Predicting visibility of interference fringes in X-ray grating interferometry

Aimin Yan; Xizeng Wu; Hong Liu

doi:10.1364/OE.24.015927

1. Introduction

X-ray grating interferometry is a differential x-ray phase-contrast imaging technique that has many potential applications in medical imaging and material science [1–24]. Figure 1 shows a schematic of an x-ray interferometer, which consists of a two-dimensional checkerboard phase grating, a micro-source-array, and a high-resolution imaging detector. The phase grating serves as a beam splitter and divides the incident beam into different diffraction orders. The interference between diffraction orders generates intensity fringes. The shape and modulation of the interference fringes depend actually on grating’s phase shift, x-ray spectrum, spatial coherence degree of x-ray illumination, and system geometry such as the phase grating to detector distance. The intensity modulation of a fringe pattern usually is characterized by the visibility V of the fringe:

V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} .

With spatially coherent illumination, interference fringes of the maximal modulation are formed only at certain discrete grating-to-detector distances, which are called the self-image Talbot distances [1–4, 20]. For example, with monochromatic and plane wave illumination, following values of grating-to-detector distances correspond to the 2D self-image Talbot distances:

Z_{T - 2 D} = \frac{n p_{1}^{2}}{8 λ},

where p₁ is the phase grating period, λ is the x-ray wavelength, n takes odd integer values for a π-gratings, and otherwise n = 4q + 2, where q is an integer [25].

Fig. 1 Schematic of an x-ray phase grating interferometer with a microfocus source

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The pattern and modulation of interference fringes depend on the spatial coherence degree of x-ray illumination as well. Spatially coherent illumination, such as generated by using synchrotron radiation or micro-focus tubes, allows high visibility of intensity fringes. However, in medical imaging applications micro-focus tube cannot offer sufficient x-ray flux to shorten exposure times. In order to fulfill the requirements of spatial coherence and adequate radiation output, one may employ a periodic array G₀ of mutually incoherent micro-anode-sources embedded in the anode (Fig. 1) [26, 27]. Equivalently one may utilize an x-ray tube equipped with an absorbing source grating G₀, which serves as an aperture mask to break the focus spot into an array of mutually incoherent micro-sources [4, 8, 26, 28]. With appropriate geometrical arrangement, as is explained below, the diffraction fringes generated by these micro-sources are all superimposed on each other. This kind of system setup is called the Talbot-Lau interferometer [4,28].

In x-ray grating interferometry fringe visibility V, as is defined in Eq. (1), is a common figure of merit in designs of x-ray grating-based interferometers. This is because the formation of high-modulation fringe pattern is a prerequisite for robust grating interferometry. When x-ray sources are spatially coherent and monochromatic, the self-image Talbot distances given by Eq. (2) may provide guidance for designs of grating interferometers. But in many applications such as medical imaging, broad-band polychromatic x-ray from x-ray tubes renders the concept of Talbot distance ill-defined. Firstly, Talbot distances are wavelength-dependent. A grating-detector distance may fall onto a Talbot distance for one wavelength but it just mismatch the Talbot distances for other wavelengths. Secondly, Talbot distances depend on grating phase shift values as well. But phase-shift value of a grating varies with x-ray wavelength. For example, a π-grating for 15-keV x-ray becomes π/2-grating for 30-keV x-ray, assuming the absence of absorption edges in the energy interval. Consequently, with a broad band x-ray beam the sharp discrete Talbot distances of the self-image regime would be stretched into several downstream intervals of relatively high fringe modulations. The locations and lengths of these intervals depend on the x-ray spectrum and the grating’s phase shift values [25, 26]. On the other hand, the broadening of self-image regime with polychromatic x-ray offers design optimization opportunities. Broadband x-ray makes fringe modulations comparably high for a large range of grating-to-detector distances, which can be selected for system optimization [25]. Presently in design optimization of grating interferometers one has to resort to laborious computer simulations to predict the fringe visibility values for interferometers with polychromatic x-ray sources [26, 27, 29, 30]. Even in these computer simulation studies, the spatially coherence effect with the source grating or micro-source array had not been included because of the complexity in simulating spatial coherence effect. Hence for x-ray interferometry design optimization there is pressing need for a quantitative theory that is able to predict the fringe visibility attainable with any given Talbot-Lau configuration and x-ray spectrum. In this work we present such a general theory to predict the fringe visibility in closed-form formulas.

