† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 61101175, 61571305, and 61227802).

A fast and simple method to extract phase-contrast images from interferograms is proposed, and its effectiveness is demonstrated through simulation and experiment. For x-ray differential phase contrast imaging, a strong attenuation signal acts as an overwhelming background intensity that obscures the weak phase signal so that no obvious phase-gradient information is detectable in the raw image. By subtracting one interferogram from another, chosen at particular intervals, the phase signal can be isolated and magnified.

X-ray phase-sensitive technique is an important extension of x-ray radiographic imaging that is widely used in medicine, nondestructive testing, and security. Besides conventional absorption images, phase-sensitive imaging can offer two other images: phase distribution and dark field distribution.^{[1–4]} Various phase-sensitive techniques have been proposed and developed over the past decade.^{[5–21]} Among them, the differential phase contrast (DPC) imaging can be used in a broad range of applications due to its low requirements for source coherence, mechanical stability and detector resolution.^{[6]}

Without specimen, interferograms of phase gratings consist of many regular fringes. An object introduced in the light path distorts the fringes, and the resulting variations in the intensity pattern encode the wavefront phase. The obtaining of the phase distribution of the sample requires a phase retrieval procedure that calculates the phase image from multiple intensity measurements. In the visible light region, Fourier-transform and phase-stepping algorithms are common phase retrieval methods.^{[8,9]} Naturally, those techniques were introduced in DPC imaging system and yielded excellent results^{[8,10,11]} No less than three images are needed for a phase-stepping algorithm; usually, the number of intensity measurements is in a range of 5 ∼ 11.^{[12]} Multiple exposures cause excess exposure time and increased x-ray absorption, which are disadvantageous in medical and biological applications. Bennett *et al.* proposed a fast and simple algorithm that requires no movement of the grating.^{[10]} However, it requires individual harmonic spectrum, resulting in the loss of high spatial frequency.

In this paper, we deduce the intensity distribution in a detecting plane for a finite-size x-ray source. The formula of intensity shows that the analysis grating parameters and x-ray source spatial coherence have the greatest effects on phase signals. The attenuation signals have more contributions than the phase shift signals in intensity image. Thus, usable phase information requires a map of phase retrieval. We propose a fast and simple method based on the intensity formula to extract the phase-gradient signals from the raw data without losing the high spatial frequency. Relative numerical calculations and experimental results demonstrate our method effectiveness.

We consider a common x-ray interferometer with a finite-size x-ray source as shown in Fig. *R* and *l* denote the distances between the source grating (*G*_{0}) and phase grating (*G*_{1}), which is downstream of the sample, and between *G*_{1} and the analysis grating (*G*_{2}), respectively. First, we consider only a monochromatic point x-ray source of wavelength *λ*, then we extend the result to a multi-slit x-ray source.

The complex transmission function *T*(*x*) of *G*_{1} can be expressed with a Fourier series as

*a*

_{n}is the amplitude of the

*n*-th harmonic and

*p*

_{1}is the period of the grating. At a distance

*l*downstream of

*G*

_{1}, the complex amplitude of the propagating field is given by

*G*

_{1}, where

*λ*is the x-ray wavelength and

*M*= (

*R*+

*l*)/

*R*is the magnification. For

*R*≫

*l*,

*M*≈ 1; hence, the phase factor of Eq. (

When a sample is placed just before *G*_{1}, *G*_{1} will split the incident light carrying the complex amplitude of the sample into different diffraction orders. As a consequence, the amplitude transmission function of the sample *A*(*x,y*)e^{iϕ(x,y)} is shifted by *ns* along the *x*-axis, such that

*s*=

*lλ*/

*p*

_{1}. Therefore, from Eq. (

*I*(

*x,y;l*) =

*U*(

*x,y;l*)

*U** (

*x,y;l*) is given by

Assuming that the shearing amount *ns* is sufficiently small,^{[5]} equations (

According to the van Cittert–Zernike theorem, the intensity distribution of a source will smoothen the interferogram *I*(*x,y;l*), so in the detecting plane, the intensity *I*′(*x,y;l*) is calculated as^{[15]}

*I*(

_{s}*x,y*) denotes the source intensity, which is the convolution of rectangle function rect(

*x/a*) with a sum of delta functions,

The multi-slit source has periodicity *p*_{0} and the slit width *α*, as illustrated in Fig.

The final intensity distribution on the detector is determined by taking into account the effect of the third grating *G*_{2} with periodicity *p*_{2}. Assuming that *G*_{1} and *G*_{2} are parallel, the intensity signal detected by one pixel, according to the model in Fig.

*G*

_{2}is

*χ*,

*G*

_{2}has 100% absorption efficiency, the aperture width of one periodicity is

*D*, and one pixel can collect

*k*periodicities of x-ray fringes.

