Contrast Generation in Stepped-Grating Interferometry

The absorption contrast generation is straightforward as the average intensity in a pixel corresponds to the transmitted signal. With the sample and reference values the absorption is calculated as follows,

𝐴𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 = − ln 𝑇 = − 𝑙𝑛 𝐼 𝐼0

= 𝜇𝑧 , [3.10]

where 𝐼0and 𝐼 are the incident and transmitted beams, respectively. 𝑧 is the sample thickness and 𝜇 is referred to as the linear attenuation coefficient.

The absorption signal is the projection of the attenuation coefficient across the sample,

3.2.2 Phase Contrast

The interference pattern experiences a shift due to the refractive effect of an object. To determine this shift, the pattern is scanned by translating a grating in a direction transverse to the grating structure and beam propagation direction as described in Section 3.1. The lateral shift, 𝑠, is related to the angular deviation of a beam, ∆𝛼, by 𝑠 = ∆𝛼 × 𝑑 and translates into a phase shift of 𝜑 = 2𝜋 × 𝑠/𝑝2.8

The relation between the angular deviation of a beam ∆𝛼 and the differential phase shift ((𝜕Φ (x, y))/𝜕𝑥) is given by the equation

∆𝛼 = 𝜆 2𝜋

𝜕Φ (x, y)

𝜕𝑥 , [3.12] where x and y are the Cartesian coordinates perpendicular to the optical axis, Φ (x, y) represents the phase shift of the wavefront, and 𝜆 is the wavelength of the radiation.1 _By substituting for ∆𝛼 in Equation 3.12 above, the differential phase shift is calculated from the equation:

𝜑 = 𝜆𝑑 𝑝2

𝜕Φ (x, y)

𝜕𝑥 . [3.13]

Tomographic reconstruction of Φ (x, y) from projection images different angles around the sample yield the volumetric distribution of the X-ray refractive index decrement, 𝛿, of the sample,

Φ(𝑥, 𝑦) =2𝜋

𝜆 ∫ 𝛿(𝑥, 𝑦, 𝑧)𝑑𝑧 , [3.14] where the optic axis is parallel to z.3

3.2.3 Dark-Field Contrast

A third contrast mechanism, the dark-field contrast, based on small-angle X-ray scattering from sub-pixel sized microstructures has been identified and can provide structural information that is inaccessible from the absorption and differential-phase

images.7 _{In grating-based interferometry, it is detected as a relative decrease in the}

visibility of the fringe pattern. The visibility is calculated as:18

𝑉𝑝 = 𝐼_𝑚𝑎𝑥 − 𝐼_𝑚𝑖𝑛 𝐼𝑚𝑎𝑥 + 𝐼𝑚𝑖𝑛 =𝑎1𝑝 𝑎0𝑝 , [3.15]

where 𝐼_𝑚𝑎𝑥 and 𝐼_𝑚𝑎𝑥are the maximum and minimum intensities, respectively, for the recorded sinogram in pixel 𝑝. 𝑎_1𝑝 is the amplitude of the sinusoid and 𝑎0𝑝 is the midline

value which is the average intensity for a given pixel. The relative decrease in visibility due to the sample or the dark-field signal is obtained by dividing the sample visibility by the reference visibility i.e.

𝐷𝐹 =𝑉𝑝 𝑠_{(𝑚, 𝑛)} 𝑉_𝑝𝑟(𝑚, 𝑛)= 𝑎1𝑝 𝑠 _𝑎 0𝑝 𝑟 ⁄ 𝑎_0𝑝𝑠 _𝑎 1𝑝𝑟 ⁄ . [3.16]

When the sample is homogeneous, small-angle X-ray scattering contribution is negligible

so the visibility remains unchanged (V=1).18 _{Specimens with structural anisotropy on}

micrometer length scales produce strong small-angle X-ray scattering thus cause a

significant decrease in the visibility.19_{Grating-based interferometers are only sensitive to}

scattering in the direction of scanning i.e. perpendicular to the grating structure.20

It has been demonstrated that the dark-field signal of the interferometer exponentially decays with sample thickness similar to the attenuation of X-rays in a sample (Beer-Lambert law), and that this decay is mathematically related to the width of

the scattering distribution 𝜎.21 _{Thus a material-dependent parameter, the diffusion} coefficient 𝜖 was described and related to the dark-field signal by the expression,21

𝐷𝐹 = 𝑒𝑥𝑝 (−2𝜋2 𝑑 2

𝑝₂2 𝜖(𝑡) ) , [3.17] where 𝑑 is the distance between the G1 and G2 gratings, 𝑝₂ is the period of the G2 grating and 𝑡 is the sample thickness. 𝜖 is expressed in terms of 𝜎2 and 𝑡 as 𝜖 ≡ 𝜎2⁄ .𝑡 22 _This dependence of the dark-field signal on sample thickness enables quantitative dark-field imaging computed tomography to be performed.

Based on the scanning technique, the dark-field signal has been shown to have a direct relationship with the autocorrelation function of the sample.20 _{The expression is} given by,20

𝐷𝐹(𝜉_{𝑐𝑜𝑟𝑟}) = 𝑒𝑥𝑝[𝛴𝑡 (𝐺 (𝜉_{𝑐𝑜𝑟𝑟}) − 1)] , [3.18] where

𝐺

is the autocorrelation function of the sample and is a function of the correlation length, 𝜉_{𝑐𝑜𝑟𝑟} , probed by the interferometer.

𝛴

is the scattering cross section and 𝑡 is the sample thickness. 𝜉𝑐𝑜𝑟𝑟 is calculated as 𝜆𝑙𝑠⁄ , where 𝜆 is the wavelength of operation, 𝑝2 𝑙_𝑠 is the sample to G2 grating distance and 𝑝₂ is the period of the G2 grating.23 _However, for a cone beam and sample positioned between G0 and G1, 𝜉_{𝑐𝑜𝑟𝑟} becomes

𝜉𝑐𝑜𝑟𝑟 = 𝜆𝑙_𝑠′

𝑝2

[3.19]

where 𝑙_𝑠′ = (𝑙 + 𝑑 − 𝑙_𝑠)𝑑/𝑙.23 _{𝑙 is the G0 to G1 distance and 𝑑 is the G1 to G2 distance.}

Since 𝜉_{𝑐𝑜𝑟𝑟} can be easily be tuned, the correlation function G(𝜉_{𝑐𝑜𝑟𝑟}) can be determined by

Correlation functions for a number of shapes have been determined.24 _{For spheres, it is}

approximated by the Gaussian function

𝐺𝑠𝑝ℎ𝑒𝑟𝑒(𝜉𝑐𝑜𝑟𝑟) ≈ 𝑒𝑥𝑝 [− 9 8( 𝜉_{𝑐𝑜𝑟𝑟} 𝑅 ) 2 ] , [3.20]

where R is approximately the radius of the sphere.20

Using the correlation function for spheres, the above relationship between the

dark-field signal and 𝐺 (Equation 3.20) have been shown to be in good agreement with

theoretical expectation.20

In document Application of X-ray Grating Interferometry to Polymer/Flame Retardant Blends in Additive Manufacturing (Page 76-80)