This work expands our previous work of Fourier expansion of interference fringes in grating interferometry. In that work, assuming spatially coherent illumination from synchrotron radiation or microfocus tubes, we evolved the Wigner distribution to obtain closed-form expressions of Fourier coefficients of intensity fringes for any grating-to-detector distances [25]. The purpose of this work is to derive closed-form formulas of the coherence-weighted and spectrum-averaged Fourier coefficients of fringes in Talbot-Lau interferometry with polychromatic x-ray sources. Moreover, in order to predict the intensity fringe patterns, these coherence-weighted and spectrum-averaged Fourier coefficients will be further weighted by detector pixel rebinning effects. In this way these closed-form formulas of coherence-weighted and spectrum-averaged Fourier coefficients enable us to predict the fringe visibility of the Talbot-Lau interferometers. We organize the presentation as follows. In section 2, starting from the Fourier analysis of fringe intensity, we briefly review the general theoretical formalism that underlines our new theory. Particularly we discuss how to compute the coherence degree to incorporate partial spatial coherence effects for Talbot-Lau interferometers. In section 3, we present the closed-form formula of the coherence-weighted Fourier coefficients of fringe intensity, which incorporated spatial coherence effects and x-ray spectral average. Furthermore, through two examples of an inverse-geometry Talbot-Lau interferometer we show how to predict fringe visibility values by using these closed-form formulas. We validate the predicted visibility values by comparing them to computer simulation results. We conclude the paper in section 4.

2. Materials and methods

Consider a grating interferometer as schematically depicted in Fig. 1. The wave-splitting phase grating is denoted by G₁, which is a two-dimensional checkerboard phase grating. The phase grating of a bi-directional period p₁ is a periodic array of phase shift modulation between values of Δϕ and zero. Mathematically we model such a checkerboard phase grating G₁ by:

\begin{array}{l} G_{1} (\vec{r}) = \exp (i Δ ϕ \cdot h (\vec{r} / p_{1} - ⌊ \vec{r} / p_{1} ⌋ - 1 / 2)), \\ h (\vec{s}) = {\begin{array}{l} 1, & if \vec{s} \in {[- 1 / 2, 0)}^{2} \cup {[0, 1 / 2)}^{2}, \\ 0, & otherwise, \end{array} \end{array}

where the so-called floor-function ⎿x⏌ is defined as the largest integer that is less or equal to x. In order to provide spatially coherent illumination, the interferometer employs a rectangular micro-source-array G₀ of bi-directional period p₀. As is reported in literature, such a micro-source-array G₀ can be realized, for example, by employing a periodic array G₀ of micro-tungsten targets embedded in a synthetic diamond-substrate [27], and the micro-tungsten targets emit x-ray when they are irradiated by accelerated electrons from the cathode. With the dominant bremsstrahlung generated from the tungsten dots as compared to that from the diamond substrate, this tungsten-dots array effectively functions as an array of microfocus sources. Equivalently one may utilize an x-ray tube combined with a 2D absorbing source grating to effectively break the tube’s focus spot into an array of micro-sources, as each aperture of the source grating functions as a micro-source, and the source grating as an array of micro-sources [27]. A high-resolution detector is assumed such that it well resolves the intensity fringe pattern. The detector pixel rebinning effect on the fringe pattern will be discussed in section 3. The interferometer configuration with such a micro-sources array is also called the Talbot-Lau interferometer [4,28]. The distance between gratings G₀ and G₁ is denoted by R₁, and that between G₁ and the detector is R₂.

Our theoretical formulation of grating interferometer starts from the Wigner distribution formalism for x-ray Fresnel diffraction [31, 32]. The Wigner distribution in the phase space (position space and ray-direction space) is defined as Fourier transform of the mutual intensity of wavefields with respect to the displacement vector [33]. The advantages of using Wigner distribution formalism for x-ray phase-contrast imaging are two-fold. First, Wigner distribution directly specifies the partial coherence of the x-ray wavefield. Second, evolution of Wigner distribution in wave diffraction process is especially simple owing to its covariance in phase-space [31, 32]. As is shown in our previous work [25], for monochromatic x-ray sources of wavelength λ, Fourier transform of the fringe intensity at a distance (R₁ + R₂) from the phase grating G₁ is given by:

{\tilde{I}}_{λ} (\frac{\vec{u}}{M_{g}}; R_{1} + R_{2}) = I_{in, λ} μ_{in} (\frac{λ R_{2} \vec{u}}{M_{g}}) \cdot \int G_{1} (\vec{s} + \frac{λ R_{2} \vec{u}}{2 M_{g}}) \cdot G_{1}^{*} (\vec{s} - \frac{λ R_{2} \vec{u}}{2 M_{g}}) \cdot \exp (i 2 π \vec{s} \cdot \vec{u}) d \vec{s},

where

\tilde{I} λ (\vec{u} / M_{g})

denotes the fringe intensity’s Fourier transform valued at

\vec{u} / M_{g}

,

\vec{u}

is the spatial frequency vector at the phase grating plane and M_g = (R₁ + R₂)/R₁ is the geometric magnification. In Eq. (4) the constant factor I_in_,λ denotes the incident x-ray intensity, and the factor

μ_{in} (λ R_{2} \vec{u} / M_{g})

denotes the reduced complex degree of x-ray spatial coherence at the entrance of G₁ [33]. For sake of convenience we simply call

