# X-ray Interferometry Using a Modulated Phase Grating: Theory and Experiments

Hunter Meyer Department of Physics and Astronomy, Louisiana State University,
Baton Rouge, LA, 70803  Joyoni Dey Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, LA, 70803  Sydney Carr Department of
Physics and Astronomy, Louisiana State University, Baton Rouge, LA, 70803
Kyungmin Ham Center for Advanced Microstructures and Devices, Louisiana State
University, Baton Rouge, LA, 70806  Leslie G. Butler Department of Chemistry,
Louisiana State University, Baton Rouge, LA, 70803  Kerry M. Dooley Cain
Department of Chemical Engineering, Louisiana State University, Baton Rouge,
LA, 70803  Ivan Hidrovo Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA, 70803  Department of Radiation Therapy, Solón
Espinosa Ayala Oncological Hospital, Quito, Ecuador.  Markus Bleuel Adelphi
Technology, Inc., Redwood City, CA, 94063  Tamas Varga The Environmental
Molecular Sciences Laboratory, Pacific Northwest National Laboratory,
Richland, WA  Joachim Schulz Microworks GmbH, Schnetzlerstr. 9, 76137
Karlsruhe, Germany  Institute of Microstructure Technology, Karlsruhe
Institute of Technology, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-
Leopoldshafen, Germany.  Thomas Beckenbach Microworks GmbH, Schnetzlerstr. 9,
76137 Karlsruhe, Germany  Konradin Kaiser Microworks GmbH, Schnetzlerstr. 9,
76137 Karlsruhe, Germany

###### Abstract

X-ray grating interferometry allows for the simultaneous acquisition of
attenuation, differential-phase contrast, and dark-field images, resulting
from X-ray attenuation, refraction, and small-angle scattering, respectively.
The modulated phase grating (MPG) interferometer is a recently developed
grating interferometry system capable of generating a directly resolvable
interference pattern using a relatively large period grating envelope function
that is sampled at a pitch that allows for X-ray spatial coherence using a
microfocus X-ray source or by use of a source G0 grating that follows the Lau
condition. We present the theory of the MPG interferometry system for a
2-dimensional staggered grating, derived using Fourier optics, and we compare
the theoretical predictions with experiments we have performed with a
microfocus X-ray system at Pennington Biomedical Research Center, LSU. The
theoretical and experimental fringe visibility is evaluated as a function of
grating-to-detector distance. Quantitative experiments are performed with
porous carbon and alumina samples, and qualitative analysis of attenuation and
dark-field images of a dried anchovy are shown.

Keywords: X-ray interferometry ; modulated phase grating ; diffraction
grating; dark-field; porosity

## 1\. Introduction

X-ray grating interferometry allows for the simultaneous acquisition of
attenuation, differential-phase contrast (DPC), and dark-field images. In
contrast to traditional X-ray radiography systems, one or more diffraction
gratings are placed in the path of the X-ray beam so that a periodic
interference pattern is produced, commonly referred to as interference
fringes, which is approximately sinusoidal. With no object in the path of the
X-ray beam, the interference pattern is typically referred to as the reference
or blank image. When an object is placed in the path of the X-ray beam, it’s
physical properties are imaged by measuring the perturbation to the reference
fringe pattern. The pattern is perturbed in three ways, resulting in images
with three distinct contrast mechanisms [1]. Attenuation causes a reduction in
the average value of the fringe pattern, producing the attenuation image.
Refraction results in a phase shift of the pattern, producing the DPC image.
Small angle scattering reduces a parameter known as fringe visibility, which
is simply the height of the fringes relative to the average value, producing
dark-field images. Interferometry has potential for a variety of applications
in science and medicine, including lung imaging [2, 3, 4, 5], breast imaging
[6, 7, 8, 9], arthritis imaging [10, 11], osteoporosis imaging [12], pore size
analysis [13], additive manufacturing quality assurance [14, 15], etc.

There are presently several grating interferometers in the literature,
including the Talbot-Lau Interferometer (TLI) [16, 17, 18], Dual Phase Grating
Interferometer (DPGI) [19, 20], and Modulated Phase Grating Interferometer
(MPGI) [21, 22, 23, 24, 25]. The TLI has a phase grating, G1, that produces
interference fringes that are not directly resolvable by typical detectors,
meaning an analyzer grating, G2, is required to create visible Moiré patterns,
resolvable at the detector. The analyzer grating’s pitch is determined by the
geometric magnification of the G1 grating, meaning the geometry of the system
is fixed. While this interferometer is highly sensitive due to the low-period
fringes produced, the requirement of an analyzer grating, which is an
absorption grating, doubles the dose per image for similar fluence at the
detector. The DPGI achieves directly resolvable patterns using two phase
gratings separated by a few millimeters, without the need for an analyzer
grating. The Moiré pattern produced has a beat pattern directly resolvable by
the detector, with the high-frequency components being washed out by blur from
the detector and X-ray source. The DPGI configurations are typically far-field
geometries and are often called a Far-Field interferometry systems [19]. The
DPGI’s two phase gratings must be co-aligned for proper fringe formation,
which often requires time-consuming alignment procedures. This problem exists
for the TLI as well.

Figure 1: Schematic of the modulated phase grating interferometer with a
micro-focus X-ray source. Here the 1-dimensional RectMPG parameters are shown,
where $W$ is the period of the grating’s envelope function, $p$ is the grating
pitch, $h_{1}$ and $h_{2}$ are the phase heights, and the geometry is defined
by the source-to-grating distance, $L_{1}$, the grating-to-detector distance,
$z$, and the object-to-detector distance, $D_{od}$. $x_{g}$ denotes the phase
stepping direction.

The modulated phase grating interferometer (MPGI), originated by our group
[21, 22, 23, 24], consists of a single grating where the heights of the
grating bars follows an envelope function, with a relatively large period,
$W$. The envelope function is sampled at a high-frequency pitch, $p$, to meet
the coherence requirements similar to that of the TLI and DPGI systems. In
this context, the envelope period, $W$, and the sampling pitch, $p$, should
not be confused, since this terminology is used interchangeably for a binary
diffraction grating such as those in the TLI or DPGI systems. The fringes
produced have many high-frequency harmonics that are washed out by the
detector and low-frequency harmonics that result from the envelope function,
analogous to the beat pattern produced by the DPGI. Images can be calculated
using either a single-shot [26, 27] or a phase stepping procedure [28]. While
the single-shot methodology is simpler, the fringe resolution is not as high,
so for this study, phase stepping was used.

