental ones. Thus, insufficient temporal coherence well explained the experimentally observed reduction in the visibility.

Fig. 6 θ_in-dependences of (a) the unwrapped moiré-phase and (b) normalized visibility averaged along the horizontal middle lines passing through the center of the top surface for p_s = 0.24.

Download Full Size | PDF

5. Discussion

As described in Section 4, we experimentally showed that insufficient temporal coherence reduces the normalized visibility of the moiré fringes obtained in X-ray grating interferometry. We explained the reduced visibility in terms of an additional temporal coherence length cτ_s,_n (see Fig. 2(b)) that is necessary for the Talbot effect to occur. In fact, it can also be explained qualitatively in terms of spectral dispersion. The angle of deflection caused by refraction by a sample is inversely proportional to the frequency ν of X-rays. A large angle of deflection drastically shifts a large moiré phase, which is also inversely proportional to ν, and even phase wrapping in the ν space occurs. As a result, the visibility of the moiré fringes integrated in the ν space decreases. The experimental result for an effective Talbot order p_s of 0.24 (see Fig. 6(b)) can also be qualitatively explained by the ν-dependence of the moiré-phase shift: because a smaller p_s induces a smaller moiré-phase shift, a larger refraction is necessary to observe the reduced visibility. We formulated in this paper the relation between the reduced visibility and temporal coherence and showed that the reduced visibility can be even quantitatively explained. This formulation may lead to a method to quantitatively determine the temporal coherence length of X-rays.

It may also be possible to perform X-ray imaging using the reduced visibility for a large refraction. In X-ray grating interferometry, it is normally assumed that the moiré-phase shift ΔP by a sample is linearly proportional to the differential phase $2 π \partial \hat{Φ} / \partial x_{s}$ , but, in a previous paper [62], we showed a criterion that defines the range within which this linearity fails:

| Δ P | > \frac{π}{2} \frac{ν_{0}}{Δ ν},

where Δν is the band width of the spectrum detected by the detector. This means that quantitative phase imaging, which is essential for quantitative tomographic reconstruction, is not possible for a sample giving a large |ΔP|. By using the

| 2 π \partial \hat{Φ} / \partial x_{s} |

-dependence of not only the moiré-phase shift but also the reduced visibility, as shown in Figs. 5 and 6, it may be possible to quantitatively determine a large differential phase even when a continuous-spectrum X-ray source with a wide bandwidth is used.

6. Conclusion

We reported on a cause of reduced visibility in X-ray grating interferometry, the effect of insufficient temporal coherence, which can be observed when continuous-spectrum X-rays are used. We theoretically showed that insufficient temporal coherence reduces the visibility of the self-image in X-ray grating interferometry (Section 2). The results of an experiment using a white X-ray beam and an ABS cylinder with a diameter of 40 mm and a height of 20 mm as a sample were consistent with our theoretical description and showed that the reduced visibility evident in Figs. 4–6 is caused by insufficient temporal coherence when X-rays are largely refracted, and not due to the other known causes such as unresolvable random structures, the beam-hardening effect, or total external reflection (Section 3). Thus, we concluded that insufficient temporal coherence is a cause of the reduced visibility in X-ray grating interferometry.

In the experiment, we used a white synchrotron X-ray beam, but this phenomenon can also occur in the X-ray grating interferometry using a laboratory source with a continuous spectrum for clinical investigations and non-destructive inspections and in the neutron grating interferometry [76–91] with a continuous spectrum integrated to increase the neutron flux used for obtaining images. We have to carefully interpret the moiré-phase shift and visibility contrast especially for quantitative phase imaging in tomographic reconstruction.

Acknowledgments

The experiment was performed at SPring-8 (proposal numbers: 2017A). This research was partly supported by a grant from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number JP15H03590).