μ_{in} (λ R_{2} \vec{u} / M_{g})

the coherence degree. Using the property of the Fourier transform of convolution product and Poisson summation formula [25], we found that Fourier transform of the intensity fringes are given by:

{\tilde{I}}_{λ} (\frac{\vec{u}}{M_{g}}; R_{1} + R_{2}) = I_{in, λ} \sum_{\vec{m} \in ℤ^{2}} μ_{in} (\frac{λ R_{2} \vec{m}}{M_{g} p_{1}}) \cdot C (\vec{m}; λ) \cdot δ (\vec{u} - \frac{\vec{m}}{p_{1}}),

where δ is the Dirac delta function. Therefore, this equation shows that the fringe intensity is a superposition of diffraction orders with discrete spatial frequencies

{\vec{m} / p_{1}; \vec{m} = (m, n) \in ℤ^{2}}

. In Eq. (5),

μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})

is the coherence degree for the diffraction order of spatial frequency

\vec{m} / p_{1}

. For perfectly coherent illumination, the coherence degree

μ_{in} (λ R_{2} \vec{u} / M_{g}) = 1

for any spatial frequencies. Hence

C (\vec{m}; λ)

is the Fourier coefficient of diffraction order of

\vec{m} / p_{1}

with coherent illumination. For partially coherent illumination with micro-source array, the modulus of the coherence degree

| μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) | < 1

. Apparently in these cases Fourier coefficients of diffraction orders are further weighted down by their respective coherence factors. Therefore, for interferometers with monochromatic sources, the key step to the derivation is to derive closed-form formulas for the coherence-weighted Fourier coefficients

μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) \cdot C (\vec{m}; λ)

. Let us first examine the coefficients

C (\vec{m}; λ)

. They were derived by substituting Eq. (3) into the integral in Eq. (4), though the computation is tedious [25]:

C (\vec{m}; λ) {\begin{array}{l} 1, & if m = n = 0, \\ 0, & if m + n = odd, \\ 2 (1 - \cos Δ ϕ) \cdot f (k) \cdot f (l), & if m = 2 k, n = 2 l, \\ - \sin Δ ϕ \cdot \frac{\sin [(4 π λ R_{2} / M_{g} p_{1}^{2}) \cdot ({(k + 1 / 2)}^{2} + {(l + 1 / 2)}^{2})]}{π^{2} (k + 1 / 2) (l + 1 / 2)}, & if m = 2 k + 1, n = 2 l + 1, \end{array}

where f (k) is given by

f (k) = {(- 1)}^{⌊ 4 k λ R_{2} / M_{g} p_{1}^{2} ⌋} \cdot {\begin{array}{l} - \frac{1}{2}, & if k = 0, \\ \frac{\sin (4 k^{2} π λ R_{2} / M_{g} p_{1}^{2})}{k π}, & otherwise . \end{array}

Since the floor-function ⎿x⏌ is defined as the largest integer that is less or equal to x, the factor

{(- 1)}^{⌊ 4 k λ R_{2} / M_{g} p_{1} ⌋}

swings between 1 and −1 as its exponent changes. From Eqs. (6) and (7), it is clear that the Fourier coefficient

C (\vec{m}; λ)

depends on the grating phase shift Δϕ, system geometry setting R₁ and R₂, x-ray wavelength λ, and its diffraction order

\vec{m} = (m, n)

. Particularly

| C (\vec{m}; λ) |

decreases relatively slowly as 1/mn with increasing order

\vec{m} = (m, n)

. The efforts in derivation of Fourier coefficients

C (\vec{m}; λ)

can be traced back to earlier optics literatures [34–36]. But no general closed-form formula was found until recently [25]. The details of the derivation can be found in the Appendix of [25].

In order to derive the coherence-weighted Fourier coefficients, we need to derive the formula for coherence degree $μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})$ as well. Spatial coherence degree depends on the source intensity distribution $I_{source} (\vec{s})$ and system configuration geometry. The coherence degree can be found from the mutual intensity evolution of wavefields [33]. Since micro-sources in an array emit x-ray incoherently, the Van Citters-Zernike theorem of the mutual intensity propagation is applicable [33]. Using this theorem, we found that the reduced complex degree of the transverse coherence is given by

μ_{in} (\frac{λ R_{2} \vec{u}}{M_{g}}) = \frac{\int I_{source} (\vec{s}) exp (i 2 π R_{2} \vec{u} \cdot \vec{s} / M_{g} R_{1}) d \vec{s}}{\int I_{source} (\vec{s}) d \vec{s}} .

Substitute the source intensity distribution into Eq. (8), we will be able to derive

μ_{in} (λ R_{2} \vec{u} / M_{g})

, as is shown below in the next section.