A key benefit of the MPGI is the simplicity of using a single grating, with
system geometry shown in Figure 1. Since no analyzer grating is present, MPG
placement is not limited by Talbot distance constraints as in the TLI. The MPG
can be moved continuously, allowing for a range of system autocorrelation
lengths. In the TLI, the autocorrelation length can only be changed by moving
the object, so an extra degree of freedom is introduced for the MPGI. For the
DPGI, the autocorrelation length can be changed by changing the inter-grating
distance, but grating alignment must be maintained. Another more subtle
advantage is that the MPGI system is always first harmonic dominant. In
contrast, the TLI and DPGI systems are often second harmonic dominant; the
first harmonic disappears if they are illuminated by a monochromatic source.
The first harmonic reappears when illuminated by a polychromatic source [20],
causing the interference fringes to not be as sinusoidal, leading to
ambiguities in the systems’ phase sensitivity, autocorrelation length, and
visibility.

In this study, we present the theory of fringe formation for the MPGI for a
polychromatic microfocus X-ray tube source. We then compare the results with
experiments performed in the Keck Imaging Laboratory at Pennington Biomedical
Research Center (PBRC). In a previous study, we presented the theory of the
1-dimensional MPG with comparisons to simulations performed using the
Sommerfeld Rayleigh Diffraction Integral (SRDI) simulator, with several orders
of magnitude of decreased simulation time and comparable results [24]. In this
paper, we will extend the 1-dimensional MPG theory to account for the
staggering of the grating bars that are implemented during fabrication for
grating stability.

The presented theory models diffraction from the phase and attenuation of the
grating bars using Fourier optics, as well as the effects of using a
polychromatic source, finite focal spot size, and detector point spread
function (PSF). The theory allows for the rapid calculation of the fringe
profile and fringe visibility as a function of system geometry and MPG
parameters. The theory can also be used to aid in developing future MPG
interferometry systems by allowing for rapid visibility calculations for a
wide range of system geometries, energies, and grating designs. The
experiments performed at PBRC include a measurement of the fringe visibility
as a function of grating-to-detector distance and images taken of a variety of
samples. Experiments were performed using rectangular envelope MPGs, known as
RectMPGs, with a schematic shown in Figure 1. Additionally, we measured the
mean attenuation and dark-field signal for several porous carbon and alumina
samples and acquired images of a dried anchovy.

## 2\. Methods

### 2.1. Theory

The theory of the 1-dimensional MPG was originally derived in [24], which will
hold for a 2-dimensional MPG if the grating is constant in one dimension.
However, for this study, multiple MPGs were produced by Microworks GmbH,
Germany, where the grating bars were staggered for improved structural
stability, following a Bridge design, [29], as shown in Figure 2. Here we
present a revised version of the MPG theory that accounts for the staggered
grating bars. This study will compare the theoretically calculated visibility
with experimental measurements of fringe visibility as a function of grating-
to-detector distance.

#### 2.1.1. A Modified Transmission Function in 1 Dimension

First, we will briefly review the theory of the 1-dimensional MPG [24], then
detail the modifications necessary for 2 dimensions. For the 1-dimensional
MPG, the grating transmission function is given as

$\displaystyle
T(x)=\left[g(x)\sum_{n=-\infty}^{\infty}\delta(x-np)+\sum_{n=-\infty}^{\infty}\delta(x-np-p/2)\right]\star\operatorname{rect}\left(\frac{x}{\alpha
p}\right)$ (1) $\displaystyle
g(x)=\sum_{m=\infty}^{\infty}g_{m}\exp\left(\frac{2\pi jmx}{W}\right)$ (2)

The envelope function is represented by $g(x)$ and is periodic. It can be
represented as a Fourier Series of period $W$, with Fourier coefficients,
$g_{m}$. In Equation 1, the first series of $\delta$ functions samples the
envelope function at a sampling period of $p$, and the convolution with the
$\operatorname{rect}$ function places a grating bar at each sample, since
$f(x)\star\delta(x-x_{0})=f(x-x_{0})$, where $\star$ represents the
convolution. $\alpha$ represents the duty cycle of the grating and $j$ is the
imaginary number. The second series of $\delta$ functions is responsible for
the gaps between the grating bars, where the grating transmission equals 1,
not 0.

The envelope function’s Fourier coefficients, $g_{m}$, are energy-dependent
and account for the transmission function’s amplitude reduction and phase
change. The physical heights of the grating bars are designed to follow the
desired phase envelope at the design energy, $E_{D}$. The grating attenuation
is determined by the physical heights of the grating bars resulting from that
design. The attenuation and phase profile of the grating are energy-dependent,
therefore we represent the grating coefficients as $g_{m}(\lambda)$.

The polychromatic intensity is simply the incoherent superposition of each
monochromatic intensity weighted by the energy spectrum. This means the energy
spectrum affects the detector intensity in two ways: by diffraction and by the
grating coefficients. The field amplitude can be derived using the angular
spectrum method under the Fresnel approximation (Goodman [30]). The detector
intensity can be calculated as simply the square of the field amplitude,
$I(x,z)=|U(x,z)|^{2}$.

It is difficult to generalize Equation 1 to 2 dimensions, since defining the
regions between the grating bars becomes cumbersome. Instead, an equivalent
transmission function, written in another form, is introduced. Here, the
grating bars result from sampling, and a constant transmission of 1 is
included to account for the regions between the grating bars. This constant
transmission is then subtracted from $g(x)$ so that the transmission of the
grating bars follows the envelope function. The equivalent transmission
function in 1 dimension is shown in Equation 3. It is straightforward to show
that this function is exactly the same as Equation 1.

The new transmission function is used to derive the field amplitude shown in
Equation 4 using the angular spectrum method [30]. Though there are notational
differences, the calculated field amplitude is exactly the same as what is
found in our previous work [24] (using the first transmission function
Equation 1).

$\displaystyle
T(x)=1+\left[(g(x)-1)\sum_{n=-\infty}^{\infty}\delta(x-np)\right]\star
rect\left(\frac{x}{\alpha p}\right)$ (3) $\displaystyle
U(x,z)=1+\alpha\sum_{m}\sum_{n}b_{1}(m,n,z)\exp\left(j2\pi
x\left(\frac{m}{W}+\frac{n}{p}\right)\right)-\alpha\sum_{n}b_{2}(n,z)\exp\left(j2\pi
x\left(\frac{n}{p}\right)\right)$ (4) $\displaystyle
b_{1}(m,n,z)=g_{m}\operatorname{sinc}\left(\alpha
p\left(\frac{m}{W}+\frac{n}{p}\right)\right)\exp\left(-j\pi\lambda
z\left(\frac{m}{W}+\frac{n}{p}\right)^{2}\right)$ (5) $\displaystyle
b_{2}(n,z)=\operatorname{sinc}\left(\alpha n\right)\exp\left(-j\pi\lambda
z\left(\frac{n}{p}\right)^{2}\right)$ (6)

Figure 2: An SEM image of the staggered 2D modulated phase grating, with
$p_{x}$ and $p_{y}$ labeled. $\alpha_{x}$ and $\alpha_{y}$ are the duty cycles
of the grating in the x- and y-dimension, respectively.

