Quantitative theory of X-ray interferometers based on dual phase grating: fringe period and visibility

Aimin Yan; Xizeng Wu; Hong Liu

doi:10.1364/OE.26.023142

1. Introduction

In recent years, X-ray grating interferometry has attracted a lot of efforts in research and development [1–20]. In this phase contrast imaging technique, one usually utilizes a phase grating as a beam splitter to divide the incident beam into different diffraction orders. The interference between the diffracted orders generates intensity fringes. The sample in the beam path will generate three modes of image contrast, namely the attenuation, fringe phase shift proportional to sample phase gradient, and the fringe-visibility loss induced by small-angle x-ray scattering. Hence, this differential X-ray phase-contrast imaging technique has many potential applications in medical imaging and material science. In these applications it is desirable to utilize common imaging detectors, whose pixels are of a few tens micrometers. However, to increase the grating interferometer’s sensitivity, one usually must use fine-pitch phase gratings of periods as small as few micrometers. Hence it is important to develop methods to resolve the fringes generated by such fine-pitch phase gratings with detectors of pixels as large as a few tens micrometers.

One way to resolve the fringe patterns is to place an absorbing grating as the analyzer in front of the detector. The analyzer grating has a period that is similar in size to the fringe period to be resolved. Even the size of a detector pixel is much larger than the fringe’s periods, one can still retrieve the sample-induced fringe shift by scanning the absorbing grating through the fringe periods. This technique is also called the phase stepping [1–4]. The use of the absorbing grating for fringe-shift measurement has some disadvantages as well. The absorbing grating blocks more than half information-carrying x-ray, and consequently significantly reduces radiation dose efficiency of the imaging system. The phase stepping procedure itself is especially cumbersome for tomography, as multiple phase steppings are required for each of angular views in tomography. Moreover, fabrications of absorbing gratings with large areas or use with high-energy x-ray are costly and challenging [1–4]. Another way to resolve high-resolution fringe patterns is to adopt the so-called inverse geometry setup. In this setup the phase grating is placed very close to the source such that the fringe period will be geometrically magnified by a factor of tens to hundreds [14–16]. This approach gets rid of the need of the absorbing grating analyzer, but often results in large footprints of the interferometer [14–16].

Quite recently a dual phase grating technique for x-ray interferometry has been demonstrated [21]. Figure 1 shows a typical setup of this new technique. Different from conventional Talbot X-ray interferometry, which uses only a single phase grating, the new technique employs two phase gratings G₁ and G₂ as the beam splitters, as is shown in Fig. 1. For high-sensitivity the phase gratings G₁ and G₂ are made of small periods. The experiments demonstrated several important advantages of this new technique. Firstly, different from the inverse geometry Talbot interferometry, with the new technique one can generate intensity fringes resolvable by a common imaging detector of few tens micrometers, while the system’s foot print is kept compact. Secondly, the fringe visibility and the periodicity of resolved fringe are adjustable by simply varying the G₁-G₂ spacing [21].

Fig. 1 Schematic of an x-ray dual-phase grating interferometer with a micro-spot X-ray source.

Download Full Size | PDF

However, dual phase grating interferometry lacks a sufficient quantitative analysis for optimization of the experimental geometry. For example, the experimental work of [21] did not provide any formulas to predict fringe period and fringe visibility as functions of setup geometry and x-ray energy.

For design optimization of the dual phase grating interferometry, one needs to be able to predict the fringe periodicity, visibility as functions of the G₁-G₂ spacing through closed-forms expressions for both π and half-π gratings. In this work we set out to fill this gap. Formulated in Wigner distribution in the phase space, we developed a quantitative theory of dual phase grating interferometry. We derived closed-form formulas for fringe periodicity, and fringe visibility. In this quantitative theory, we also incorporate the partial spatial coherence and detector pixel re-binning effects. In section 2, we introduce the Wigner distribution formalism, and outline how to derive the intensity fringe patterns at the detector entrance. The derivation details are presented in the Appendix. In section 3, we discuss the formation mechanism of the resolved fringe patterns. We derived the formulas of the fringe period and fringe visibility, as useful tools for design optimization. In section 4, in addition to summarizing the main results of this work, we also point out the limitations in the scope of this paper.

2. Materials and methods

Consider a dual phase grating interferometer as schematically depicted in Fig. 1. In this setup two binary phase gratings with duty cycle of 0.5 are employed. The first phase grating G₁ has a period p₁ and a phase shift ∆ϕ₁, and the second phase grating G₂ is of period p₂ and phase shift of ∆ϕ₂. We model the phase gratings by their Fourier expansions as:

\begin{array}{l} G_{1} (x) = \sum_{l \in ℤ} a_{l} \exp [i 2 π \frac{m x}{p_{1}}], \\ G_{2} (x) = \sum_{r \in ℤ} b_{r} \exp [i 2 π \frac{n x}{p_{2}}], \end{array}

where a_l denotes the Fourier coefficients of the G₁ grating and their values depend on the grating period p₁ and phase shift ∆ϕ₁ of the grating. Similarly, b_r is the Fourier coefficients of the G₂ grating. In Fig. 1, S marks the source and D the detector. To trace x-ray wavefront evolution, one in principle can trace the Fresnel diffraction from the source downstream, through the two gratings, reaching to the detector entrance. The x-ray irradiance at detector entrance is the squared modulus of the evolved wavefront. To facilitate x-ray irradiance calculation, in this work we adopt the Wigner function formalism to compute the x-ray irradiance at the detector entrance. The Wigner function is defined as the Fourier transform of the mutual intensity of X-ray wavefront [22–24]:

W (x, u) = \int J (x + \frac{Δ}{2}, x - \frac{Δ}{2}) \cdot \exp [- i 2 π u Δ] d Δ,

where J denotes the mutual intensity of x-ray wave front. Wigner function is a function in the phase space. It represents the joint x-ray wave’s probability densities in space coordinate x and spatial frequency u. According to Eq. (2), the Wigner function at G₁-plane, which is a distance R₁ from the source, is given by:

W_{R_{1}} (x, u; y) = \int J_{in} (x + \frac{Δ}{2}, x - \frac{Δ}{2}) \cdot G_{1} (x + \frac{Δ}{2}) \cdot G_{1}^{*} (x - \frac{Δ}{2}) \cdot \exp [- i 2 π u Δ] d Δ,

where J_in(x + ∆/2, x − ∆/2) is the mutual intensity of impinging x-ray wavefront at G₁-plane. For sake of clarity, we mark the Wigner function at a given plane by its distance from the source. As wave propagates from G₁ to the entrance of G₂ grating, the Wigner function undergoes a shearing in phase space [22–24]

W_{R_{1} + R_{2} - 0} (x, u; y) = W_{R_{1}} (x - λ R_{2} u, u; y) .

Implementing the Wigner function evolution from the G₂ grating to the detector entrance in the same way as is described in Eqs. (3) and (4), we can obtain Wigner function

W_{R_{1} + R_{2} + R_{4}} (x, u; y)

at the detector entrance. The x-ray irradiance

I_{R_{1} + R_{2} + R_{4}} (x, y)

at detector entrance can then be found by integrating the Wigner function

W_{R_{1} + R_{2} + R_{4}} (x, u; y)

over the spatial frequencies [22–24]:

I_{R_{1} + R_{2} + R_{4}} (x, y) = \int W_{R_{1} + R_{2} + R_{4}} (x, u; y) d u .

Using x-ray irradiance

I_{R_{1} + R_{2} + R_{4}} (x, y)

, we can investigate the fringe periods and fringe visibility.

3. Results

Following the method described above in section 2, we found the x-ray irradiance $I_{R_{1} + R_{2} + R_{4}} (x, y)$ at detector entrance as follows:

\begin{array}{l} I (x, y) = \frac{I_{in}}{M_{5}^{2}} \cdot \sum_{l, n, r, s \in ℤ} μ (- \frac{λ R_{4}}{M_{1}} \cdot [\frac{l}{M_{5} p_{1}} + \frac{r}{M_{D} p_{2}}] - l \frac{λ R_{2}}{M_{1} p_{1}}) \cdot a_{l + n} a_{n}^{*} b_{r + s} b_{s}^{*} \times \\ \times \exp [- i 2 π (\frac{l (l + 2 n) λ R_{2}}{2 M_{1} p_{1}^{2}})] \cdot \exp [- i 2 π (\frac{(l + 2 n) λ R_{4}}{2 M_{1} p_{1}}) \cdot (\frac{l}{M_{5} p_{1}} + \frac{r}{M_{D} p_{2}})] \times \\ \times \exp [- i 2 π (\frac{(r + 2 s) λ R_{4}}{2 p_{2}}) \cdot (\frac{l}{M_{5} p_{1}} + \frac{r}{M_{D} p_{2}})] \cdot \exp [i 2 π (\frac{l}{M_{5} p_{1}} + \frac{r}{M_{D} p_{2}}) x] . \end{array}

The derivation of Eq. (6) is tedious, interesting readers are referred to Appendix for details. In Eq. (6)R₁ denotes the source-to-G₁ distance, R₂ the spacing between G₁ and G₂ gratings, and R₄ is the G₂-to-detector distance (Fig. 1). In this equation several magnification factors are defined:

M_{1} = \frac{R_{1} + R_{2}}{R_{1}}; M_{D} = \frac{R_{1} + R_{2} + R_{4}}{R_{1} + R_{2}}; M_{5} = \frac{R_{1} + R_{2} + R_{4}}{R_{1}} .

While Eq. (6) looks complicated, but its physical meaning can be understood as follows. Since the gratings G₁ and G₂ serve as beam splitters in the setting, the incident x-ray beam is first splitted by G₁ into various tilting waves, with their amplitudes proportional the G₁’s Fourier coefficients a_l of Eq. (1). Undergoing Fresnel diffraction over a distance R₂, these tilting waves are splitted again by G₂ with amplitudes proportional to the G₂’s Fourier coefficients b_r. Undergoing Fresnel diffraction over a distance R₄ toward the detector, these tilting waves interfere with each other and result in various diffracted orders which are indexed by pairs of integers (l, r) and represented by the terms such as exp[i2π[l/(M₅p₁) + r/(M_Dp₂)] x]. Among the coefficients associated with a diffraction order, the constant phase factor such as

\exp [- i 2 π [l (l + 2 n) λ R_{2} / (2 M_{1} p_{1}^{2})]]

or exp[−i2π[(r + 2s)λR₄/(2p₂)] · [l/(M₅p₁) + r/(M_Dp₂)]] reflects the phase factor accrued in Fresnel diffractions over distance R₂ or R₄ respectively. Moreover, for a given diffraction order (l, r) in Eq. (6), the summation over integers n and s of the product of the grating Fourier coefficients and those constant Fresnel phase factors determines the fringe visibility of the diffracted order (l, r). In Eq. (6) the term µ (−λR₄/M₁ ·[l/(M₅p₁) + r/(M_Dp₂)] –l(λR₂)/M₁p₁)) is the coherence degree associated with the diffracted order (l, r). It represents the reduction of the visibility owing to partial spatial coherence of x-ray illumination. Equation (6), although looked formidable, provides a comprehensive framework for quantitative analysis of the dual phase grating interferometers.