3. Results

According to the method explained in above section, we found that Fourier expansion of the intensity fringe of a Talbot-Lau interferometer with monochromatic source is given by:

I_{λ} (\vec{r}; R_{1} + R_{2}) = \frac{I_{in, λ}}{M_{g}^{2}} \sum_{\vec{m} \in ℤ^{2}} μ_{in} (\frac{λ R_{2} \vec{m}}{M_{g} p_{1}}) \cdot C (\vec{m}; λ) \cdot exp (i 2 π \frac{\vec{m} \cdot \vec{r}}{M_{g} p_{1}}) .

While the coefficients

C (\vec{m}; λ)

in Eq. (9) are given by Eqs. (6) and (7), the spatial coherence degree

μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})

is determined by the source intensity distribution and system geometry. Consider a micro-source array consisting of (2N_x +1) × (2N_y +1) identical micro-sources. The centers of these micro-sources form a (2N_x + 1) × (2N_y+ 1) lattice of bidirectional periodicity of p₀. As for the intensity distribution of an individual micro-source, it could be uniform circular or Gaussian distribution, or other forms. For sake of discussion, we assume that each of the micro-sources is a small disk of diameter 2a with uniform intensity. We will discuss Gaussian micro-sources as well in section 4. This being so, we model the source intensity distribution

I_{source} (\vec{s})

generated from this micro-source array as

I_{source} (\vec{s}) = I_{s 0} \sum_{k = - N_{x}}^{N_{x}} \sum_{l = - N_{y}}^{N_{y}} Circ (\frac{| \vec{s} - k p_{0} {\vec{e}}_{x} - l p_{0} {\vec{e}}_{y} |}{a}),

where

Circ (r) = {\begin{array}{l} 1, & if | r | \leq 1, \\ 0, & otherwise, \end{array}

and I_s₀ is a constant specifying the intensity of each of the micro-sources. Substituting Eqs. (10) and (11) into Eq. (8), according to Van Citters-Zernike theorem, we found that the coherence degree is given by:

μ_{in} (λ R_{2} \vec{u} / M_{g}) = \frac{2 J_{1} (2 π a \cdot (M_{g} - 1) | \vec{u} | / M_{g})}{2 π a \cdot (M_{g} - 1) | \vec{u} | / M_{g}} \cdot W (λ R_{2} \vec{u} / M_{g}),

where J₁(x) is the Bessel function of the first kind, and J₁(x)/x is the so-called jinc function, and a is the radius of the micro-source. The second factor

W (λ R_{2} \vec{u} / M_{g})

in above equation denotes the interference factor, which represents the effective contribution from the interference of x-ray wave fields emitted from all the micro-sources. Through a lengthy calculation we found that

W (λ R_{2} \vec{u} / M_{g}) = \frac{\sin ((2 N_{x} + 1) π \cdot (M_{g} - 1) p_{0} \frac{u_{x}}{M_{g}}) \cdot \sin ((2 N_{y} + 1) π \cdot (M_{g} - 1) p_{0} \frac{u_{y}}{M_{g}})}{(2 N_{x} + 1) (2 N_{y} + 1) \cdot \sin (π (M_{g} - 1) p_{0} \frac{u_{x}}{M_{g}}) \cdot \sin (π (M_{g} - 1) p_{0} \frac{u_{y}}{M_{g}})} .

The derivation details can be found in the appendix. Since the fringe intensity is a superposition of diffraction orders with discrete frequencies

{\vec{u} = \vec{m} / p_{1}; \vec{m} = (m, n) \in ℤ^{2}}

, we found from Eq. (13) that

W (λ R_{2} \vec{m} / M_{g} p_{1})

achieves its maximal value, namely

W (λ R_{2} \vec{m} / M_{g} p_{1}) = 1

, when the following constructive interference condition is satisfied for any phase shift Δϕ

p_{0} = \frac{R_{1}}{R_{2}} M_{g} p_{1} .

The constructive interference condition Eq. (14) has a simple geometric interpretation: the fringe displacement generated by any off-center micro-source should be equal to a multiple of the fringe period.

It should be pointed that some authors wrote the constructive interference condition as p₀ = (R₁/R₂) · (M_gp₁/2) for the cases of π-gratings [4, 8, 26, 28]. This would be correct only if monochromatic x-ray sources were employed. In these cases, according to Eq. (6), there are no odd diffraction orders generated because sin(π) = 0. Therefore the fringe period is M_gp₁/2. But for polychromatic x-ray cases the grating phase-shift varies with x-ray photon energy, so the fringe period should be M_gp₁, as is given by Eq. (6) and (9). Hence the constructive interference condition is generally given by Eq. (14). For polychromatic x-ray cases, if the incorrect condition were used, the odd diffraction orders would be greatly suppressed, and the fringe visibility would be reduced.