The signal *I*″ would be a very complex expression if we implement Eq. (*G*_{1} at the Talbot distance, we can obtain a compact expression of *I*″. For example, when the duty cycle of *G*_{1} is 0.5 and the imaging distance of *G*_{1} satisfies

*I*(

*x,y;l*) in Eq. (

*θ*denotes the phase shift of

*G*

_{1}, and

*p*

_{2}=

*M*

*p*

_{1}/2. We assume that

*A*(

*x,y*)e

^{iϕ(x,y)}varies slowly with

*x*, so the effect of convolving

*A*(

*x,y*) with

*ϕ*(

*x,y*) in Eq. (

^{[15]}That is to say, we neglect the scatting contribution to the intensity distribution

*I*(

*x,y;l*). The same assumption in Eq. (

*x*= sin

*πx*/

*πx*. At the same time,

*G*

_{1}is located at special positon such that

Now we can discuss the contributions of *A*(*x,y*) and *ϕ*(*x,y*) to the intensity signal *I*″(*x,y*). The value of *γ* determines the contribution of *A*(*x,y*) and small apertures of *G*_{0} and *G*_{2} can increase the occupancy rate of *ϕ*(*x,y*) in *I*″(*x,y*). However, *a* and *D* cannot be very small. In general, we choose parameters such that half the incident light passes though *G*_{0} and *G*_{2}, which means that *a*/*p*_{0} = *D*/*p*_{2} = 0.5 and *γ* ≈ 0.4. Thus, the phase information amplitude only contributes 15% of the measured intensity, and most energy is attributed to the attenuation term *Γ*(*x,y*). The phase contrast signals, which provide more structural information, are consequently hard to detect for the attenuation signal.

We estimate the values of *ϕ*(*x,y*) and *A*^{2}(*x,y*) from the following equations:

*δ*is the real part of the complex refractive index and

*μ*is the quality factor of the attenuation. For light material,

*δ*is typically on the order of 10

^{−6}–10

^{−7}for 20 keV–30 keV x-rays, so that

*ϕ*(

*x,y*) is usually quite small(a few arc seconds), except is areas near edges and boundaries. We assume again that

*ϕ*(

*x,y*) varies slowly varying spatially and is sufficiently small so that equation (

^{[15]}

If the strong background signal *Γ*(*x,y*) can be eliminated from Eq. (*ϕ*(*x,y*). A simple approach can delete this background signal by using only two raw images. A first interferogram *α*, and a second *α* + *π*. Subtracting

*k*

_{1}and

*k*

_{2}are constants that do not affect image contrast. Since

*ϕ*(

*x,y*) of an object is generally a spatially random distribution, we can reasonably expect

*k*

_{2}can be estimated and removed from the

*I*″. On the boundaries, due to the large gradient, equation (

*I*‴ can still give structural information. Our simple method removes the background signals and isolates phase signals.

In this section, we show the results of numerical calculations of one carbon ball on the basis of Eqs. (

Figures *G*_{1} placed at different positions, and figures *λ* = 0.041 nm, 1-mm carbon ball diameter, *δ* = 5.2 × 10^{−6}, *μ* = 0.584 cm^{−1}, *p*_{1} = 5.6 μm, *l* = 105 mm, *R* = 1470 mm, *p*_{0} = 42 μm, *p*_{2} = 3 μm, and 27-μm pixels. Except in the line profiles in Figs.

A fresh pig foot purchased from a supermarket was imaged using a non-absorption grating imaging system described in Ref. [14]. Two raw data images and a phase gradient image are shown in Figs.

The x-ray source is a common x-ray tube with no filter, at a 70-kV acceleration voltage, and with a 2-mA anode current. The distance between sample and detector is 105 mm. The phase grating is made of silicon, has a 5.6-μm period, a 0.5-duty ratio, and a 40-μm depth, corresponding to a *π* phase shift for 31-keV x-rays. The scintillator is directly coupled with an ANDOR 2048×2048-pixel (13.5 μm/pixel) CCD camera through a fiber optical tape, with 2× demagnification.

We present intensity formulae for x-ray Talbot interferometry with a partial spatial coherent source. On the basis of these formulae, we find that the structural parameters of the source and analyzer gratings greatly affect the phase-gradient signal intensity. For *G*_{0} and *G*_{2}, a larger aperture in one periodicity dramatically reduces the amplitude of phase information to the extent that the phase-gradient image is difficult to detect in raw data. We propose a simple approach to extracting the phase image from raw signals and demonstrate its effectiveness through numeral calculation and experiment. Compared with the existing stepping methods using no less than three interferograms, our method is simpler and much faster.

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