References and links

1. C. David, B. Nöhammer, and H. H. Solak, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002). [CrossRef]

2. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003). [CrossRef]

3. T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005). [CrossRef]

4. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005). [CrossRef] [PubMed]

5. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006). [CrossRef]

6. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006). [CrossRef]

7. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mat. 7, 134–137 (2008). [CrossRef]

8. W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of Capturing a Phase Image using Cone-beam X-ray Talbot Interferometry,” J. Opt. Soc. Am. A 25, 2025–2039 (2008). [CrossRef]

9. A. Momose, W. Yashiro, and Y. Takeda, “Sensitivity of X-ray Phase Imaging Based on Talbot Interferometry,” Jpn. J. Appl. Phys. 47, 8077–8080 (2008). [CrossRef]

10. A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-Based X-ray Phase Imaging Using Multiline X-ray Source,” Jpn. J. Appl. Phys. 48, 076512 (2009). [CrossRef]

11. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17, 12540–12545 (2009). [CrossRef] [PubMed]

12. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009). [CrossRef]

13. W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, “Hard-X-Ray Phase-Difference Microscopy Using a Fresnel Zone Plate and a Transmission Grating,” Phys. Rev. Lett. 103, 180801 (2009). [CrossRef] [PubMed]

14. M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for X-ray radiology: Phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010). [CrossRef] [PubMed]

15. A. F. Stein, J. Ilavsky, R. Kopace, E. E. Bennett, and H. Wen, “Selective imaging of nano-particle contrast agents by a single-shot x-ray diffraction technique,” Opt. Express 18, 13271–13278 (2010). [CrossRef] [PubMed]

16. G.-H. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970 (2010). [CrossRef] [PubMed]

17. T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol. 55, 3317–3323 (2010). [CrossRef] [PubMed]

18. V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instr. 81, 073709 (2010). [CrossRef]

19. W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010). [CrossRef] [PubMed]

20. M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010). [CrossRef] [PubMed]

21. T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010). [CrossRef]

22. A. Momose, W. Yashiro, and Y. Takeda, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, 2010).

23. A. Momose, W. Yashiro, S. Harasse, and H. Kuwabara, “Four-dimensional X-ray phase tomography with Talbot interferometry and white synchrotron radiation: dynamic observation of a living worm,” Opt. Express 19, 8423–8432 (2011). [CrossRef] [PubMed]

24. H. Kuwabara, W. Yashiro, S. Harasse, H. Mizutani, and A. Momose, “Hard-X-ray Phase-Difference Microscopy with a Low-Brilliance Laboratory X-ray Source,” Appl. Phys. Express 4, 062502 (2011). [CrossRef]

25. A. Momose, H. Kuwabara, and W. Yashiro, “X-ray Phase Imaging Using Lau Effect,” Appl. Phys. Express 4, 066603 (2011). [CrossRef]

26. S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50, 4310–4319 (2011). [CrossRef] [PubMed]

27. V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011). [CrossRef]

28. W. Yashiro, S. Harasse, K. Kawabata, H. Kuwabara, T. Yamazaki, and A. Momose, “Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry,” Phys. Rev. B 84, 094106 (2011). [CrossRef]

29. M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011). [CrossRef]

30. W. Han, J. A. Eichholz, X. Cheng, and G. Wang, “A Theoretical Framework of X-ray Dark-Field Tomograpy,” Siam. J. Appl. Math. 71, 1557–1577 (2011). [CrossRef]

31. P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett. 108, 048101 (2012). [CrossRef] [PubMed]

32. W. Cong, F. Pfeiffer, M. Bech, and G. Wang, “X-ray dark-field imaging modeling,” J. Opt. Soc. Am. A 29, 908–912 (2012). [CrossRef]

33. G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012). [CrossRef] [PubMed]

34. A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based X-ray interferometry,” EPL 99, 48001 (2012). [CrossRef]

35. S. Matsuyama, H. Yokoyama, R. Fukui, Y. Kohmura, K. Tamasaku, M. Yabashi, W. Yashiro, A. Momose, T. Ishikawa, and K. Yamauchi, “Wavefront measurement for a hard-X-ray nanobeam using single-grating interferometry,” Opt. Express 20, 24977 (2012). [CrossRef] [PubMed]

36. Y. Yang and X. Tang, “The second-order differential phase contrast and its retrieval for imaging with x-ray Talbot interferometry,” Med. Phys. 39, 7237–7253 (2012). [CrossRef] [PubMed]

37. V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective X-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012). [CrossRef]

38. T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013). [CrossRef] [PubMed]

39. A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61268 (2013). [CrossRef] [PubMed]

40. T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013). [CrossRef] [PubMed]

41. F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 23, 236–242 (2013). [CrossRef]

42. F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013). [CrossRef]

43. V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by X-ray scatter dark field imaging with a grating interferometer,” NDT&E International 58, 64–71 (2013). [CrossRef]