#### 2.1.2. The 2-Dimensional Transmission Function

Equation 3 is adapted to two-dimensions, shown in Equation 7. A noticeable
change is the reintroduction of a second sampling function, (not to be
confused with the double sampling in Equation 1). Additionally, there is now a
pitch in each direction, $p_{x}$ and $p_{y}$, as well a duty cycle in each
direction, $\alpha_{x}$ and $\alpha_{y}$. Here, the two sampling terms both
follow $g(x)$ but are offset by $p_{x}/2$ and $p_{y}/2$, to account for the
staggering of the grating bars. Note that the staggering is removed if
$\alpha_{y}=1$, $\alpha_{x}=\alpha/2$, $p_{x}=2p$, and the 2D transmission
function will be equivalent to the 1D transmission function.

$T(x,y)=1+\Bigg{\\{}(g(x)-1)\biggl{[}\sum_{n}\sum_{l}\delta(x-np_{x})\delta(y-lp_{y})+\sum_{n}\sum_{l}\delta\left(x-np_{x}-\frac{p_{x}}{2}\right)\delta\left(y-lp_{y}-\frac{p_{y}}{2}\right)\biggr{]}\Bigg{\\}}\star
rect\left(\frac{x}{\alpha_{x}p_{x}}\right)rect\left(\frac{y}{\alpha_{y}p_{y}}\right)$
(7)

In Appendix A, the field amplitude is derived using the angular spectrum
method [30], followed by the Fresnel scaling theorem [31] to scale the
intensity from that of a plane-wave to that of a point-source. The intensity
at the detector is calculated as

$I(x,y,z)=1+\sum_{l}\biggl{[}c(l,x,z)+c^{*}(-l,x,z)+d(l,x,z)\biggr{]}\exp\left(j2\pi
y\frac{l}{Mp_{y}}\right)$ (8)

where $c(l,x,z)$ is as defined in Appendix A and $d(l,x,z)=c(l,x,z)\star
c^{*}(-l,x,z)$. $c(l,x,z)$ and $d(l,x,z)$ contain the x-harmonics.

Next, as detailed in Appendix A (Equations 40-41), to simplify calculations,
the 2D intensity is approximated by the $l=0$ harmonic, where $l$ represents
the harmonic number in the y-dimension. This means that we keep only the zero-
th harmonic (mean) in the y-direction. This approximation is valid when the
detector and source blur are sufficiently high enough to blur the non-zero
y-harmonics that result from the staggering of the grating bars necessary for
fabrication stability. Notably, while this approximation removes the
dependence of $p_{y}$, the fringe profile still depends on the duty cycle in
the y-direction $\alpha_{y}$. The staggering of the grating bars still affects
the fringe visibility, even when there is sufficient blur to approximate the
2-dimensional intensity profile into a single dimension.

$I(x,y,z)\approx 1+c(0,x,z)+c^{*}(0,x,z)+d(0,x,z)$ (9)

Appendix A Equation 39 is the intensity calculated under the $l=0$
Approximation and is reproduced in Equation 9. The $l=0$ approximation
provides a powerful tool to make 2D intensity calculations fast under some
often realistic conditions when the detector point spread function (PSF) and
source blur combination is larger than the (magnified) grating pitch in the
y-dimension, $p_{y}$. This approximation greatly reduces the computation
required, since only one dimension needs to be calculated, instead of two.

The theory is evaluated in several ways in the Results, Section 3.1. The
intensity profile calculated for the 2D MPG under the $l=0$ approximation is
shown to be equivalent to a full 2D MPG simulation with sufficient blur in the
y-direction and a sufficiently small $p_{y}$. It is also shown that the $l=0$
approximation is further reduced to the true 1D MPG—where it is constant in
one dimension and the field amplitude is shown in Equation 4—under the
conditions of $\alpha_{x}=\alpha/2$, $p_{x}=2p$, $\alpha_{y}=1$.

#### 2.1.3. Post-Processing and Visibility Calculations

The intensity derived in Appendix A Equation 40 is the intensity profile when
the source is a monochromatic point source. To properly model the fringe
profile and visibility measured in an experiment, we must include a
polychromatic source, finite focal spot size, and detector point spread
function. Additional post-processing includes modeling the phase stepping
procedure and downsampling to the detector sample rate.

The intensity of a polychromatic point source can be found by integrating over
the energy spectrum, $S(E)$.

$I_{poly}(x,y,z)=\int_{E}I(x,y,z;E)S(E)dE$ (10)

The effect of finite focal spot size can be found by convolving the point-
source intensity profile with the magnified source profile. It’s important to
note that the source is magnified by a factor of $M-1=\frac{z}{L1}$, as shown
in Appendix B of [24],

$I_{poly,spot}(x,y,z)=I_{poly}(x,y,z)\star\sigma(\frac{x}{M-1})$ (11)

Finally, the detector point spread function (PSF) is convolved to get the
final intensity profile,

$I_{final}(x,y,z)=I_{poly,spot}(x,y,z)\star PSF(x,y)$ (12)

The phase stepping procedure is modeled by shifting the intensity profile by
the phase step size, $x_{s}$, multiplied by the magnification factor, $M$.
Following this, downsampling is performed to match the detector pixel size.
The intensity is initially calculated at a sample rate much higher than the
pixel size. For the purposes of this study, the intensity profile is initially
calculated at a rate of $0.1\>\mu m$, and the Dexela 1512 detector used in
this study has a pixel size of $75\>\mu m$. It’s important to note that the
detector PSF and pixel size are not the same. Additionally, in the theoretical
simulations, downsampling should occur after phase stepping. The visibility is
measured following the methods of Marathe et al. [28].

### 2.2. Experimental Methods

Several experiments were performed in the Keck Imaging Laboratory at
Pennington Biomedical Research Center (PBRC) for the purposes of this study.
Two MPGs were used, both with rectangular phase modulation and the same design
parameters, listed in Table 1. The gratings, referred to as MPG7 and MPG8,
were manufactured by Microworks, GmbH. They provided the measured heights of
the grating structures, listed in Table 2, where it is seen that MPG8 has a
larger difference in the heights than MPG7. Due to the novelty of the grating
fabrication process, the height of the grating structures ($h_{1}$ and
$h_{2}$) is not highly reproducible, which can explain the observed
differences. A Hamamatsu L9181-02 microfocus X-ray tube was used with a Dexela
1512 X-ray Detector. The X-ray source was consistently run at $45\>kVp$ and
$55\>\mu A$, under the small focus spot mode ($5-8\>\mu m$).

