A rotating random-phase-screen diffuser is sometimes employed on synchrotron x-ray imaging beamlines to ameliorate field-of-view inhomogeneities due to electron-beam instabilities and beamline optics phase artifacts. The ideal result is a broader, more uniformly illuminated beam intensity for cleaner coherent x-ray images. The spinning diffuser may be modeled as an ensemble of transversely random thin phase screens, with the resulting set of intensity maps over the detector plane being incoherently averaged over the ensemble. Whilst the coherence width associated with the source is unaffected by the diffuser, the magnitude of the complex degree of second-order coherence may be significantly reduced [K. S. Morgan, S. C. Irvine, Y. Suzuki, K. Uesugi, A. Takeuchi, D. M. Paganin, and K. K. W. Siu, Opt. Commun. 283, 216 (2010)]. Through use of a computational model and experimental data obtained on x-ray beamline BL20XU at SPring-8, Japan, we investigate the effects of such a diffuser on the quality of Fresnel diffraction fringes in propagation-based x-ray phase contrast imaging. We show that careful choice of diffuser characteristics such as thickness and fiber size, together with appropriate placement of the diffuser, can result in the ideal scenario of negligible reduction in fringe contrast whilst the desired diffusing properties are retained.
©2010 Optical Society of America
Propagation based phase contrast imaging [2–4] using a synchrotron x-ray source is a powerful and flexible tool in the study of biological soft tissues . To achieve quality phase contrast images, a transversely uniform field of incident coherent x-rays is desired. Division of the phase-contrast image by a flat field image (i.e., division by an image taken in the absence of the sample) can remove some transverse intensity variations due to a non-uniform incident beam. However, frame-to-frame movement of the beam image will not be corrected. Additionally, this correction does not necessarily remove image inhomogeneities created by undesired transverse phase variations in the x-ray beam illuminating the sample, for example those introduced by optical elements. The use of a diffuser to remove such phase defects and to even out the illuminating beam intensity may therefore be required. This must be balanced with coherence requirements, as the presence of a diffuser will simultaneously decrease the observed degree of coherence of the illuminating beam . Notwithstanding their utility in the context of propagation-based x-ray phase contrast imaging, little attention has been paid to developing a physical model for phase contrast image formation in the presence of a diffuser. Here we develop such a model, which is readily implemented computationally, and which we anticipate to be of particular utility in computer simulations of propagation-based phase contrast imaging employing a phase diffuser. We apply this model to investigate the effect of the spinning paper diffuser used at synchrotron x-ray beamline BL20XU at SPring-8, Japan. Our simple computational model correlates well with our experimental data.
As mentioned previously, the need for a diffuser stems from the difficulty in achieving a perfectly uniform x-ray beam for the full field of view using a synchrotron source. This is particularly evident when the area to be illuminated is relatively large (e.g. several millimeters to centimeters in extent), as required in biomedical small animal in vivo imaging , where the frame-to-frame movement of the beam is particularly problematic. Further to this, short exposure times (e.g. on the order of 100ms) are helpful in avoiding image blur due to animal movement, and those experiments which attempt to observe rapid biomedical dynamics such as blood flow [6,7], require even shorter exposures. With such short exposures it becomes more apparent that fluctuations in the electron-beam stability are causing a time-varying beam, which imposes undesirable inhomogeneities in propagation-based x-ray phase contrast images.
Further image inhomogeneities may be imposed by imperfections in various beamline components [8,3]. Indeed, it is precisely because propagation-based x-ray phase contrast is so sensitive to transverse phase variations, that even minute variations (due e.g. to beryllium windows, insufficiently flat reflecting optics, inhomogeneities in beam curvature etc.) may lead to artifacts in the image of a sample. The introduction of a diffuser will both minimize the presence of these phase artifacts in the final image of the sample, as well as creating a more stable area of uniform intensity [2,3,9]. One thereby remains sensitive to transverse phase gradients imposed by the sample, whilst simultaneously being insensitive to transverse phase gradients which are not due to the sample.