3.1. Resolvable intensity fringes

It is important to figure out how to adjust the periods of the resolvable fringes. From Eq. (6), the resulting fringes from the interference are represented by the terms such as exp[i2π[l/(M₅p₁) + r/(M_Dp₂)]x], where the diffracted orders are indexed by two integers (r, l). These fringes have periods of (M₅M_Dp₁p₂)/(lM_Dp₂ + rM₅p₁). However, the detected intensity fringes depend on the pixels size of the detector as well. In fact, for a fringe of period p_fr, the l-th Fourier coefficient of the measured intensity, i.e. the intensity averaged over the pixel p_D, is the original l-th Fourier coefficient multiplied to a factor

a_{v} = \frac{1}{p_{D}} \int_{- p_{D} / 2}^{p_{D} / 2} \exp [i 2 π l \frac{x}{p_{fr}}] d x = \frac{\sin (π l p_{D} / p_{fr})}{π l p_{D} / p_{fr}} = sinc (l \frac{p_{D}}{p_{fr}}) .

Apparently, this average is diminishing if the fringe period is much smaller than the detector pixel size, p_fr ≪ p_D. As mentioned in the introduction, one of the goals of the dual phase grating setting is to enable the use of a detector of large pixels in the sensitive interferometry with small period gratings. Equation (6) provides a guide of how to achieve this goal. Without loss of generality, we consider the geometry settings that satisfy the following constraints:

p_{1} = p_{2} = p ≪ p_{D}, and R_{1} = R_{4} ≫ R_{2} .

We will comment the cases with R₁ ≠ R₄ later in section 4. Under these constraints, the fringes resolved by the detector are those of large periods that span several p_D, since the fringes with their periods smaller than p_D are not resolved and rendered as the background intensity. Inspecting the irradiance of Eq. (6), we found that the interference fringes generated by the diffracted orders of l = −r have much large fringe periods compared to those orders of l ≠ −r. In fact, that the interference terms of l = −r becomes: exp[i2π[l/(M₅p₁) + r/(M_Dp₂] x]_l_{= −}_r = exp[i2π(lx)/(M₆p)], where M₆ = (R₁ + R₂ + R₄)/R₂ is the magnification factor associated with resolvable intensity fringes. With the setup geometry constrained by Eq. (9), the magnification factor can be as large as few hundred while the setup keeps compact.

The period of the resolved l-th harmonics of the fringe is obviously given by p(l) = M₆p/l. Hence, the period of the intensity fringe is in general given by

p_{fr} = M_{6} p = (1 + \frac{R_{1} + R_{4}}{R_{2}}) p .

On the other hand, a π-grating does not generate odd differential orders at the design energy, and the lowest diffracted order is the second order, so the fringe period of a π-grating at the design energy is given by p_fr = M₆p/2. With the setup geometry constrained by Eq. (9), the magnification factor can be as large as few hundred while the setup keeps compact. Especially, tuning down grating spacing R₂, one can easily make resolvable fringe pitch spanning over a few pixels of a common detector. This is an important advantage as compared to the inverse geometry technique, where the phase grating must be placed very close to the source that severely limits the field of view. Moreover, a small adjustment of the source-grating distance, in the inverse geometry setup, will cause huge change of the magnification factor and the total system length. On the other hand, the other fine fringes, which are represented by the terms exp[i2π[l/M₅p) + r/(M_Dp)] x] with l ≠ −r in Eq. (6), are not resolvable by the large pixel size p_D of the detector. These unresolvable fringes are rendered as the background intensity in the measurement.

Extracting the diffracted orders of l = −r from Eq. (6), the resolvable intensity fringe pattern I_res can be found as:

\begin{array}{l} I_{res} (x, y) = \frac{I_{in}}{M_{5}^{2}} \cdot \sum_{l, n, s, \in ℤ} μ (- \frac{λ R_{2}}{M_{5} p}) \cdot sinc (\frac{l p_{D}}{M_{6} p}) \cdot a_{l + n} a_{n}^{*} b_{s - l} b_{s}^{*} \times \\ \times \exp [- i 2 π \cdot \frac{l (l + 2 n) λ R_{1}}{2 M_{6} p^{2}}] \cdot \exp [i 2 π \cdot \frac{l (- l + 2 s) λ R_{4}}{2 M_{6} p^{2}}] \cdot \exp [- i 2 π \frac{l x}{M_{6} P}], \end{array}

where the term sinc ((lp_D)/(M₆p)) accounts for the pixel-averaging effect. Here we made use of R₂/M₅ = R₁/M₆. From above derivation of Eqs. (6) and (8), we reveal the fringe formation mechanism: the resolved fringe is formed through the beating oscillation of the cross-modulated intensity fringes generated by the dual phase grating, and also the low-pass filtering enabled by detector pixel averaging. To simplify Eq. (11), we need to figure out how to carry out the triple summation of infinite series in Eq. (11). For a given diffracted order l in Eq. (11), we can carry the series summation over the indices n and s separately. First consider the series summation over n, which we denote as

D_{G_{1}} (l)

. It involves only the products of the Fourier coefficients of the phase gating G₁:

D_{G_{1}} (l) = \sum_{n = - \infty}^{\infty} a_{l + n} a_{n}^{*} \cdot \exp [- i 2 π \frac{l (l + 2 n) λ R_{1}}{2 M_{6} p^{2}}] .

To sunderstand the physical meaning of

D_{G_{1}} (l)

, look back at a case of an interferometer based on a general phase grating. The grating has a period p_g and phase shift ∆ϕ_g, the Fourier coefficient of its transmittance is denoted by g_n. Assume that d is the reduced Fresnel diffraction distance from the grating to the image plane. With spatially coherent x-ray illumination of wavelength λ, the fringe intensity is determined by [25]:

\begin{array}{l} I (x) = \sum_{l = - \infty}^{\infty} \sum_{n = - \infty}^{\infty} g_{l + n} g_{n}^{*} \cdot \exp [- i 2 π \frac{l (l + 2 n) λ d}{2 p_{g}^{2}}] \cdot \exp [i 2 π \frac{l x}{p g}] \\ = \sum_{l = - \infty}^{\infty} C_{l} (d, λ, p_{g}, Δ ϕ_{g}) \cdot \exp [i 2 π \frac{l x}{p_{g}}] . \end{array}