MPG Parameter | MPG7/MPG8
---|---
$\bm{W\>(\mu m)}$ | 120
$\bm{p_{x}\>(\mu m)}$ | 1.96
$\bm{p_{y}\>(\mu m)}$ | 5.75
$\bm{\alpha_{x}}$ | 0.25
$\bm{\alpha_{y}}$ | 0.86
Design $\bm{h_{1}}$ | $\pi/8$
Design $\bm{h_{2}}$ | $\pi/2$
Material | Gold (Au)
Table 1: Modulated phase grating parameters used in visibility measurements. $h_{1}$ and $h_{2}$ are listed as their corresponding phase-shift in radians. |  Design $\bm{\phi}$ |  Design Height $\bm{(\mu m)}$ |  MPG7 Height $\bm{(\mu m)}$ |  MPG8 Height $\bm{(\mu m)}$
---|---|---|---|---
$\bm{h_{2}}$ | $\pi/2$ | $2.41$ | $2.17\pm 0.57$ | $2.50\pm 0.41$
$\bm{h_{1}}$ | $\pi/8$ | $0.602$ | $0.69\pm 0.17$ | $0.62\pm 0.17$
$\bm{\Delta h}$ | $3\pi/8$ | $1.81$ | $1.48$ | $1.88$
Table 2: Comparison of the average measured height of the grating structures,
$h_{1}$ and $h_{2}$, with the designed heights, for MPG7 and MPG8.
Uncertainties shown are the standard deviation. The heights were measured by
the manufacturer, Microworks, GmbH.

The fringe visibility produced by each grating was measured as a function of
grating-to-detector distance, $z$, for a fixed source-to-detector distance,
$L=110\>cm$. For the purposes of comparing the experimentally acquired
visibility with that predicted by the presented theory, the detector PSF was
measured by imaging the TO MTF tungsten edge phantom by Leeds Test Objects
[32] at a small angle close to the detector. A forward model was produced of
an angled edge and a generalized Gaussian was fit to find the blur induced by
the detector, yielding a PSF with generalized Gaussian parameters of
$\sigma=57.87\>\mu m$ and $k=0.9$. The source profile was measured following
the methods of Nishiki et al. [33], again using the TO MTF phantom, yielding a
Gaussian source profile with $\sigma=3.14\>\mu m$.

Next, images of several porous carbon and alumina samples were acquired over a
range of autocorrelation lengths of approximately $20-80\>nm$ and attenuation,
dark-field, and DPC images were calculated. Lastly, an anchovy was imaged at
approximately the maximum visibility positions for the two gratings and the
attenuation and dark-field images were calculated. Image parameters were
measured using the methods of Marathe et al., [28], but log-scale attenuation
and dark-field images were used in our analysis, as explained in Section
2.2.2.

#### 2.2.1. Visibility Measurements

The visibility was measured as a function of grating-to-detector distance,
$z$, for a fixed source-to-detector distance, $L=110\>cm$. Two MPGs, labeled
MPG7 and MPG8, each with rectangular phase modulation, were used. The MPG
parameters are listed in Table 1. The grating-to-detector distance ranged from
$56-98\>cm$ in $2\>cm$ increments. The grating was phase stepped 7 times, at
$30\>\mu m$ increments.

#### 2.2.2. Carbon and Alumina Samples: Image Acquisition and Analysis

Images of several porous carbon and alumina samples were taken, over an
autocorrelation length (ACL) range of approximately $20-80\>nm$ for the
purposes of measuring how the dark-field signal changed as a function of ACL.
This was achieved by taking one set of images with a source-to-grating
distance of $L_{1}=20\>cm$ and grating-to-object distances of
$D_{GO}=10,\>25,\>40,\>55\>cm$ and another set with $L_{1}=30\>cm$ and
$D_{GO}=10,\>20,\>30,\>40\>cm$. The ACL was calculated for each geometry using
a $25\>keV$ peak energy. For each acquisition, MPG7 was phase stepped 24
times, at $12\>\mu m$ increments.

The carbon samples were OMC-6-600, Nuchar, and Calgon-PCB. The alumina samples
were ASM-385, SAS-90, and a silica-alumina porosimeter reference standard.
Each carbon sample was in powder form, so images were acquired with each
sample in a plastic capillary tube. ASM-385 and SAS-90 were in the form of
compact spheres larger than a capillary tube, so they were placed in a larger
plastic tube for imaging. The silica-alumina was in the form of tiny,
approximately cylindrical particles and was also placed into a larger plastic
tube. Because of the relative size difference in the size of the cylinders and
the tube, some overlap was inevitable over the path of the ray.

Porosimetry data of each sample was acquired using an ASAP 2020 Plus
porosimeter. Nitrogen adsorption-desorption measurements were taken, and the
differential pore volume distribution with respect to pore diameter, $dV/dD$,
was computed by the Barrett-Joyner-Halenda (BJH) algorithm [34]. From this,
the total pore volume can be calculated by integrating the differential pore
volume with respect to pore diameter. Since our interferometer was only
sensitive to pores around the ACL at acquisition, we limited the integration
range to only pores between $10-120\>nm$ and called it the partial pore
volume. The range of pore sizes, pore shapes, and polychromatic X-ray energies
make the scattering response of our samples difficult to predict
quantitatively, but qualitative predictions can be made. We expect to see
higher scattering signals for samples with a higher partial pore volume in the
range of $10-120\>nm$, since the interferometer is designed to be sensitive to
scattering structures approximately the size of the ACL.

Example images of the carbon and alumina samples with their regions of
interest (ROI) are shown in Figure 3, with additional images shown in Results
Section 3.2.2. For the carbon samples, a tall rectangular ROI was chosen in
the center of each tube, to approximate constant thickness through the path of
X-ray. For the alumina spheres, a small square ROI was chosen at the center of
a single sphere, again to approximate constant thickness. The same sphere was
followed for each image, to avoid minor variations between spheres. For the
silica-alumina, overlap was inevitable due to the size of the cylinders
relative to their container, but approximately the same ROI was chosen for
each geometry.

(a)

(b)

Figure 3: Example dark-field images with ROIs highlighted of (a) OMC-6-600
(left), Nuchar (middle), Calgon-PCB (right) and (b) ASM-385, acquired with
$L_{1}=20cm$, $D_{GO}=40cm$, with $ACL=37.57\>nm$

For each ROI, the attenuation and dark-field signals were averaged to be
plotted versus ACL. Generally speaking, the attenuation signal is expected to
not change due to the geometry, but the dark-field signal may change as the
ACL is changed, depending on the size and shape of the scattering structures
within the sample. Additionally, the mass attenuation coefficient, $\mu/\rho$,
will be approximately constant within each set of carbon and alumina samples.
This is clearly true for the carbon samples, but for the set of alumina
samples there is more nuance. The pure alumina samples (ASM-385 and SAS-90)
obviously have the same mass attenuation coefficient, but this is not obvious
for the silica-alumina compound. However, according to NIST’s XCOM database,
[35], silica and alumina have approximately the same $\mu/\rho$, meaning the
compound also has approximately the same mass attenuation coefficient as the
other alumina samples.