As an example of the non-uniform illumination which will be present in even the best of third-generation synchrotrons, a 25keV (as used in biomedical imaging ) flat field image observed at the bio-medical imaging hutch of beamline 20XU at SPring-8, Japan, is shown in Fig. 1 . Without a diffuser present (Fig. 1a), the beam shows a high intensity peak (visible as a red stripe in the false-color map), which represents the temporally-varying incident intensity averaged over a single exposure. As shown in the movie supplementary to Fig. 1 (Media 1), consecutive image frames show significant frame-to-frame variation. There is also propagation-based phase contrast introduced by the non-ideal optics, as indicated by the arrows. As shown in Fig. 1b, when a stationary diffuser (see Sec. 2 for details) is introduced, the illuminating area becomes significantly more uniform, but still shows some movement with time (supplementary Fig. 1 (Media 1)). The flat field has phase images and intensity variations characteristic of the part of the diffuser positioned in the beam. Rotating the diffuser, so that the illuminated area changes over time, will mean that these variations are averaged out over the exposure time. This will produce a field that is even in space and over time, provided that the image exposure time is long compared to the characteristic timescale of the rotating diffuser (Fig. 1c and supplementary Fig. 1 (Media 1)). The placement of this random phase screen directly after the optics means that the phase artifacts in the flat-field image are blurred out.
As seen in the inset to Fig. 1c, the impulse-like power spectrum of the spinning diffuser flat-field map clearly indicates a uniform intensity. However with either the stationary diffuser or no diffuser, further components are introduced in the low frequency (beam shape), medium frequency (diffuser pattern) and high frequency (beam-line optics phase artifacts) spatial regions of the power spectra. Note that the dark rings, present in the insets to Figs. 1a and 1b, are zeros in the phase contrast transfer function for the coherent x-ray imaging system .
While the series of flat field images in Fig. 1 (Media 1) suggests that the use of a diffuser will be beneficial during synchrotron x-ray imaging, the effect on the transverse coherence of the beam should also be considered. The ensemble-averaged intensity due to many phase screens over the exposure time (cf. Sec. 3) means that the magnitude of the complex degree of second order spatial coherence  will be decreased, although the coherence width (the width is dominant given the large horizontal size of the undulator beam compared to the vertical size) will be unchanged . In this case, the coherence width is defined as the minimum distance over which two transversely separated points of a wavefield may be considered as spatially incoherent . When a sample is placed in the beam for propagation-based phase contrast imaging, the phase-distorted incident beam will produce a speckled, ‘warped’ phase contrast image dependent on the instantaneous position of the diffuser (cf. Fig. 5b), which is incoherently summed over time. Consequently, the visibility and number of fringes in the phase contrast fringe set from a single interface in the sample will be significantly reduced . The link between unresolved phase variations (resulting in speckle) and observed decoherence effects has been previously discussed (e.g. , ) in the context of spatial ensemble averaging by the detector (in our case it is predominantly an effect of temporal averaging).
In some bio-medical imaging applications, a single bright/dark phase contrast fringe is all that is desired, to reveal the position of an interface . Other applications require multiple fringes, either to permit use of a phase retrieval algorithm , or so that the fringe pattern may be easily identified amid surrounding intensity variations . It is for this reason that we study the optimal diffuser set-up in this paper.
This analysis looks at both simulated and experimental propagation-based phase contrast images with 25keV energy and an object-to-detector propagation distance of between 0.5 and 1.45 meters. The computer simulation is based on a diffuser model described in , evaluating the incoherent sum of a number of simulated phase contrast images, each produced with an incident field that has incurred random transverse phase variations of a specified characteristic depth and width at the diffuser position. The incoherent ensemble average, due to these many diffuser positions over a complete exposure, forms the phase contrast image.