Comparing Eq. (12) to Eq. (13), we conclude the sum

D_{G_{1}} (l)

in Eq. (12) is indeed the Fourier coefficient of the intensity generated by G₁ grating. However, the intensity Fourier coefficient C_l(d, λ, p_g, ∆ϕ_g) in Eq. (13) is a complicated function of the grating period p_g, phase shift ∆ϕ_g, wavelength λ and reduced Fresnel diffraction distance d. In a previous work we have derived a closed form expression for C_l(d, λ, p_g, ∆ϕ_g) to facilitate quantitative analysis of grating interferometry [17, 18]:

C_{l} (d, λ, p_{g}, Δ ϕ_{g}) = {\begin{array}{l} 1, & if l = 0, \\ - (1 - \cos Δ ϕ_{g}) \cdot {(- 1)}^{⌊ 4 k λ d / p_{g}^{2} ⌋ \cdot \frac{\sin (4 k^{2} π λ d / p_{g}^{2})}{k π}}, & if l = 2 k \neq 0, \\ - i 2 π \sin Δ ϕ_{g} \cdot \frac{\sin (4 k λ d / p_{g}^{2} \cdot {(k + 1 / 2)}^{2})}{π (2 k + 1)}, & if l = 2 k + 1 . \end{array}

From Eq. (14) it is easy to see that

C_{l} (d, λ, p_{g}, Δ ϕ_{g}) = C_{- l}^{*} (d, λ, p_{g}, Δ ϕ_{g})

. Comparing Eq. (12) to Eqs. (13) and (14) we found the sum

D_{G_{1}} (l) = C_{l} (R_{1} / M_{6}, λ, p, Δ ϕ_{1})

. Similarly, we denote the summation over s in Eq. (11) by

D_{G_{2}} (- l)

:

D_{G_{2}} (- l) = \sum_{s = - \infty}^{\infty} b_{- l + s} b_{s}^{*} \cdot \exp [- i 2 π \frac{- l (- l + 2 s) λ R_{4}}{2 M_{6} p^{2}}] .

One is now easy to recognize that

D_{G_{2}} (- l)

is the Fourier coefficient of the intensity generated by G₂ grating. Comparing Eq. (15) to Eqs. (13) and (14) we found the sum

D_{G_{2}} (- l) = C_{- l} (R_{4} / M_{6}, λ, p, Δ ϕ_{2})

. Putting above results together, the resolvable intensity fringe pattern I_res can be found as:

\begin{array}{l} I_{res} (x, y) = \frac{I_{in}}{M_{5}^{2}} \cdot \sum_{l = - \infty}^{\infty} μ (- \frac{λ R_{1}}{M_{6} p}) \cdot sinc (\frac{l p_{D}}{M_{6} p}) \times \\ \times C_{l} (\frac{R_{1}}{M_{6}}, λ, p, Δ ϕ_{1}) \cdot C_{- l} (- \frac{R_{4}}{M_{6}}, λ, p, Δ ϕ_{2}) \cdot \exp [- i 2 π \frac{l x}{M_{6} P}] . \end{array}

Equation (16) shows that the resolvable intensity fringe consists of diffracted orders indexed by a single integer l. In Eq. (16) µ (−l(λR₁)/(M₆p)) is the spatial coherence degree associated with the order l, while the term sinc (lp_D/(M₆p)) is the intensity averaging factor as is shown in Eq. (8).

We performed computer simulation to verify the fringe period formula of Eq. (10) and the resolvable fringe equation of Eq. (16). In the simulation we assume a 20 keV point x-ray source. The simulated dual π-grating is with period of 2µm. The geometry is set to R₁ = R₄ = 300mm, and R₂ = 15mm. The solid blue lines in Fig. 2 represent the plot of the simulated intensity fringes, while the dashed red line is the sum of all the resolvable diffraction orders as given in Eq. (16). In Fig. 2(a), when detector pixel size is small (0.5µm), various high-frequency fringes are present. In Fig. 2(b), when the detector pixel size is large (20.5µm), the resolved fringe demonstrates a period of M₆p/2, as is predicted earlier. The fringes of small periods are diminished due to the pixel average effect given by Eq. (8).

Fig. 2 Effects of detector pitch size on fringe period. In this simulation, we present the effects of detector pitch on the resolved fringes. In the simulation we assume a 20 keV point x-ray source. The simulated dual π-grating is with period of 2µm. The geometry is set to R₁ = R₄ = 300mm, and R₂ = 15mm. The solid blue lines represent the plot of the simulated intensity fringes, while the dashed red line, in Fig. 2(b), is the sum of all the resolvable diffraction orders given in Eq. (16). It can be seen that when detector pixel size is sufficiently small, the fringes of all periods are present (Fig. 2(a)). When detector pixel size is large enough, Fig. 2(b), only the resolvable fringes, of period p_fr/2 = M₆p/2 = 41µm, are visible, and the fringes of small periods are diminished due to the average effect given by Eq. (8). The simulation result (the blue line in Fig. 2(b)), agrees well with theoretical values (the red line in Fig. 2(b)).

Download Full Size | PDF

3.2. Fringe visibility

As is well known, formation of high-modulation fringe pattern is crucial to robust grating interferometry, the intensity modulation of a fringe pattern is characterized by the visibility V of the fringe pattern, namely the visibility is defined as V = (I_max − I_min)/(I_max + I_min), where I_max and I_min are the maximum and minimum intensities respectively. Fringe visibility is a common figure of merit in design of a grating-based x-ray interferometer [1–10]. Equation (16) shows that fringe visibility depends on dual grating phase shifts, spatial coherence of the illumination, system geometry setup, and x-ray spectrum.