However, their mass thickness, $\rho t$, varies due to the differences in the
packing density (for the carbon samples), the porosity differences between the
samples, and the pixel-to-pixel thickness variation (however small). This can
be corrected if we normalize the log-scale dark-field image by the log-scale
attenuation image. Assuming the mass attenuation coefficient, $\mu/\rho$, is
approximately constant over the projection of one pixel, the attenuation
contrast can be represented as

$-\ln\left(\frac{a_{0,sample}}{a_{0,blank}}\right)=\frac{\mu}{\rho}\times\rho
t$ (13)

The dark-field contrast can be represented in a similar manner, using the
linear diffusion coefficient, $\epsilon/\rho$, which depends on the pore
microstructures within the sample:

$-\ln\left(\frac{V_{sample}}{V_{blank}}\right)=\frac{\epsilon}{\rho}\times\rho
t$ (14)

Thus, normalizing the dark-field signal by the attenuation signal corrects for
the mass thickness, $\rho t$, and all left is the linear diffusion coefficient
scaled by the mass attenuation coefficient. Since the mass attenuation
coefficient is expected to be the same for the three carbon samples and the
same for the three alumina samples, we expect the average normalized dark-
field signal to follow the trend of the partial pore volume, even when the
dark-field contrast curve without normalization may not.

## 3\. Results

### 3.1. Theory Results

The theory presented in Section 2.1 is evaluated in several ways. The
intensity profile is calculated for the 2D MPG, and the 2D MPG under the $l=0$
approximation is compared to the case with sufficient detector blur in the
y-direction and a sufficiently small $p_{y}$. The $l=0$ approximation is then
compared to the 1D MPG under the conditions of $\alpha_{x}=\alpha/2$,
$p_{x}=2p$, $\alpha_{y}=1$, which corresponds to the removal of the staggering
of the grating bars. Lastly, the visibility as a function of grating-to-
detector distance, $z$, is calculated for an ideal MPG with $(\pi,0)$ phase
heights and an MPG with phase heights that match MPG7 and MPG8,
$(\pi/2,\pi/8)$. The same setup conditions from our experiments were used in
our simulations.

#### 3.1.1. The Validity of the $l=0$ Approximation

The $l=0$ approximation is validated by showing that the analytical 2D field
calculations are well approximated by the $l=0$ approximation when the blur in
the y-dimension is sufficiently high. The 2D field amplitude at the detector
is found from Equation 19. The intensity is then calculated as
$I=|U(x,y,z|^{2}$. A monochromatic case of $25\>keV$ is simulated for a
realistic MPG with a rectangular envelope with a source-to-detector distance
of $L=110\>cm$ and grating-to-detector distance of $z=70\>cm$. A realistic
Gaussian source with $\sigma=3.14\>\mu m$ and generalized Gaussian PSF with
$\sigma=57.87\>\mu m$ and $k=0.9$ were used. These parameters were determined
experimentally for the Keck imaging system, as explained in Section 2.2,
leading to Figure 4(a). Under the same conditions, the intensity using the
$l=0$ approximation from Equation 40 is calculated and is overlaid in Figure
4(b). There is excellent agreement between the blurred fully calculated 2D
intensity and the intensity calculated using the $l=0$ approximation,
verifying that this approximation is appropriate for realistic MPG and setup
conditions since the source and detector blur in the y-dimension is
sufficiently high enough to remove the other harmonics. The $l=0$
approximation required about 3000 times less computations for a sample rate of
$0.1\>\mu m$ and a vertical field-of-view (FOV) of $0.3\>mm$. This FOV was
necessary for the convolution of the Gaussian source (magnified) and detector
PSF over 3 standard deviations.

(a)

(b)

Figure 4: Verification of the $l=0$ approximation using a RectMPG (Table 1) at
$25\>keV$, $L=110\>cm$, $z=70\>cm$. (a) Theoretical 2D intensity calculation
after smoothing (b) Overlay of a single line through the smoothed 2D intensity
profile and the $l=0$ Approximation.

Furthermore, as seen in Figure 5, our methodology is shown to be consistent
with the 1D MPG from previous work, [24], when $\alpha_{y}=1$, $p_{x}=2p$ and
$\alpha_{x}=\alpha/2$ where the 1D pitch is $p$ and $\alpha$ is the grating’s
duty cycle. Over a range of geometries, the fringe profile of the true 1D MPG
was calculated using Equation 4, and the fringe profile of the equivalent 2D
MPG was calculated using the $l=0$ approximation, Equation 40. The visibility
was calculated as a function of grating-to-detector distance, $z$, for a fixed
source-to-detector distance, $L$, and good agreement was shown between the two
methods. It is evident that the 2D MPG is reduced to the 1D MPG when the
staggering of the grating bars is removed when $\alpha_{y}=1$ and by scaling
$p_{x}$ and $\alpha_{x}$ to the equivalent 1D MPG parameters.

(a)

(b)

Figure 5: Verification of the $l=0$ approximation with the equivalent 1D
RectMPG with parameters $\alpha=0.5$, $p=1\>\mu m$, $W=120\>\mu m$,
$h_{2}=\pi/2$, $h_{1}=\pi/8$, $E_{D}=25\>keV$, made of Gold. Calculated using
a monochromatic $25\>keV$ source, a fixed source-to-detector distance of
$L=110\>cm$, and realistic source and detector blur (a) Overlay of the 1D
intensity with the 2D equivalent found using the $l=0$ approximation (b)
Visibility versus grating-to-detector distance comparison between the true 1D
and 2D equivalent found using the $l=0$ approximation.

#### 3.1.2. Theoretical Visibility

Simulations using the theory presented in Section 2.1 were performed for the
case that matched the RectMPG parameters of MPG7 and MPG8 with heights
$(h_{2},h_{1})$ (Figure 1) tuned to a phase shift of $(\pi/2,\pi/8)$ at
$25\>keV$. Other simulation parameters matched our experimental setup, with a
fixed source-to-detector distance of $L=110\>cm$ and a range of grating-to-
detector distances of $z=56-98\>cm$. We used a $45\>kVp$ tungsten anode
source, with a $0.2\>mm$ beryllium filter, calculated using the TASMIP spectra
calculator, [36]. A Gaussian source of $3.14\>\mu m$ and generalized Gaussian
PSF with $\sigma=57.87\>\mu m$ were used. These parameters were determined
experimentally, as explained in Section 2.2.