We close the introduction with a summary of the remainder of the paper. In section 2, we outline our phase contrast x-ray imaging experiment and describe the key observations relating to the effect of a phase diffuser on our phase contrast images. The numerical simulation, based on our model for the phase diffuser, is detailed and implemented in section 3. The final section compares theory and experiment, quantifying the effect of a diffuser on the visibility and number of observable fringes in a phase-contrast x-ray image of an edge. We conclude with some practical recommendations for future experiments. The most important recommendation is to place the diffuser as close as possible to the x-ray source, in order to retain desired fringe information while simultaneously suppressing phase-contrast artifacts due to imperfections in the illuminating beam.
2. Experiment on propagation-based phase contrast imaging with a diffuser
X-ray images were acquired on BL20XU, an undulator-source beamline at the SPring-8 synchrotron in Japan. We worked at the downstream hutch, which gives a source-sample distance of 240m. A double crystal Si-111 monochromator was used to select x-rays at an energy of 25keV. The high resolution detector system used comprised a 4008 × 2672 pixel CCD camera coupled to a Hamamatsu AA50 x-ray converter with a 50× objective lens, yielding an effective pixel size of 0.18μm.
The diffuser utilized at this beamline consists of a ≈12cm diameter circular disk of conventional office-grade white copy paper (80 g/m2) which was made to rotate within a plane perpendicular to the x-ray optical axis. A set of slits created a 300 μm × 300 μm aperture which acted as a secondary source. The diffuser used on this beamline was located approximately 78m from the source and 29m from the aperture (see Fig. 2 ).
Samples for this experiment were a cylindrical perspex rod (diameter 3.0mm), and two perspex spheres of 1.5mm diameter which in projection overlapped very slightly (region of overlap was approximately 30μm at its widest point, see inset to Fig. 3a ). On this beamline, at sample-to-detector propagation distances ranging from 0.5 to 1.45m, these objects produce tens of Fresnel-regime phase contrast fringes. In the case of the overlapping spheres, these fringes intersect, forming lattice-like sets of intensity peaks whose visibility is highly sensitive to the effect of a diffuser. Such a region is shown in Figs. 3b and 3c, clearly demonstrating the reduction in area and visibility of fringes when a diffuser is introduced in the experiment.
3. Model for x-ray diffuser in propagation-based phase contrast imaging
The effect of the spinning diffuser is computationally modeled as a series of random transverse phase perturbations, each added to the phase of the simulated x-ray wavefield at the position of the diffuser, propagated and the resulting intensity map summed incoherently at the detector plane. The random transverse phase distribution is characteristic of the thickness and fiber size of the paper. An optical microscopic image of the paper surface (at 12.5× magnification) is shown in Fig. 4 , demonstrating a range of features ranging from coarse (e.g. α) to fine (e.g. β). The azimuthally-averaged power spectrum of this image is well approximated by a two-dimensional Lorentzian whose full width at half maximum (FWHM) corresponds in real space to 7µm. For each term in the time-sum, a particular random phase screen realization is generated beginning with a normally-distributed white-noise random two-dimensional array which is then spatially smoothed by Fourier-space multiplication with the normalized Lorentzian. The smoothed transversely random array is multiplied by the mean paper thickness Tpaper, the vacuum x-ray wavenumber (2π/λ) and the refractive index phase decrement δ according to the projection approximation (see ) (Eq. (1)). This phase shift is then added to the diffuser-incident phase , to give the diffuser-exit phase as a function of coordinates perpendicular to the optic axis:
We used δ = 2.75 x 10−7 for paper, producing phase depths with a standard deviation of approximately 1.25 radian at 25keV, illustrated by the example profile in Fig. 4b.