To demonstrate how to use Eq. (16) for fringe visibility analysis, we consider an interferometer consists of dual phase grating of period p and phase shift π at design energy E_D, a monochromatic source of the same energy and width a, and a detector of pixel size p_D. The geometry is set to R₁ = R₄, and we want to derive a formula of fringe visibility. To apply Eq. (16) to this setup, firstly we need to find the coherence degree µ (l(λR₂)/(M₅p)). Applying Van Citters-Zernike theorem to a focal spot of width a, as is shown in the Appendix (Eq. (26), we found that the coherence degree µ(s) = sinc(as/(λR₁)), where sinc(x) = sin(πx)/(πx). Hence µ(l(λR₂)/(M₅p)) = sinc µ(la(λR₂)/(M₅p · λR₁)) = sinc (la)/(M₆p)), since R₂/M₅ = R₁/M₆. Secondly, using Eq. (14), we found that for even diffraction orders:

C_{l} (\frac{R_{1}}{M_{6}}, E_{D}, p, π) \cdot C_{- l} (\frac{R_{4}}{M_{6}}, E_{D}, p, π) = \frac{16}{π^{2} l^{2}} \sin^{2} (l^{2} π \frac{R_{1} / M_{6}}{8 Z_{π}}) .

Here Z_π = p²/(8λ_D) is the first fractional Talbot distance for a single π grating, and λ_D is x-ray wavelength at the design energy [1–4]. Note that here C_l = C₋_l = 0 for all odd diffraction orders. Substituting Eq. (17) into Eq. (16), we found that, for even orders, the Fourier coefficients in Eq. (16) diminish rapidly with increasing diffaction order (l):

sinc (\frac{l a}{M_{6} p}) \cdot C_{l} (\frac{R_{1}}{M_{6}}, E_{D}, p, π) \cdot C_{- l} (\frac{R_{4}}{M_{6}}, E_{D}, p, π) \cdot sinc (\frac{l p_{D}}{M_{6} p}) \leq \frac{16}{l^{4} π^{4}} \cdot \frac{{(M_{6} p)}^{2}}{a p_{D}},

for l = 2, 4, 6, ⋯. Equation (18) shows that the lowest order (l = 2) dominates the contribution to the intensity fringes for the setups with finite focal spots. This being so, the intensity fringe formula of Eq. (16) is then simplified to:

I_{res} (x) = \frac{I_{in}}{M_{5}^{2}} [1 + \frac{8}{π^{2}} sinc (\frac{2 a}{M_{6} p}) \cdot sinc (\frac{2 p_{D}}{M_{6} p}) \cdot \sin^{2} (π \frac{R_{1} / M_{6}}{2 Z_{π}}) \cdot \cos (2 π \frac{x}{M_{6} p / 2})] .

Equation (19) shows that the fringe has a period of M₆p/2, and the fringe visibility, namely V = (I_max − I_min)/(I_max + I_min), is found to be:

V = \frac{8}{π^{2}} sinc (\frac{2 a}{M_{6} p}) \cdot sinc (\frac{2 p_{D}}{M_{6} p}) \cdot \sin^{2} (π \cdot \frac{R_{1} / M_{6}}{2 Z_{π}}), Z_{π} = \frac{p^{2}}{8 λ_{D}} .

Figure 3 plots the derived visibility of Eq. (20) as a function of the G₁-G₂ spacing R₂. In this plot, we assumed that the interferometer consists of dual phase grating of period p = 1µm and phase shift π at design energy E_D = 20keV, a 20keV-source of width a = 40µm, and a detector of pixel size p_D = 25µm. The geometry is set symmetrically such that R₁ = R₄ = 450mm. The blue curve in Fig. 3 shows how the (theoretical) fringe visibility rapidly changes with R₂. The theoretical visibility curve based on Eq. (20) reaches its maximum 0.62 when R₂ = 3.6 mm, and it has its minimum 0 at R₂ = 8.14mm. According to Eq. (10), adjustment of R₂ is accompanied with fringe period change. For example, the fringe period is 251µm for R₂ = 3.6mm, and becomes 181µm when R₂ = 5mm.

Fig. 3 Plot of visibility curve with respect to the G₁-G₂ spacing R₂. In the figure, we assume the dual phase grating has period p = 1µm and phase shift π at design energy E_D = 20keV. The 20keV source is a focal spot of width a = 40µm, and the detector pixel size is p_D = 25µm. The geometry is set symmetrically with R₁ = R₄ = 450mm and the dual-grating spacing R₂ changes from 2mm to 10mm. The blue line is computed directly from Eq. (20). While the black squares represent the visibility values of the fringe patterns, which were obtained by numerical simulations of Fresnel diffraction at various spacing R₂. We can see the simulation results fit the theoretical values very well.

Download Full Size | PDF

To validate this theoretical prediction of the fringe visibility, we conducted computer simulation of the interferometry based on Fresnel wave propagation through the simulated dual π grating. For each source point of the focal spot, we obtained the simulated intensity distribution at the detector entrance. The sum of the intensity fringes generated by each source point in the 40µm focal width is then re-binned for the pixels 25µm in size. We then calculated the fringe visibility from the resulting intensity pattern from the simulation. We repeated the simulation procedure for different R₂ settings. In Fig. 3, the black squares depict the fringe visibility values obtained from the simulation. These squares fit the blue theoretical curve of visibility very well. This agreement shows that the fringe visibility formula of Eq. (19) provides a useful tool for design optimization of interferometers based on dual π-grating setups.

Now we turn to interferometers of dual π/2-phase grating. In these cases, the dominant diffraction order is that of order l = 1. Considering the symmetric setup R₁ = R₄, using the general formula of Eq. (14), one is easy to find, for odd integer l,

C_{l} (\frac{R_{1}}{M_{6}}, E_{D}, p, \frac{π}{2}) \cdot C_{- l} (\frac{R_{4}}{M_{6}}, E_{D}, p, \frac{π}{2}) = \frac{4}{l^{2} π^{2}} \sin^{2} (l^{2} π \frac{R_{1} / M_{6}}{2 Z_{π / 2}}),

where Z_π_/2 ≡ p²/(2λ_D) denotes the 1^st fractional Talbot distance of single π/2-grating [1–4]. Similar to the arguments of dual π-grating, we have

I_{res} (x) = \frac{I_{in}}{M_{5}^{2}} [1 + \frac{8}{π^{2}} sinc (\frac{a}{M_{6} p}) \cdot sinc (\frac{p_{D}}{M_{6} p}) \cdot \sin^{2} (π \frac{R_{1} / M_{6}}{2 Z_{π / 2}}) \cdot \cos (2 π \frac{x}{M_{6} p})] .