We’ve previously shown the best visibility for a RectMPG is when the height
$h_{1}=0$ and $h_{2}$ corresponds to a $\pi$ shift at the design energy for
the grating material (in our case, Gold at $25\>keV$) [24]. To test this ideal
case, theoretical simulations were also performed for the case of an ideal
RectMPG, with $(\pi,0)$ phase heights. The results are compared with the case
of the $(\pi/2,\pi/8)$ RectMPG in Figure 6. The shape of the curves is
approximately the same, with the primary difference being the visibility of
the $(\pi/2,\pi/8)$ grating is lower than the $(\pi,0)$ grating, highlighting
the importance of maximizing the difference in the phase heights for a
RectMPG. However, this is limited by cost and fabrication stability. The error
bars represent the standard deviation in the visibility, which results from
the oscillations in the visibility on a pixel-to-pixel basis due to a finite
number of phase steps.

(a)

(b)

Figure 6: Theoretical Visibility vs. grating-to-detector distance, $z$, for
(a) an ideal $(\pi,0)$ RectMPG and (b) a $(\pi/2,\pi/8)$ RectMPG, as listed in
Table 1. All parameters are the same between the two simulations except for
the phase heights of the grating envelope function, $h_{2}$ and $h_{1}$.

### 3.2. Experiment Results

#### 3.2.1. Experimental Visibility Compared With Theory

The visibility produced by MPG7 and MPG8 was experimentally measured as a
function of grating-to-detector distance, $z$, for a fixed source-to-detector
distance, $L=110\>cm$, as described in Section 2.2.1. The theoretical
visibility, as described in Section 3.1.2 and shown in Figure 6(b), was
directly compared with the experimentally obtained MPG7 and MPG8 visibility
curves, as shown in Figure 7. It is seen that the theoretical visibility lies
between the measured visibility of each grating. We suspect the differences
seen in the visibility produced by each grating results from variabilities in
the height of the grating structures for MPG7 and MPG8, as shown in Table 2.
It is seen that MPG8 has a larger difference in the height of the grating
structures, $\Delta h$, leading to a higher visibility.

Figure 7: Comparison of Visibility vs. Grating-to-Detector Distance, $z$, for
MPG7, MPG8, and the presented theory. The visibility was measured as as
function of grating-to-detector distance, $z$ for a fixed source-to-detector
distance, $L=110\>cm$, with details in Section 2.2.1.

#### 3.2.2. Carbon and Alumina Samples: Analysis Results

Attenuation, dark-field, and differential-phase contrast images of several
carbon and alumina samples were acquired using the methods described in
Section 2.2.2. Figure 8 shows examples of the SAS-90 alumina sample. The DPC
images are visible for the ASM-385 and SAS-90 alumina samples, where the
spheres’ characteristic bright and dark sides are seen. The other samples did
not have a significant DPC signal.

(a)

(b)

(c)

Figure 8: Attenuation (a), dark-field (b), and differential-phase contrast (c) images of SAS-90, taken at a source-to-detector distance of $L=110\>cm$, the source-to-grating distance of $L_{1}=20\>cm$, and grating-to-object distance of $D_{GO}=40\>cm$. Sample | Total Pore Volume (cc/g) | Partial Pore Volume (cc/g)
---|---|---
OMC-6-600 | 0.464 | 0.0168
Nuchar | 0.352 | 0.104
Calgon-PCB | 0.0899 | 0.00711
ASM-385 | 0.474 | 0.445
SAS-90 | 0.408 | 0.334
Silica-Alumina | 0.6699 | 0.220
Table 3: Total Pore Volume and Partial Pore Volume within the range of
$10-120\>nm$ for each carbon and alumina sample.

The mean normalized dark-field signal—the dark-field image divided by the
attenuation image—was measured for each carbon and alumina sample, to be
qualitatively compared with the partial pore volume of the samples, with
details in Section 2.2.2. For the purposes of calculating the partial pore
volume, BJH adsorption measurements were used for the carbon samples while BJH
desorption was used for the alumina samples, due to the differences in pore
shapes commonly seen between these samples [37, 38]. Figure 9 shows the
differential pore volume measurements. Integration was performed from
$10-120\>nm$ (or the maximum pore size measured) to calculate the partial pore
volume.

(a)

(b)

Figure 9: Differential Pore Volume Measurements of the (a) carbon and (b)
alumina samples. The partial pore volume is calculated by integrating from
$10-120\>nm$.

Table 3 shows the total and partial pore volume measurements. For the carbon
samples, it is seen that Nuchar has the highest partial pore volume within the
range of $10-120\>nm$ and OMC-6-600 and Calgon-PCB are both lower by about an
order of magnitude. ASM-385 has a higher partial pore volume for the alumina
samples than SAS-90, and the silica-alumina sample has the lowest partial pore
volume.

The same trends are seen when looking at the normalized dark-field signal
versus ACL curves in Figures 10 and 11. For the carbon samples, the normalized
dark-field signal is highest for Nuchar, which had the highest partial pore
volume. The OMC-6-600 and Calgon-PCB samples both have lower mean signal, with
OMC-6-600 being higher on the lower end of the ACL range. This is likely due
to the partial pore volume being slightly higher in the OMC-6-600, especially
if the pore width range of interest was expanded to include the peak
differential pore volume. The mean signal of the Calgon-PCB sample is about
the same as the OMC-6-600 sample at the higher end of the ACL range, which may
be due to the differential pore volume for Calgon-PCB being larger at the
higher end of the pore width range, as seen in Figure 9(a). For the alumina
samples, the normalized dark-field signal follows the trend of the partial
pore volume, with the ASM-385 sample having the highest signal, followed by
SAS-90, then the silica-alumina sample.

Figure 10: Mean attenuation and mean normalized dark-field signal as a
function of ACL for the carbon samples. The ACL was calculated using
$E=25\>keV$. Figure 11: Mean attenuation and mean normalized dark-field signal
as a function of ACL for the alumina samples. The ACL was calculated using
$E=25\>keV$.

#### 3.2.3. Anchovy Images

Images of an anchovy were acquired with both MPG7 and MPG8, shown in Figures
12 and 13. At first glance, the dark-field images look like noisier versions
of the attenuation images, but on closer inspection, some key differences in
structure are seen, which are highlighted. In addition to the attenuation and
dark-field images, a filtered dark-field image is shown, where a 3x3 median
filter was taken in addition to an anisotropic diffusion filter, with a
gradient threshold of 0.2 and 4 iterations. In the filtered image, the noise
is reduced and the differences in structure are more visible. The differences
in the attenuation and dark-field images are consistent between MPG7 and MPG8.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 12: Anchovy images acquired using MPG7. Images were acquired at a
source-to-detector distance of $L=110\>cm$, a source-to-grating distance of
$L_{1}=20\>cm$, and a grating-to-object distance of $D_{GO}=13.5\>cm$, for an
autocorrelation length of $ACL=57.5\>nm$. The zoomed regions are ROI1 (yellow)
and ROI2 (cyan). Note colorbar differences in original and zoomed images.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 13: Anchovy images acquired using MPG8. Images were acquired at a
source-to-detector distance of $L=110\>cm$, a source-to-grating distance of
$L_{1}=20\>cm$, and a grating-to-object distance of $D_{GO}=13.5\>cm$, for an
autocorrelation length of $ACL=57.5\>nm$. The zoomed regions are ROI1 (yellow)
and ROI2 (cyan). Note colorbar differences in original and zoomed images