Under this constructive interference condition, $μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})$ becomes just a jinc function:

μ_{in} (\frac{λ R_{2} \vec{m}}{M_{g} p_{1}}) = \frac{2 J_{1} (2 π \sqrt{m^{2} + n^{2}} \cdot a / p_{0})}{2 π \sqrt{m^{2} + n^{2}} \cdot a / p_{0}},

where the diffraction order is labeled by

\vec{m} = (m, n)

. It is important to note that

μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})

does not depend on x-ray wavelength, because each of the micro-sources in the array emits x-ray independently from each other, so the wavelength factors get canceled out according to the VanCitters-Zernike theorem of Eq. (8). The spatial coherence degree derived above is obviously equally applicable to the cases of the absorbing source gratings. To apply Eq. (15) to a source grating, p₀ is its bidirectional period, and 2a is the diameter of the individual apertures. As is shown in Fig. 2,

| μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) | \leq 1

and

| μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) |

generally decreases with increasing diffraction orders and diameters of individual micro-sources. Since the modulation of a given diffraction order is reduced by the factor

| μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) |

, hence the higher the diffraction order is, the larger the loss in fringe modulation incurs.

Fig. 2 Value of $μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})$ with different $\vec{m}$ with the constructive interference condition Eq. (14), where p₀ is the period of the micro-source array, and a is the radius of the individual micro-source.

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For polychromatic x-ray, the fringe intensity is a sum of its spectral components given by Eq. (9). Hence with a polychromatic source the fringe intensity is given by

\begin{array}{l} I (\vec{r}; R_{1} + R_{2}) = \frac{I_{in}}{M_{g}^{2}} \sum_{\vec{m} \in ℤ^{2}} V (\vec{m}) \cdot exp (i 2 π \frac{\vec{m} \cdot \vec{r}}{M_{g} p_{1}}), \\ V (\vec{m}) \equiv \int S (λ) \cdot μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) \cdot C (\vec{m}; λ) d λ, \end{array}

where S(λ) is the normalized source x-ray fluence spectrum, and we call

V (\vec{m})

as the spectrum-averaged and coherence-weighted Fourier coefficients of intensity fringes. As we explained earlier,

μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1})

is indeed independent of wavelength.

On the other hand, we may rewrite Eq. (16) as:

\begin{array}{l} I (\vec{r}) = \frac{I_{in}}{M_{g}^{2}} \cdot {1 + 2 \sum_{(m > 0, n \neq 0)} V (\vec{m}) \cdot \cos (2 π \frac{m x + n y}{M_{g} p_{1}}) + \\ + 2 \sum_{(m > 0, n = 0)} V (\vec{m}) \cdot [\cos (2 π \frac{m x}{M_{g} p_{1}}) + \cos (2 π \frac{m y}{M_{g} p_{1}})]} . \end{array}

For sake of discussion, we assume that the detector has pixel size p_d and a filling factor of unity without pixel cross talk. Incorporating the “blurring” effect of pixel binning by multiplying the sinc functions (sinc(x) ≡ sin(πx)/πx), we can express the fringe intensity with the detector pixel re-binning effect as:

\begin{array}{l} I (\vec{r}) = \frac{I_{in}}{M_{g}^{2}} \cdot {1 + 2 \sum_{(m > 0, n \neq 0)} V (\vec{m}) \cdot sinc (\frac{m p_{d}}{M_{g} p_{1}}) \cdot sinc (\frac{n p_{d}}{M_{g} p_{1}}) \cdot \cos (2 π \frac{m x + n y}{M_{g} p_{1}}) + \\ + 2 \sum_{(m > 0, n = 0)} V (\vec{m}) \cdot sinc (\frac{m p_{d}}{M_{g} p_{1}}) \cdot [\cos (2 π \frac{m x}{M_{g} p_{1}}) + \cos (2 π \frac{m y}{M_{g} p_{1}})]} . \end{array}

Combining Eq. (18), with Eqs. (6) and (15), we derived the closed-form formulas for computing the intensity fringes. These formulas incorporate the effects of partial spatial coherence, spectral average and detector pixel re-binning. These formulas, Eqs. (6), (15) and (18) are the central results of this work.

Since the quantities $| μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) |$ , $| C (\vec{m}; λ) |$ and |sinc(mp_d/M_gp₁)| all decrease with increasing |m| and |n|, only several low diffraction orders with small |m| and |n| are able to make non-negligible contributions to the intensity fringes. This feature facilitates the computation of the fringe visibility, as is demonstrated in the following example.

Let us consider the design of a grating imaging system with the inverse geometry configuration [26–28, 28], which was briefly mentioned in section 1. In the inverse geometry configuration, the phase grating G₁ is placed close to the micro-source array such that R₂ ≥ R₁ and the magnification factor M can be as high as a few tens. With such high-magnification settings, the resulting interference fringes can be directly resolved by the imaging detector without using an additional absorbing grating to mask the detector. Consequently, the interferometers with such inverse geometry will significantly reduce radiation dose to the subjects. In design of an inverse geometry interferometer, matching R₂ to a Talbot distance requires long source-detector distances of few meters, owing to the high-magnification factor required in such a system [28]. However, with polychromatic sources, a broadband x-ray spectrum allows a large range of R₁ and R₂ values available for comparable fringe visibility values. One can then select shorter R₁ and R₂ distances to reduce the system size. In the literature researchers employed laborious computer simulations to compute and compare the visibility values attainable with different system geometries [26, 27, 37]. In contrast to resorting to computer simulations, we show here that the closed-form formulas of Eqs. (6), (15) and (18) provide a versatile tool for computing fringe visibilities.