That is the dual π/2-grating has resolvable fringe period p_fr = M₆p and visibility

V = \frac{8}{π^{2}} sinc (\frac{a}{M_{6} p}) \cdot sinc (\frac{p_{D}}{M_{6} p}) \cdot \sin^{2} (π \cdot \frac{R_{1} / M_{6}}{2 Z_{π / 2}}), Z_{π / 2} = \frac{p^{2}}{8 λ_{D}} .

In Fig. 4 we plot the visibility curve of dual π/2-grating with respect to the dual-grating spacing R₂. The only difference from Fig. 3 in the system setup is that the dual gratings are changed to a π/2 phase shift at design energy E_D. As is shown in Fig. 4, the maximal visibility occurs at R₂ = 10.89mm, as compared to R₂ = 3.6mm in the dual π-grating case. This difference can be traced back to the difference in Talbot distances for these gratings.

Fig. 4 We plot the visibility curve of dual π/2-grating with respect to the dual-grating spacing R₂. The only difference from Fig. 3 in the system setup is that the dual gratings are of π/2 phase shift at design energy E_D. One can see that the maximal visibility occurs at R₂ = 10.89mm, as compared to R₂ = 3.6mm for the dual π-grating case.

Download Full Size | PDF

4. Discussion and conclusions

In medical imaging and material science applications, it is desirable to utilize common imaging detectors, whose pixels are of a few tens micrometers. However, to increase the grating interferometer’s sensitivity, one needs to use fine-pitch phase gratings of periods as small as few micrometers. Hence a challenging task is to develop methods to resolve the fringes generated by such fine-pitch phase gratings with detectors of pixels as large as a few tens micrometers.

From the discussion in section 3, the dual phase grating technique provides another solution to the challenge, in addition to the absorbing grating analyzer technique and the inverse geometry technique as described in the introduction. The pioneer work [21] demonstrated that the dual phase grating technique that can use a common detector to resolve interference fringes generated by two phase gratings of micrometer-periods. In addition, with this new technique one can conveniently adjust fringe period and fringe visibility with almost no change of the total system length. This facilitates system implementations. However, this experimental work of [21] did not provide any formulas to predict fringe period and fringe visibility as functions of setup geometry and x-ray energy. Besides, the work [21] evaluated only the setup of a π-grating. For interferometer design optimization, one needs to know quantitatively how to control the fringe periods and how to achieve high visibility of the fringes. Different from the work presented in [21], in this work, we elucidate the fringe-formation mechanism, and derived closed-forms formulas for answering these two important questions critical for dual phase grating interferometry design, not only for setups with π gratings, but also for setups with π/2 gratings.

Understanding the fringe-formation mechanism is critical to develop quantitative theory of dual phase grating interferometry. In that work [21], it is claimed that the fringe is formed by the G₂ grating from an extended virtual source generated by the first grating G₁ at its fractional Talbot distance. This explanation, while is plausible, lacks of support based on a rigorous theory [21]. Consequently, the work [21] does not provide any quantitative formulas for the fringe period and fringe visibility. Different from the work [21], we based our explanation of the fringe formation on wave evolution process. Using the Wigner distribution formalism, we derived Eq. (6), the formula that determines all the intensity fringes at the detector entrance. This formula, together with Eq. (8), enables us to reveal the fringe formation formalism. The resolved fringe is formed through two processes. The first is the beating oscillation of the cross-modulated intensity fringes generated by the dual phase gratings, and the second is the low-pass filtering enabled by detector pixel averaging. Based on our understanding of this mechanism, we can answer the questions of how to control the fringe period and fringe visibility. For example, in the interferometer design, it is necessary to match the resolved-fringe period with a given pixel size of detector. One usually wants a fringe period spans over several pixels. In this regard, Eq. (10) provides a formula of fringe period that determines fringe period for given grating pitch p and the geometry distances R₁, R₂ and R₄. In the special case of π-gratings, the fringe period is shorted to one half of the value given by Eq. (10). Moreover, Eq. (10) shows that even a small adjustment of the G₁-G₂ spacing could induce a large increase of the fringe period, while keeping the total system length almost unchanged. This is an obvious advantage over the inverse geometry technique, in which a change in fringe period necessitates a significant change of the system length. Secondly, in the interferometer design, it is desirable to be able quantitatively predict fringe visibility. This is because fringe visibility is a figure of merit for interferometer performance. High fringe visibility reduces quantum noise in differential phase images and increases interferometry sensitivity. The work [21] experimentally studies fringe visibility of a dual π-grating interferometer, but it does not provide any formula of fringe visibility. Different from the work in [21], in this work we derived the fringe visibility formulas for both the dual π-grating and dual π/2-grating. In the case of dual π-grating, the formula Eq. 20 shows how the G₁-G₂ spacing, together with grating pitch, focal spot width and detector pixel-size, determine the fringe visibility. Moreover, this fringe visibility formula is validated by a good agreement of the formula with the wave simulation result, as is shown in Fig. (3). A similar visibility formula for the setup with π/2 gratings is given in Eq. (23). Hence, different from the work [21], the derived fringe-visibility formulas Eqs. (20) and (23) derived in this work provide useful tools for design optimization of dual phase grating interferometry.