## 4\. Discussion

We presented the theory for staggered 2D gratings fabricated using a Bridge
technique. We showed that under realistic detector PSF and source blur, an
approximation can be made to greatly reduce computational load known as the
$l=0$ approximation, where all y-harmonics except $l=0$ are removed. Using
this approximation, we simulated our experimental setup for measuring the
visibility of two RectMPGs, labeled MPG7 and MPG8. In these experiments, MPG8
was shown to have a higher visibility at all distances, which is consistent
with the fact that MPG8 also has a larger difference in the height of the
grating structures, $\Delta h$, as we suspect the differences seen between the
two gratings result from variabilities in the height of the grating
structures. The theoretical visibility was shown to be comparable to the
experimental results, with the theoretical visibility falling between the
experimental visibility of the two gratings, consistent with the measured
differences in the height of the grating structures, $\Delta h$, as shown in
Table 2. This indicates the potential for this model to be used in aiding the
design of future MPGI systems. Additional potential sources of error include
improper modeling of the source spectrum or detector response function and
diffraction artifacts resulting from the MPGs not exactly following the
envelope function, non-rectangular grating bars, or other imperfections in the
quality of the gratings. The tolerance for height control has to be determined
for future applications.

We also imaged several carbon and alumina samples and an anchovy. The
differential-phase contrast images of the ASM-385 and SAS-90 samples show the
characteristic sides of the alumina spheres. The mean normalized dark-field
signal of the carbon and alumina samples was shown to follow the trend of the
partial pore volume obtained from BJH adsorption and desorption differential
pore volume measurements. The dark-field anchovy images were noisy, but we did
observe zones where different structures were visible in the dark-field images
compared to the attenuation images, which were consistent for both MPG7 and
MPG8. For the visibility experiments, fewer phase steps were used than during
the image acquisition of samples, since we were only interested in the average
fringe visibility, which should not be biased by the number of phase steps if
the signal-to-noise ratio is sufficiently high [39].

## 5\. Conclusion

The modulated phase grating interferometer (MPGI) is a phase sensitive imaging
system that simultaneously acquires attenuation, differential-phase contrast,
and dark-field images, which show X-ray attenuation, refraction, and small
angle scattering, respectively. We have successfully modeled the 2-dimensional
MPG fabricated using a Bridge technique, where staggering of the grating bars
is present, and we have employed an approximation known as the $l=0$
approximation to reduce the 2-dimensional calculation to only a single
dimension, greatly reducing simulation time. We have shown fringe visibility
predictions that are comparable with experimental results and have shown that
the model presented has the potential to be used to rapidly iterate when
designing future MPGI systems. We have imaged several porous carbon and
alumina samples using an MPGI and have shown that the normalized dark-field
signal trends well with the partial pore volume of the samples when an
appropriate pore width range is used. Lastly, we imaged a dried anchovy and
showed multiple regions where unique scattering information is present.

## 6\. Acknowledgements

This work is funded in part by NIH NIBIB Trail-blazer Award
1-R21-EB029026-01A1. Parts of this work were presented at the South West
Regional Chapter of the American Association of Physics in Medicine (SWAAPM)
in Feb 2024 and will be presented at the Graduate Research Conference, LSU in
April 2024. Initial experiments were performed by author S.C. for her MSc
thesis [40].

## 7\. Conflict of Interest

The authors have no conflicts to disclose.

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##  Appendix A Appendix: 2D MPG Intensity Derivation

### A.1. The Field Amplitude in 2 Dimensions

For a single system geometry and energy, the angular spectrum method, [30], is
used to derive the field amplitude, $U(x)$, for a plane wave source. The
Fresnel scaling theorem, [31], is used to scale the field from that of a plane
wave source to that of a point-source. Following the angular spectrum method,
the angular spectrum immediately following the MPG can be found by taking the
Fourier transform of Equation 7,

$\begin{split}A(f_{x},f_{y},z=0)&=\mathcal{F}\left(T(x,y)\right)\\\
A(f_{x},f_{y},z=0)&{}={}\delta(f_{x})\delta(f_{y})+\alpha_{x}\alpha_{y}\operatorname{sinc}(\alpha_{x}p_{x}f_{x})\operatorname{sinc}(\alpha_{y}p_{y}f_{y})\left(\beta(f_{x},f_{y})-\gamma(f_{x},f_{y})\right)\end{split}$
(15)

where,

$\beta(f_{x},f_{y})=\sum_{m}\sum_{n}\sum_{l}\>g_{m}\delta\left(f_{x}-\frac{m}{W}-\frac{n}{p_{x}}\right)\delta\left(f_{y}-\frac{l}{p_{y}}\right)\times\bigg{(}1+\exp(-j\pi\left(f_{x}-\frac{m}{W}\right)p_{x})\exp(-j\pi
f_{y}p_{y})\bigg{)}$ (16)

$\gamma(f_{x},f_{y})=\sum_{n}\sum_{l}\delta\left(f_{x}-\frac{n}{p_{x}}\right)\delta\left(f_{y}-\frac{l}{p_{y}}\right)\bigg{(}1+\exp(-j\pi
f_{x}p_{x})\exp(-j\pi f_{y}p_{y})\bigg{)}$ (17)

The angular spectrum can then be propagated to the detector,

$A(f_{x},f_{y},z)=A(f_{x},f_{y},z=0)\>e^{jkz}\>\exp\left(-j\pi\lambda
z(f_{x}^{2}+f_{y}^{2})\right)$ (18)

And the field is calculated,

$\begin{split}U(x,y,z)&=\mathcal{F}^{-1}\left(A(f_{x},f_{y},z)\right)\\\
&=e^{jkz}(1+U_{1}(x,y,z)-U_{2}(x,y,z))\end{split}$ (19)

$U_{1}(x,y,z)$ and $U_{2}(x,y,z)$ are derived from the propagation of
$\beta(f_{x},f_{y})$ and $\gamma(f_{x},f_{y})$, respectively. The $x$\- and
$y$-components of each term can be represented separately using matrix
multiplication,