44. F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. 29, 478–486 (2013). [CrossRef] [PubMed]

45. M. P. Olbinado, P. Vagovič, W. Yashiro, and A. Momose, “Demonstration of Stroboscopic X-ray Talbot Interferometry Using Polychromatic Synchrotron and Laboratory X-ray Sources,” Appl. Phys. Express 6, 096601 (2013). [CrossRef]

46. Z. Wang and M. Stampanoni, “Quantitative x-ray radiography using grating interferometry: a feasibility study,” Phys. Med. Biol. 58, 6815–6826 (2013). [CrossRef] [PubMed]

47. M. Endrizzi, P. C. Diemoz, T. P. Millard, J. L. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard X-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104, 024106 (2014). [CrossRef]

48. F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of X-Ray Vector Radiography to Bone Micro-Architecture,” Sci. Rep. 4, 3695 (2014). [CrossRef] [PubMed]

49. A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, K. T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014). [CrossRef]

50. W. Yashiro and A. Momose, “Grazing-incidence ultra-small-angle X-ray scattering imaging with X-ray transmission gratings: A feasibility studey,” Jpn. J. Appl. Phys. 53, 05FH04 (2014).

51. T. Lauridsen, E. M. Lauridsen, and R. Feidenhans’l, “Mapping misoriented fibers using X-ray dark field tomography,” Appl. Phys. A 115, 741–745 (2014). [CrossRef]

52. P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple Scattering Tomography,” Phys. Rev. Lett. 113, 020801 (2014). [CrossRef] [PubMed]

53. P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014). [CrossRef]

54. F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in X-ray dark-field tomography,” PNAS 111, 12699–12704 (2014). [CrossRef] [PubMed]

55. W. Yashiro and A. Momose, “Effects of unresolvable edges in grating-based X-ray differential phase imaging,” Opt. Express 23, 9233–9251 (2015). [CrossRef] [PubMed]

56. W. Yashiro, P. Vagovič, and A. Momose, “Effect of beam hardening on a visibility-contrast image obtained by X-ray grating interferometry,” Opt. Express 23, 23462–23471 (2015). [CrossRef] [PubMed]

57. J. Wolf, J. I. Sperl, F. Schaff, M. Schüttler, A. Yaroshenko, I. Zanette, J. Herzen, and F. Pfeiffer, “Lens-term- and edge-effect in X-ray grating interferometry,” Biol. Opt. Express 6, 4812–4824 (2015). [CrossRef]

58. T. Koenig, M. Zuber, B. Trimborn, T. Farago, P. Meyer, D. Kunka, F. Albrecht, S. Kreuer, T. Volk, and M. Fiederle, “On the origin and nature of the grating interferometric dark-field contrast obtained with low-brilliance x-ray sources,” Phys. Med. Biol. 61, 3427 (2016). [CrossRef] [PubMed]

59. G. Pelzer, G. Anton, F. Horn, J. Rieger, A. Ritter, J. Wandner, T. Weber, and T. Michel, “A beam hardening and dispersion correction for x-ray dark-field radiography,” Med. Phys. 43, 2774–2779 (2016). [CrossRef] [PubMed]

60. Y. Bao, Q. Shao, R. Hu, S. Wang, K. Gao, Y. Wang, Y. Tian, and P. Zhu, “Effect of particle-size selectivity on quantitative X-ray dark-field computed tomography using a grating interferometer,” Radiat. Phys. Chem. 137260–263 (2017). [CrossRef]

61. W. Yashiro, D. Noda, and K. Kajiwara, “Sub-10-ms X-ray tomography using a grating interferometer,” Appl. Phys. Express 10, 052501 (2017). [CrossRef]

62. W. Yashiro, R. Ueda, K. Kajiwara, D. Noda, and H. Kudo, “Sub-10-ms X-ray tomography using a grating interferometer,” Jpn. J. Appl. Phys. 56, 112503 (2017). [CrossRef]

63. R. Fitzgerald, “Phase-sensitive x-ray imaging,” Phys. Today 53, 23–26 (2000). [CrossRef]

64. A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005). [CrossRef]

65. K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59, 1–99 (2010). [CrossRef]

66. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

67. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics XXVII (Elsevier, 1989). [CrossRef]

68. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University, 1995). [CrossRef]

69. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999). [CrossRef]

70. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974). [CrossRef] [PubMed]

71. H. Schreiber and J. H. Bruning, Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 2007), Chap. 14

72. E. Hack and J. Burke, “Invited Review Article: Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011). [CrossRef] [PubMed]

73. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-Transform Method of Fringe-Pattern Analysis for Computer-based Topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]

74. http://www.spring8.or.jp/en/about_us/whats_sp8/facilities/accelerators/storage_ring/

75. T. Tanaka and H. Kitamura, “SPECTRA: a synchrotron radiation calculation code,” J. Synchrotron Rad. 8, 1221–1228 (2001). [CrossRef]

76. C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008). [CrossRef] [PubMed]

77. C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008). [CrossRef]

78. M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron Dark-Field Tomography,” Phys. Rev. Lett. 101, 123902 (2008). [CrossRef] [PubMed]

79. S. W. Lee, D. S. Hussey, D. L. Jacobson, C. M. Sim, and M. Arif, “Development of the grating phase neutron interferometer at a monochromatic beam line,” Nuc. Inst. Meth. A 605, 16–20 (2009). [CrossRef]

80. M. Strobl, A. Hilger, N. Kardjilov, O. Ebrahimi, S. Keil, and I. Manke, “Differential phase contrast and dark field neutron imaging,” Nuc. Inst. Meth. A 605, 9–12 (2009). [CrossRef]

81. I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010). [CrossRef] [PubMed]

82. S. W. Lee, Y. K. Jun, and O. Y. Kwon, “A Neutron Dark-field Imaging Experiment with a Neutron Grating Interferometer at a Thermal Neutron Beam Line at HANARO,” J. Korean Phys. Soc. 58, 730–734 (2011). [CrossRef]

83. C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013). [CrossRef]

84. S. W. Lee, Y. Zhou, T. Zhou, M. Jiang, J. Kim, C. W. Ahn, and A. K. Louis, “Visibility studies of grating-based neutron phase contrast and dark-field imaging by using partial coherence theory,” J. Kor. Phys. Soc. 63, 2093–2097 (2013). [CrossRef]

85. J. Kim, S. W. Lee, and G. Cho, “Visibility evaluation of a neutron grating interferometer operated with a polychromatic thermal neutron beam,” Nucl. Instrum. Meth. A 746, 26–32 (2014). [CrossRef]

86. F. Yang, F. Prade, M. Griffa, I. Jerjen, C. Di Bella, J. Herzen, A. Sarapata, F. Pfeiffer, and P. Lura, “Dark-field X-ray imaging of unsaturated water transport in porous materials,” Appl. Phys. Lett. 105, 154105 (2014). [CrossRef]

87. M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4, 7243 (2014). [CrossRef] [PubMed]

88. B. Betz, R. P. Harti, M. Strobl, J. Hovind, A. Kaestner, E. Lehmann, H. Van Swygenhoven, and C. Grünzweig, “Quantification of the sensitivity range in neutron dark-field imaging,” Rev. Sci. Instrum. 86, 123704 (2015). [CrossRef]

89. F. Prade, A. Yaroshenko, J. Herzen, and F. Pfeiffer, “Short-range order in mesoscale systems probed by X-ray grating interferometry,” EPL 112, 68002 (2015). [CrossRef]

90. T. Reimann, S. Mühlbauer, M. Horisberger, B. Betz, Peter Böni, and M. Schulz, “The new neutron grating interferometer at the ANTARES beamline: design, principles and applications,” J. Appl. Crystr. 49, 1488–1500 (2016). [CrossRef]

91. Y. Seki, T. Shinohara, J. D. Parker, W. Yashiro, A. Momose, K. Kato, H. Kato, M. Sadeghilaridjani, Y. Otake, and Y. Kiyanagi, “Development of Multi-colored Neutron Talbot-Lau Interferometer with Absorption Grating Fabricated by Imprinting Method of Metallic Glass,” J. Phys. Soc. Jpn. 86, 044001 (2017). [CrossRef]

Effect of insufficient temporal coherence on visibility contrast in X-ray grating interferometry

Abstract

1. Introduction

2. Effect of temporal coherence

3. Experiment

4. Results

5. Discussion

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (67)

Optics Express