In this paper, we only present the results of symmetric setup, R₁ = R₄. In fact this theory is applicable to the setups with R₁ ≠ R₄. For sake of space limit, we only list some interesting results here. For a dual π-grating setup, in order for the intensity fringe to get maximal visibility, R₁, R₂ and R₄ must satisfy: R₁/R₂ = k₁/k₂, where k₁, k₂ are positive odd integers and R₂ = (k₁ + k₂)R₁Z_π/(R₁ + k₁ Z_π), Z_π = p²/(8/λ_D). This result is also applicable to dual π/2-grating by replacing Z_π with Z_π/₂ = p²/(2λ_D). Of cause, the two gratings need not be identical but the results will be more complicated. It should be pointed, in all the cases the symmetric setup is the most compact since all other setups will require a longer source-to-detector distance.

For sake of space limit, we plan to present several other topics on dual grating interferometry in a sequel paper. For example, should the sample be placed before the first phase grating, or behind the second phase grating? What are the differences in sensitivity for detecting sample refraction for different sample placement? We will derive corresponding formulas of fringe phase shift to answer these questions. Besides, as the most applications use polychromatic x-ray sources such as x-ray tubes, we will derive the fringe visibility formulas for the polychromatic x-ray as well in the sequel paper. Last but not the least, when one deals with a source of a large focal spot, one needs to employ the source grating. In the planned sequel, we will also discuss how to set up the source grating or micro-source array in the dual phase gratings setup. Obviously, the well-known Lau condition for coherent fringe superposition is not applicable because of the dual phase gratings. Based on the coherence degree analysis, we will derive a new condition for coherent fringe superposition in the sequel paper.

To conclude, in this work we present a quantitative theory of dual grating x-ray interferometry. The theory reveals the fringe formation mechanism in this technique. The derived formulas of fringe period and fringe visibility provide useful tools for design optimization of dual phase grating interferometers.

Appendix

According to Eq. (3), the Wigner function at G₁-plane is given as [22–24]

W_{R_{1}} (x, u; y) = \int J_{in} (x + \frac{Δ}{2}, x - \frac{Δ}{2}) \cdot G_{1} (x + \frac{Δ}{2}) \cdot G_{1}^{*} (x - \frac{Δ}{2}) \cdot \exp [- i 2 π u Δ] d Δ,

where J_in (x + ∆/2, x − ∆/2) is the mutual intensity of impinging x-ray wavefront at G₁-plane. For an incoherent source, according to Van Citters-Zernike theorem [22,26]:

J_{in} (x + \frac{Δ}{2}, x - \frac{Δ}{2}) = I_{in} \cdot \exp (i \frac{2 π Δ \cdot x}{λ R_{1}}) μ_{in} (Δ),

and

μ_{in} (Δ) = \frac{\int I_{source} (s) \cdot \exp [i 2 π Δ \cdot s / (λ R_{1})] d s}{\int I_{source} (s) d s} .

Substituting Eqs. (1), (25), and (26) into Eq. (24), we have

\begin{array}{l} W_{R_{1}} (x, u; y) = I_{in} \cdot \sum_{l, n = - \infty}^{\infty} a_{l + n} a_{n}^{*} \cdot \exp (i 2 π \frac{l x}{p_{1}}) \times \\ \times \int \exp [i \frac{2 π Δ \cdot x}{λ R_{1}}] \cdot μ_{in} (Δ) \cdot \exp [i 2 π \frac{l + 2 n}{p_{1}} Δ] \cdot \exp [- i 2 π u Δ] d Δ . \end{array}

As wave propagates from G₁-plane to the entrance plane of G₂ grating, the Wigner function undergoes a shearing in the phase space [22–24]

W_{R_{1} + R_{2} - 0} (x, u; y) = W_{R_{1}} (x - λ R_{2} u, u; y) .

Hence, at the entrance plane of G₂ grating, the mutual intensity function [22]

J_{R_{1} + R_{2}} (x + \frac{Δ_{2}}{2}, x - \frac{Δ_{2}}{2}; y) = \int W_{R_{1} + R_{2} - 0} (x, u; y) \cdot \exp [i 2 π u Δ_{2}) d u .

Substituting Eqs. (27) and (28) into Eq. (29), we found the mutual intensity at the entrance plane of G₂ grating is:

\begin{array}{l} J_{R_{1} + R_{2}} (x + \frac{Δ_{2}}{2}, x - \frac{Δ_{2}}{2}; y) = \frac{I_{in}}{M_{1}^{2}} \sum_{l, n = - \infty}^{\infty} a_{l + n} a_{n}^{*} \cdot \exp (i 2 π \frac{l x}{M_{1} p_{1}}) \times \\ \times \exp [i 2 π \frac{Δ_{2} x}{M_{1} λ R_{1}}] \times \exp [- i 2 π \frac{l (l + 2 n) λ R_{2}}{2 M_{1} p_{1}^{2}}] \cdot \exp [i 2 π \frac{(l + 2 n) Δ_{2}}{2 M_{1} p_{1}}] \times \\ \times μ_{in} (\frac{Δ_{2}}{M_{1}} - \frac{l λ R_{2}}{M_{1} p_{1}}) \cdot \end{array}

As the wavefront just behind grating G₂, its Wigner function is changed to:

\begin{array}{l} W_{R_{1} + R_{2} + 0} (x, u; y) = \int J_{z = R_{1} + R_{2}} (x + \frac{Δ_{2}}{2}, x - \frac{Δ_{2}}{2}; y) \cdot G_{2} (x + \frac{Δ_{2}}{2}) \times \\ \times G_{2}^{*} (x - \frac{Δ_{2}}{2}) \cdot \exp [- i 2 π u Δ_{2}] d Δ_{2} . \end{array}