As one example, consider a single phase grating interferometer with a source array of micro-tungsten-targets, which operates at an acceleration voltage 35 kV and the generated x-ray is filtered with 50μm rhodium filter. This tube voltage and filtration are one of typical settings used in breast imaging. The resulting broad x-ray spectrum spreading from 7 keV to 35 keV is shown in Fig. 3. The spectrum peaks at 22.5 keV and is shaped by a 50μm rhodium filter with the k-edge at 23.2 keV. The interferometer is assumed to employ a 25 × 25 array of micro-targets of 2μm in diameter each. The phase grating is a two-dimensional checkerboard phase grating of p₁ = 5μm and phase shift Δϕ = π/2 at 20-keV. We assume that the fringe pattern is to be resolved by using a detector with pixel size p_d = 25μm, and a high magnification factor of 15 is used. While there are many geometry settings can be used to implement the high-magnification setting [25, 28], here we set the source array-to-grating distance of R₁ = 8.4 cm and a grating-to-detector distance of R₂ = 117.6 cm to make the source-detector distance compact. It is easy to check that such a selection of R₁ and R₂ grossly mismatches any of the Talbot distances for 20 keV x-ray [25,28]. With such a choice we just want to demonstrate the capability of Eqs. (6), (15) and (18) to predict the fringe visibility for any geometry configuration and any x-ray spectrum. In order to satisfy the constructive interference condition of Eq. (14), we set the period of the micro-source array to p₀ = 5.36μm. Using Eqs. (6), (15) and (18), we computed the spectrum-averaged and coherence-weighted Fourier coefficients $V (\vec{m})$ and the associated sinc-functions. In this example we found that only first few diffraction orders with $| \vec{m} | \leq 2$ contribute to the fringe intensity, while the other orders with $| \vec{m} | > 2$ can be neglected within accuracy of 0.2%. Therefore, we found that the fringe intensity of the exemplary grating interferometer is given by the following expression:

\begin{array}{l} I (\vec{r}; R_{1} + R_{2}) = \frac{I_{in}}{M_{g}^{2}} {1 + 2 \times 0.168 \times [- \cos (2 π \frac{x + y}{M_{g} p_{1}}) + \cos (2 π \frac{x - y}{M_{g} p_{1}})] - \\ - 2 \times 0.3333 \times [\cos (2 π \frac{2 x}{M_{g} p_{1}}) + \cos (2 π \frac{2 y}{M_{g} p_{1}})]} . \end{array}

Equation (19) shows the composition of the fringe pattern. Obviously the first part in the curly bracket on the right hand side of Eq. (19) contributes to the constant background intensity. The second part in the curly bracket, which is a cross-sinusoidal term in x and y, represents a checkerboard pattern, which is superimposed on the background with an intensity-modulation weighting of (2 × 0.168), and the third part represents a mesh pattern, which is superimposed on the pattern with a tiny intensity-modulation weighting of (2 × 0.0333).

Fig. 3 Utilized x-ray spectrum

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As for the fringe visibility, we need to consider the down-sampling effects of the pixel re-binning as well. Since in this example M_gp₁ = 75μm and p_d = 25μm, there are three sampled points in a fringe period along each of the directions. Obviously the cross-sinusoidal term in Eq. (19) determines the locations of the maximum and minimum intensities. It is easy to see that the intensity reaches its maximum $I_{max} = 1.5706 \cdot I_{in} / M_{g}^{2}$ at (x − p_d/2, y − p_d/2) = (k, l) · M_gp₁, and (x − p_d/2, y − p_d/2) = (2/3 + k, 2/3 + l) · and (x, y) = (2/3+k, l) ·Mgp₁, where k, l ∈ ℤ. This being so, the visibility V =(I_max−I_min)/(I_max+I_min) of the fringe pattern is equal to 0.4725. Note also that Eq. (19) predicts not only the fringe visibility for the interferometer, but also the intensity values at all pixels. Hence Eq. (19) is able to predict the whole interference fringe pattern of the interferometer.

