Image Quality in Attenuation-Based and Phase-Contrast-Based X-ray Imaging

Quantitative details of the interaction of an X-ray beam with a weakly scattering non-crystalline sample can usually be evaluated in terms of the integrals of the complex refractive index, n_λ (r) = 1 − δ_λ (r) + iβ_λ (r), along the X-ray trajectories, where r = (x, y, z) are the spatial Cartesian coordinates and λ is the wavelength (Paganin 2006) (see also Section IV, Chapter 49 of this book). For hard X-rays, the refractive index decrement δ_λ for most materials is very small (of the order of 10⁻⁶ or less), hence the refraction angles are small too and it is usually possible to approximate the X-ray trajectories through a sample by straight lines. For simplicity, let us consider first the case of a monochromatic X-ray wave with unit intensity propagating along the direction of the optic axis z (Figure 15.1). Phase shifts, https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351228251/2c05ef39-5b4e-46fa-a804-7a51253d526f/content/inline-math15_1.jpg"/> φ λ ( r ⊥ , 0 ) , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351228251/2c05ef39-5b4e-46fa-a804-7a51253d526f/content/inline-math15_2.jpg"/> r ⊥ = ( x , y ) , that the sample (which lies completely in the half-space z ≤ 0) introduces into the transmitted X-ray beam upon transmission through the sample, are proportional to the projections (line integrals) of the real part of the refractive index, while the distribution of the logarithm of transmitted X-ray intensity in the object plane, https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351228251/2c05ef39-5b4e-46fa-a804-7a51253d526f/content/inline-math15_3.jpg"/> I λ ( r ⊥ , 0 ) , is proportional to projections of the imaginary part: 15.1 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351228251/2c05ef39-5b4e-46fa-a804-7a51253d526f/content/math15_1.jpg"/> φ λ ( r ⊥ , 0 ) = − ( 2 π / λ ) ∫ − ∞ 0 δ λ ( r ⊥ , z ) d z 15.2 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351228251/2c05ef39-5b4e-46fa-a804-7a51253d526f/content/math15_2.jpg"/> ln I λ ( r ⊥ , 0 ) = − ( 4 π / λ ) ∫ − ∞ 0 β λ ( r ⊥ , z ) d z