All simulations were performed using the Interactive Data Language (IDL 7.0), with a final pixel size of 0.18μm, as in the experiment (see Sec. 2). To determine only the relative effects of the diffuser and avoid prohibitively long calculation times, the model assumes a point source. The effects of an extended source on the transverse coherence properties in this set-up have been considered in a previous work by the authors . The complex monochromatic scalar wavefield Ψ from such a source was calculated directly at the plane of the square aperture, cropped in accordance with the Kirchhoff approximation to simulate the effect of the aperture, and then propagated through a distance Δ from the aperture to the diffuser position. For propagation of a coherent scalar x-ray wavefield we make use of a Fourier representation of the angular spectrum , employing the Fast Fourier Transform :
In the diffuser plane, the phase shift of the modeled paper is added to the wavefield phase, which is then propagated to the sample position. The projection approximation is again utilized to calculate the relative phase change and attenuation due to the sample of material components j:
Parameters δj and respectively quantify the refractive and absorptive properties of material j, where μj is the linear absorption coefficient.
The angular-spectrum formalism (Eq. (2)) is then used once more to simulate the propagation over the object-to-detector distance. A single intensity image over the detector plane then simulates a single position of the diffuser. The incoherent ensemble average over many such diffuser positions produces a smooth background in the final image, simulating the observed image for a complete exposure.
Note that the very long propagation distances from diffuser to object (over 150m), and an object resolution of micrometers, necessitated use of the 32GB RAM VPAC computer cluster  to sustain very large array calculations (up to 8000 × 8000 pixels) for accurate resolution of the free-space propagator (for small wavelengths, the limiting propagation distance is proportional to the array dimensions and the square of the pixel size) .
Figures 5a-5c, shown below, illustrate the effect of this process. In Fig. 5a, simulated phase contrast fringes present due to Fresnel diffraction from the edges of a 3mm diameter perspex cylinder are seen. Figure 5b shows a single realization of the diffuser model, with distorted fringes and overlying ‘image’ of the diffuser. Interestingly, the speckle from the given realization of the diffuser’s thin random phase screen, has a visibility comparable to the fringes produced by the object. (The intensity variations of Fig. 5b are greater than that seen in the observed stationary diffuser flat field of Fig. 1b due to the finite length of the exposure time and beam movement within this time.) These random undulations in the phase incident on the sample locally distort the image for each diffuser realization. Figure 5c is the incoherent sum of 1000 realizations of the diffuser model, wherein the phase contrast fringes of Fig. 5a are once again apparent but at a reduced visibility. Several of the narrowest fringes are no longer visible. The speckles due to the diffuser itself are no longer seen, as the ensemble average of many random realizations over the full exposure time becomes a smooth background of unresolved x-ray speckle.
To complete the simulations, the modelled phase contrast images were convolved with the detector point spread function (PSF). The 1D PSF was measured from experiment as the derivative of a knife-edge image , yielding a PSF of 1.2μm FWHM.
The results of this simulation modeling the same two spheres as in Fig. 3, are shown below in Fig. 6 , matching very well in terms of the fringe patterns and visibility. Again, both the contrast and number of visible fringes are greatly reduced when the diffuser is present (Fig. 6c) compared to when the diffuser is not used (Fig. 6b). The effect is exacerbated when a small degree of noise (approx 2%, consistent with experimental noise levels), is added to the simulated images (not shown).
4. Comparison between modeling and experiment, for x-ray phase contrast diffuser
In order to quantitatively determine the effect of the diffuser on propagation-based x-ray phase contrast images, in both simulation and experiment, we applied Michelson’s measure for interference fringe visibility V over the entire fringe set [24,25]:
To obtain a single phase-contrast intensity profile for this analysis, cylinder images were row-averaged, showing reasonable agreement between experimental results and our model, as shown in Fig. 7 . The diffuser is seen to decrease both the visibility and number of phase contrast fringes, in both experiment and in simulation. This detailed fringe information may be valuable in quantitative image analysis, e.g. in the context of phase-retrieval algorithms which utilize such information , however the narrowest fringes are suppressed by the diffuser.