$\begin{bmatrix}U_{1}(x,y,z)\\\
U_{2}(x,y,z)\end{bmatrix}=\begin{bmatrix}U_{1}^{Ax}(x,z)&U_{1}^{Bx}(x,z)\\\
U_{2}^{Ax}(x,z)&U_{2}^{Bx}(x,z)\end{bmatrix}\times\begin{bmatrix}U^{Ay}(y,z)\\\
U^{By}(y,z)\end{bmatrix}$ (20)

where the x-components are

$\displaystyle U_{1}^{Ax}(x,z){}={}$
$\displaystyle\sum_{m}\sum_{n}A_{1x}(m,n,z)\exp\left(j2\pi
x\left(\frac{m}{W}+\frac{n}{p_{x}}\right)\right)$ (21) $\displaystyle
U_{1}^{Bx}(x,z){}={}$
$\displaystyle\sum_{m}\sum_{n}B_{1x}(m,n,z)\exp\left(j2\pi
x\left(\frac{m}{W}+\frac{n}{p_{x}}\right)\right)$ (22) $\displaystyle
U_{2}^{Ax}(x,z){}={}$ $\displaystyle\sum_{m}\sum_{n}A_{2x}(n,z)\exp\left(j2\pi
x\frac{n}{p_{x}}\right)$ (23) $\displaystyle U_{2}^{Bx}(x,z){}={}$
$\displaystyle\sum_{m}\sum_{n}B_{2x}(n,z)\exp\left(j2\pi
x\frac{n}{p_{x}}\right)$ (24) $\displaystyle A_{1x}(m,n,z){}={}$
$\displaystyle\alpha_{x}g_{m}\exp\left(-j\pi\lambda
z\left(\frac{m}{W}+\frac{n}{p_{x}}\right)^{2}\right)\operatorname{sinc}\left(\alpha_{x}p_{x}\left(\frac{m}{W}+\frac{n}{p_{x}}\right)\right)$
(25) $\displaystyle A_{2x}(n,z){}={}$
$\displaystyle\alpha_{x}\exp\left(-j\pi\lambda
z\left(\frac{n}{p_{x}}\right)^{2}\right)\operatorname{sinc}\left(\alpha_{x}n\right)$
(26) $\displaystyle B_{1x}(m,n,z){}={}$ $\displaystyle
A_{1x}(m,n,z)\exp\left(-j\pi n\right)$ (27) $\displaystyle B_{2x}(n,z){}={}$
$\displaystyle A_{2x}\exp\left(-j\pi n\right)$ (28)

and the y-components are

$\displaystyle U_{Ay}(y,z){}={}$
$\displaystyle\sum_{l}A_{y}(l,y,z)\exp\left(j2\pi y\frac{l}{p_{y}}\right)$
(29) $\displaystyle U_{By}(y,z){}={}$
$\displaystyle\sum_{l}B_{y}(l,y,z)\exp\left(j2\pi y\frac{l}{p_{y}}\right)$
(30) $\displaystyle A_{y}(l,z){}={}$
$\displaystyle\alpha_{y}\exp\left(-j\pi\lambda
z\left(\frac{l}{p_{y}}\right)^{2}\right)\operatorname{sinc}\left(\alpha_{y}l\right)$
(31) $\displaystyle B_{y}(l,z){}={}$ $\displaystyle
A_{y}(l,y,z)\exp\left(-j\pi l\right)$ (32)

Finally, the Fresnel scaling theorem can be applied by scaling x, y, and z by
the point-source magnification factor, $M=\frac{L_{1}+z}{L_{1}}$,

$\displaystyle x\longrightarrow\frac{x}{M}$ $\displaystyle
y\longrightarrow\frac{y}{M}$ $\displaystyle z\longrightarrow\frac{z}{M}$

It should be noted that there are additional phase and amplitude
multiplicative factors introduced by the Fresnel scaling theorem that we do
not consider for the purposes of this study, since the phase factors disappear
when calculating the intensity and the amplitude factors do not affect the
visibility.

### A.2. The Field Intensity and the l = 0 Approximation

The field intensity can be calculated as simply the square of the amplitude.

$I(x,y,z)=|U(x,y,z,)|^{2}=U(x,y,z)U^{*}(x,y,z)$ (33)

Since the detector will blur the y-harmonics due to the relatively small
$p_{y}$, the detector intensity will be well approximated by the $l=0$
harmonic, analogous to the $n=0$ approximation present in [24]. To simplify
this, we will represent the field amplitude from Equation 19 using only the
y-harmonics (ignoring the $e^{ikz}$ which disappears in the intensity). The
$l=0$ approximation must be taken in the intensity, not the amplitude, but
representing the field amplitude in this way will simplify the calculations.
The $l=0$ harmonic has the primary benefit of greatly reducing the required
computations, since the intensity is reduced to 1 dimension while maintaining
the effects caused by the staggering of the grating bars.

$U(x,y,z)=1+\sum_{l}c(l,x,z)\exp\left(j2\pi y\frac{l}{Mp_{y}}\right)$ (34)

where,

$\displaystyle c(l,x,z)=U_{Ax}(x,z)A_{y}(l)+U_{Bx}(x,z)B_{y}(l)$ (35)
$\displaystyle U_{Ax}(x,z)=U_{1}^{Ax}(x,z)-U_{2}^{Ax}(x,z)$ (36)
$\displaystyle U_{Bx}(x,z)=U_{1}^{Bx}(x,z)-U_{2}^{Bx}(x,z)$ (37)

The intensity can then be calculated:

$I(x,y,z)=1+\sum_{l}\biggl{[}c(l,x,z)+c^{*}(-l,x,z)+d(l,x,z)\biggr{]}\exp\left(j2\pi
y\frac{l}{Mp_{y}}\right)$ (38)

where,

$d(l,x,z)=c(l,x,z)\star c^{*}(-l,x,z)$ (39)

The $l=0$ approximation is then taken,

$I(x,y,z)\approx 1+c(0,x,z)+c^{*}(0,x,z)+d(0,x,z)$ (40)

$c(0,x,z)$ and $c^{*}(0,x,z)$ are easy to compute by recognizing that
$A_{y}(0)=B_{y}(0)=\alpha_{y}$, whereas $d(0,x,z)$ can be easily computed,

$d(0,x,z)=U_{Ax}U^{*}_{Ax}\sum_{l^{\prime}}s(l^{\prime})+U_{Ax}U^{*}_{Bx}\sum_{l^{\prime}}s(l^{\prime})\exp(-j\pi
l^{\prime})+U^{*}_{Ax}U_{Bx}\sum_{l^{\prime}}s(l^{\prime})\exp(j\pi
l^{\prime})+U_{Bx}U^{*}_{Bx}\sum_{l^{\prime}}s(l^{\prime})$ (41)

where,

$s(l)=A_{y}(-l)\star
A^{*}_{y}(-l)=\alpha_{y}^{2}\operatorname{sinc}\left(-\alpha_{y}l\right)^{2}$
(42)

The $l=0$ approximation greatly reduces the computation required and is only
valid if $M*p_{y}$ is significantly less than the pixel size. Notably, this
does not fully remove the effect of staggering on the fringes. The intensity
profile still depends on $\alpha_{y}$, meaning the fringe visibility will also
depend on it.