Substituting Eq. (30) and Eq. (1) into Eq. (31), the Wigner function just behind grating can be found as:

\begin{matrix} W_{R_{1} + R_{2} + 0} (x, u; y) = \frac{I_{in}}{M_{1}^{2}} \sum_{l, n, r, s = - \infty}^{\infty} a_{l + n} a_{n}^{*} b_{r + s} b_{r}^{*} \times \\ \times \exp [i 2 π x (\frac{l}{M_{1} p_{1}} + \frac{r}{p_{2}} - \frac{l (l + 2 n) λ R_{2}}{2 M_{1} p^{2}})] \times \\ \times \int \exp [i 2 π \frac{(l + 2 n) Δ_{2}}{2 M_{1} p_{1}} \cdot \exp [i 2 π \frac{(r + 2 s) Δ_{2}}{2 p_{2}}]] \times \\ \times μ_{in} (\frac{Δ_{2}}{M_{1}} - \frac{l λ R_{2}}{M_{1} p_{1}}) \cdot \exp [i 2 π \frac{Δ_{2} x}{M_{1} λ R_{1}}] \cdot \exp [- i 2 π u Δ_{2}] d Δ_{2} . \end{matrix}

As Wigner function evolves from G₂ grating plane to the detector’s entrance, the Wigner function undergoes again a phase space shearing. Hence the Wigner function at the detector entrance is

W_{R_{1} + R_{2} + R_{4}} (x, u; y) = W_{R_{1} + R_{2} + 0} (x - λ R_{4} u, u; y)

. From Eq. (5) x-ray irradiance at the entrance is given by:

I_{R_{1} + R_{2} + R_{4}} (x, y) = \int W_{R_{1} + R_{2} + 0} (x - λ R_{u}, u; y) d u .

Substituting Eq. (31) into the integrals of Eq. (33), and carrying the integration over u firstly and then over ∆₂, one is finally led to Eq. (6) in section 3.

Funding

National Institutes of Health (NIH) (1R01CA193378).

References and links

1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003). [CrossRef]

2. 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]

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

4. 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]

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

6. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc Natl. Acad. Sci. USA 107, 13576–13581 (2010). [CrossRef] [PubMed]

7. N. Bevins, J. Zambelli, K. Li, Z. Qi, and G.-H. Chen, “Multicontrast x-ray computed tomography imaging using talbot-lau interferometry without phase stepping,” Med. Phys. 39, 424–428 (2012). [CrossRef] [PubMed]

8. X. Tang, Y. Yang, and S. Tang, “Characterization of imaging performance in differential phase contrast ct compared with the conventional ct: Spectrum of noise equivalent quanta neq(k),” Med. Phys. 39, 4367–4382 (2012). [CrossRef]

9. T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional gratings-based phase-contrast imaging using a conventional x-ray tube,” Opt. Lett. 36, 3551–3553 (2011). [CrossRef] [PubMed]

10. E. Bennett, R. Kopace, A. Stein, and H. Wen, “A grating-based single shot x-ray phase contrast and diffraction method for in vivo imaging,” Med. Phys. 37, 6047–6054 (2010). [CrossRef] [PubMed]

11. M. Jiang, C. Wyatt, and G. Wang, “X-ray phase-contrast imaging with three 2d gratings,” Int. J. Biomed. Imaging 2008, 827152 (2008). [PubMed]

12. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based x-ray phase-contrast imaging using fourier transform phase retrieval,” Opt. Express 19, 3339–3346 (2011). [CrossRef] [PubMed]

13. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Physics in Medicine and Biology 58, R1–R35 (2013). [CrossRef]

14. A. Momose, H. Kuwabara, and W. Yashiro, “X-ray phase imaging using lau effects,” Appl. Phys. Express 4, 066603 (2011). [CrossRef]

15. N. Morimoto, S. Fujino, K. Ohshima, J. Harada, T. Hosoi, H. Watanabe, and T. Shimura, “X-ray phase contrast imaging by compact talbot-lau interferometer with a single transmission grating,” Opt. Lett. 39, 4297–4300 (2014). [CrossRef] [PubMed]

16. N. Morimoto, S. Fujino, A. Yamazaki, Y. Ito, T. Hosoi, H. Watanabe, and T. Shimura, “Two dimensional x-ray phase imaging using single grating interferometer with embedded x-ray targets,” Opt. Express 23, 16582–16588 (2015). [CrossRef] [PubMed]

17. A. Yan, X. Wu, and H. Liu, “A general theory of interference fringes in x-ray phase grating imaging,” Med. Phys. 42, 3036–3047 (2015). [CrossRef] [PubMed]

18. A. Yan, X. Wu, and H. Liu, “Predicting visibility of interference fringes in x-ray grating interometry,” Opt. Express 24, 15927–15939 (2016). [CrossRef] [PubMed]

19. A. Yan, X. Wu, and H. Liu, “Polychromatic x-ray effects on fringe phase shifts in grating interferometry,” Opt. Express 25, 6053–6068 (2017). [CrossRef] [PubMed]

20. A. Yan, X. Wu, and H. Liu, “Beam hardening correction in polychromatic x-ray grating interferometry,” Opt. Express 25, 24690–24704 (2017). [CrossRef] [PubMed]

21. M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110, 014105 (2017). [CrossRef]

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

23. X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on wigner distributions,” Med. Phys. 31, 2378–2384 (2004). [CrossRef] [PubMed]

24. X. Wu and H. Liu, “Phase-space evolution of x-ray coherence in phase-sensitive imaging,” Appl. Opt. 47, 44–52 (2008). [CrossRef] [PubMed]

25. V. Arrizón and J. Ojeda-Castañeda, “Irradiance at fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992). [CrossRef]

26. J. Goodman, Statistical Optics (John Wiley and Sons, Inc., 1985).

Quantitative theory of X-ray interferometers based on dual phase grating: fringe period and visibility

Abstract

1. Introduction

2. Materials and methods

3. Results

3.1. Resolvable intensity fringes

3.2. Fringe visibility

4. Discussion and conclusions

Appendix

Funding

References and links

Cited By

Figures (4)

Equations (33)

Optics Express