The relative effect of the diffuser on fringe visibility is dependent on the propagation-distance between sample and detector. In the Fresnel regime, increasing propagation causes an increase in fringe contrast and spacing. The visibility of the wider fringes seen at large propagation distances is also less affected by the point spread function of the detector, when compared to the narrower fringes obtained at smaller propagation distances. The rate of increase of fringe visibility as a function of propagation distance is reduced when the diffuser is introduced. This means that the diffuser makes more of a difference at the greater sample-to-detector propagation distances, as shown in Fig. 8a . The percentage decrease in visibility incurred by a diffuser increases by a factor of around 9% when moving from 0.5 to 1.5m propagation.
This decrease in visibility is due to the smoothing effect the presence of a diffuser has on the observed phase contrast fringes (cf. Figs. 3, 5 and 6). Consequently the narrowest fringes, which are generally also the fringes with smallest local contrast, are lost when a diffuser is introduced. In a similar trend to visibility, the rate of increase, with propagation, in the width of the interference fringe area from a single edge is lessened when a diffuser is present, as seen in Fig. 8b. Measurements from simulation based on our model agree with the observed changes, showing the trends described above. There are some discrepancies at short propagation distances, where the width of the phase contrast fringes becomes comparable to the pixel size.
The phase variations which determine the efficacy of the diffuser may be described by both the phase depth and characteristic transverse length scale l of the resulting wave-field phase. Speaking only of the un-propagated wavefield (i.e., the wavefield at the exit-surface of the diffuser), the phase depth of the wavefield is determined according to the projection approximation, which incorporates x-ray energy, the phase decrement δ of the diffuser material and the variations in the projected thickness of the material. The characteristic transverse length scale l is determined by the size distribution of the diffuser constituent materials which give rise to those projected thickness variations, together with the thickness of the diffuser.
A simple descriptor which combines the properties and l is that of a ‘root-mean-square (RMS) transverse phase gradient’; roughly speaking it is given by / l. In Figs. 9a and 9b are shown the simulated effects of changing these properties, on the Fresnel-regime fringes of the Perspex cylinder image. For simplicity the phase of each cylinder-entrance wavefield was generated directly, i.e. without prior propagation, using Gaussian length-scale distributions. Figure 9a keeps l constant at 20μm FWHM and varies the phase-depth from a standard deviation of 0.2 rad to 2.0, whilst Fig. 9b keeps constant at a standard deviation of 1.0 rad and varies l from 2um to 20um.
In Fig. 9a the profile with a = 0.2 rad is almost indistinguishable from the profile simulated without any diffuser. As the phase depth is increased the profile is increasingly strongly smoothed until the = 2.0 rad profile is so strongly smoothed as to have lost all fringe information altogether. These profiles are visually comparable to convolution of the no-diffuser profile with a Gaussian whose FWHM varies with the square of the phase depth.
In Fig. 9b, where the characteristic length scale l of the diffuser material is varied, the effect at large length scales (or small phase gradient) is consistent with a slight smoothing effect. As the length scale is decreased, the fringe contrast is further reduced however the smallest fringes are still apparent. The 2μm FWHM length-scale profile has a much smaller absolute visibility than the largest phase depth profile in Fig. 9a and yet has retained significant fringe information. As such, it appears that the larger length scale variations in phase must also be present to smooth out the narrow fringes, as was observed in experiment.
Propagation of the wavefield from diffuser to sample was shown in simulations to result in negligible changes in phase depth. However, due to the relatively large propagation distances involved (see Fig. 2) and the corresponding scaling associated with Fraunhofer diffraction, the characteristic transverse length scale is increased as the whole wavefield is magnified with distance by the factor:
Figure 10 shows experimentally the effect of changing the diffuser position relative to source and sample. The resulting magnification with increased propagation means that the RMS phase gradient at the sample is decreased (and the degree of coherence increased), reducing the dampening of the fringes by the diffuser. All of the diffuser measurements described thus far were conducted with the diffuser in its customary location, i.e. the experimental hutch which is 29 m from the aperture (see Fig. 2). However, as a result of these and previous studies , the diffuser has now been permanently relocated to the closer optics hutch, and placed 1.7m from the aperture, hence the reduction in visibility is now insignificant.