In order to validate our prediction, we performed computer simulations. We simulated x-ray Fresnel diffraction through a single phase grating interferometer with a source array of micro-tungsten-targets. We set the simulated system configuration identical to that described earlier in above example. Assuming a point-like source, we first simulated Fresnel diffraction to propagate the wavefront of the phase grating G₁ to the detector plane with extremely small sampling pitch p_s = p_d/(2⁸Mg). From the resulting wavefront we calculated the intensity image I₁. To simulate the fringe intensity pattern generated by the micro-source array, we computed the convolution of intensity image I₁ with the source intensity $I_{source} (\vec{s})$ of Eq. (10). Finally we apply the ray-binning to the convoluted intensity to obtain the final binned and down-sampled intensity image that measured by the detector. Figure 4(a) shows the resulting intensity fringe pattern from the simulation, and Fig. 4(b) is the fringe pattern obtained from Eq. (19). The two agree very well to each other. The maximum and minimum values in the simulated intensity fringe pattern are 1.5746 and 0.5673, respectively. Thereby the visibility of the fringe pattern is equal to 0.4703, which is in good agreement with the analytical prediction presented earlier. This good agreement validates our general theory presented by Eqs. (6), (15) and (18).

Fig. 4 (a). The fringe image from the computer simulation. In the simulation, R₁ = 8.4cm, R₂ = 117.6cm, the phase grating period p₁ = 5μm. The source is a 25 × 25 array of micro-tungsten-targets. The source array has a bidirectional period p₀ = 5.36μm, and each micro-target has a diameter 2a = 2μm. The source array operate at 35 kV and with a 50μm rhodium filter. The maximum, minimum and thereby the visibility of the fringes are 1.5746, 0.5673 and 0.4703 respectively. (b). The fringe image obtained from Eq. (19).

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As another example, consider a design modification to the Talbot-Lau interferometer descibed in above example. Suppose the micro-target’s diameter is enlarged from 2 μm to 2.5 μm, while the design of Talbot-Lau interferometer is otherwise identical. Using Eq. (18), we found the intensity fringe pattern is changed to:

\begin{array}{l} I (\vec{r}; R_{1} + R_{2}) = \frac{I_{in}}{M_{g}^{2}} {1 + 2 \times 0.133 \times [- \cos (2 π \frac{x + y}{M_{g} p_{1}}) + \cos (2 π \frac{x - y}{M_{g} p_{1}})] - \\ - 2 \times 0.0182 \times [\cos (2 π \frac{2 x}{M_{g} p_{1}}) + \cos (2 π \frac{2 y}{M_{g} p_{1}})]} . \end{array}

Following the same analysis method as that applied to the fringe pattern of Eq. (19), we found that the fringe visibility with this design modification is significantly reduced to 0.385. The computer simulation results in a visibility value of 0.386, which is again in good agreement with our theoretical prediction.

4. Discussion and conclusions

X-ray grating interferometry has many potential applications in medical imaging and material science. The intensity modulation of a fringe pattern is characterized by the visibility of the fringe. Since formation of high-modulation fringe pattern is critical to robust grating interferometry, the fringe visibility is a common figure of merit in design of x-ray grating-based interferometer. So far there is no quantitative theory yet that is able to predict fringe visibility, which actually depends on x-ray spectrum, spatial coherence degree of x-ray illumination, system geometry, and detector pixel re-binning. In literature intensive and burdensome computer simulations were used to predict the fringe visibility values of interferometers with polychromatic x-ray sources. In this work, we developed a general theory to predict the fringe visibility values of phase grating interferometers. Our theory, as is instilled in Eqs. (6), (15) and (18), provides closed-form formulas to compute the fringe visibility of a phase grating interferometer.

In previous discussion we assume that each of the micro-sources is a small disk of uniform intensity. But our theory applies to the case with Gaussian micro-source array, in which each of the micro-sources has a Gaussian intensity distribution. Consider a micro-source array consisting of (2N_x +1)×(2N_y +1) identical Gaussian micro-sources. The source intensity distribution $I_{source} (\vec{s})$ from a Gaussian micro-source array can be modeled as:

I_{source} (\vec{s}) = \frac{Q}{2 π σ^{2}} \sum_{k = - N_{x}}^{N_{x}} \sum_{l = - N_{y}}^{N_{y}} \exp [- \frac{{(\vec{s} - k p_{0} {\vec{e}}_{x} - l p_{0} {\vec{e}}_{y})}^{2}}{2 σ^{2}}],

where σ is the standard deviation of the Gaussian distribution, and Q is a constant equal to the area integral of

I_{source} (\vec{s})

. Substituting Eq. (21) into Eq. (8), we found that, when the constructive interference condition of Eq. (14) is satisfied, the coherence degree for such a Gaussian micro-source array is given by:

μ_{in} (\frac{λ R_{2} \vec{m}}{M_{g} p_{1}}) = \exp (- \frac{2 π^{2} (m^{2} + n^{2}) σ^{2}}{p_{0}^{2}}) .

Substituting Eq. (22) into Eq. (18), one can compute the fringe visibility of an interferometer with a Gaussian micro-source array.

Our theory is a general quantitative theory that is versatile in other usage as well. For example, in the cases with spatially coherent x-ray illumination generated by synchrotron radiation or a point-like focal spot, the theory is applicable by simply letting $μ_{in} (λ R_{2} \vec{m} / M_{g} p_{1}) = 1$ in Eq. (16). Moreover, in the cases with a source of single spot, the fringe intensity pattern equation Eq. (18) remains applicable, provided p₀ in Eq. (15) or Eq. (22) is replaced by (R₁/R₂)·M_gp₁.

The formulas of Eqs. (6), (15) and (18) derived in this work can be applied to one-dimensional grating interferometers as well, provided some dimensional reduction changes are imposed. The diffraction order label $\vec{m} = (m, n)$ in these formulas will be reduced to the one-integer label m in one-dimensional (1D) cases. Hence, when apply Eq. (15) and (18) to 1D interferometers, one should let n = 0 in these formulas. More importantly, for the 1D cases the two-dimensional Fourier coefficients $C (\vec{m}; λ)$ should be replaced by the one-dimensional C(m;λ) given as follows [25]:

C (m; λ) = {\begin{array}{l} 1, & if m = 0, \\ - (1 - \cos Δ ϕ) \cdot {(- 1)}^{⌊ 4 k λ R_{2} / M_{g} p_{1}^{2} ⌋} \cdot \frac{\sin (4 k^{2} π λ R_{2} / M_{g} p_{1}^{2})}{k π} & if m = 2 k, k \neq 0, \\ i \cdot \sin Δ ϕ \cdot \frac{\sin [(4 π λ R_{2} / M_{g} p_{1}^{2}) \cdot {(k + 1 / 2)}^{2}]}{π \cdot (k + 1 / 2)}, & if m = 2 k + 1, \end{array}

where

i = \sqrt{- 1}

. With these dimensional reduction changes, the intensity fringe pattern for an one-dimensional Talbot-Lau interferometer is given by

\begin{array}{l} I (x) = \frac{I_{in}}{M_{g}^{2}} {1 + 2 \sum_{k > 0} [V (2 k) \cdot sinc (\frac{2 k p_{d}}{M_{g} p_{1}}) \cdot \cos (2 π \frac{2 k x}{M_{g} p_{1}}) + \\ + V (2 k - 1) \cdot sinc (\frac{(2 k - 1) p_{d}}{M_{g} p_{1}}) \cdot \sin (2 π \frac{(2 k - 1) x}{M_{g} p_{1}})]}, \\ V (m) \equiv \int S (λ) \cdot μ_{i n} (λ R_{2} m / M_{g} p_{1}) \cdot D (m; λ) d λ, \end{array}

where D(m;λ) = C(m;λ), if m is even and D(m;λ) = i · C(m;λ), if m is odd.

In conclusion, we developed a general quantitative theory to predict the fringe visibility values of Talbot-Lau interferometers with polychromatic x-ray sources. The derived closed-form expressions for the interference fringes provide a versatile tool for design optimization of grating-based x-ray interferometers. Since polychromatic anode sources with limited partial spatial coherence prevail in medical imaging, the presented theory will be especially useful for future medical applications of x-ray Talbot-Lau interferometry.

Funding

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

Appendix: Derivation of the interference factor in Eq. (13)

The reduced complex degree of spatial coherence $μ_{in} (λ R_{2} \vec{u} / M_{g})$ is determined by the source intensity distribution and system geometry. In the derivation one encounters a key integral as follows:

\int Circ (\frac{| \vec{s} |}{a}) \cdot \exp (i 2 π \frac{R_{2} \vec{u}}{M_{g} R_{1}}) d \vec{s} = \frac{M_{g} a R_{1}}{R_{2} | \vec{u} |} J_{1} (2 π a \frac{R_{2} | \vec{u} |}{M_{g} R_{1}}),

where J₁(x) is the Bessel function of the first kind. Hence, substituting Eqs. (10) and (11) into Eq. (8), one obtains:

\begin{array}{l} μ_{in} (\frac{λ R_{2} \vec{u}}{M_{g}}) = \frac{2 J_{1} (2 π a \cdot (M_{g} - 1) | \vec{u} | / M_{g})}{2 π a \cdot (M_{g} - 1) | \vec{u} | / M_{g}} \times \\ \times \frac{\sum_{m = - N_{x}}^{N_{x}} \sum_{n = - N_{y}}^{N_{y}} \exp [i 2 π (M_{g} - 1) p_{0} (m u_{x} + n u_{y}) / M_{g}]}{(2 N_{x} + 1) (2 N_{y} + 1)} . \end{array}

Working out the summations in above equation, one is led to Eq. (13).

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Predicting visibility of interference fringes in X-ray grating interferometry

Abstract

1. Introduction

2. Materials and methods

3. Results

4. Discussion and conclusions

Funding

Appendix: Derivation of the interference factor in Eq. (13)

References and links

Cited By

Figures (4)

Equations (26)

Optics Express