Figure 11 is a plot showing the simulated effects of changing RMS phase gradient on fringe visibility. The decrease in visibility when changing characteristic transverse length scale is shown on the same axis as when changing phase depth. As always the visibility was calculated from the global maxima and minima of the fringes. There is general overlap between the two series of points, which is consistent with the idea of the RMS phase gradient as the determining factor. The series differ most at higher phase gradients, where the length scale was smallest (and less physically likely).
The simulated visibility at the two diffuser positions are also indicated in Fig. 11. It was seen that by moving the diffuser close to the source, the magnification increase led to a relative phase length which was effectively doubled, i.e. halving the phase gradient. This change is consistent with the effect observed in Fig. 10. The additional propagation from the diffuser to the sample increases the observed coherence, in accord with the van Cittert-Zernike theorem [26–28].
Figure 11 may thus be viewed as a plot which summarizes the deleterious effect of the diffuser on the phase contrast fringe information for a given sample-detector distance. The RMS phase gradient of the x-ray wavefield at the sample entrance plane is characterized by the diffuser position and material, and indicates the strength of the diffuser. The visibility of the resulting detected fringes from a typical sample, relative to their visibility in the absence of the diffuser, is proportional to the factor decrease in the observed degree of coherence which the diffuser causes. Use of too strong a diffuser yields a part of the curve where the decrease in observed fringe visibility and hence degree of coherence may be unacceptable to the user. On the other hand, this loss of coherence may be advantageous if the user wishes to remove phase contrast.
If the user wishes to conserve the observed degree of coherence as much as possible, and yet maintain the desired effects of the diffuser on the intensity homogeneity of the detected beam, it is perhaps easiest to move the position of the diffuser, away from the sample and closer to the source.
5. Summary and Conclusion
A spinning piece of paper represents an extremely simple and low-cost method for the diffusion of hard x-ray beams in the context of propagation-based phase contrast imaging. Use of such a diffuser is of particular benefit to imaging experiments which require close to homogeneous intensity over a broad area and increasingly shorter exposure times, for example in many in vivo biomedical imaging applications. As many of these applications depend on a sufficiently high degree of partial transverse coherence for phase contrast, it is important to minimize the potential associated deleterious effects of the diffuser on fringe visibility. In this work which combines experimental propagation-based phase contrast x-ray imaging data using a diffuser with simulations, we have modeled the effect of the diffuser as the time-averaged incoherent sum of random phase perturbations to the x-ray wavefield. Simulated Fresnel-regime fringes with and without a diffuser were consistent with experimental observations, displaying a significant decrease in both the number and visibility of fringes when the diffuser was introduced into its standard position. The effect was greatest at larger sample-detector distances. The role of the phase distribution of the random screen representing the diffuser, in terms of a characteristic phase depth and length scale, was investigated and discussed in terms of an average absolute ‘phase gradient’. This leads to several practical suggestions for the optimization of a diffuser in terms of minimizing the effect on the degree of coherence at the object and detector. Primarily, it is desirable to place the diffuser as close to the source, or secondary source (e.g. aperture), as possible. The resulting increase in far-field propagation and consequent scaling serves to magnify the length scale of the phase distribution, decreasing the phase gradient and hence the effect on fringe contrast. The combination of thickness/roughness and fiber characteristics of the diffuser-material should also be considered given specific requirements for the detail and visibility of a phase contrast fringe set.
Conversely, it is also possible to apply these principles for opposite effect, in situations where a high degree of coherence is temporarily undesirable. One example of this would be on a synchrotron beamline where pure absorption-based CT is required but a contact (zero distance between sample and detector) image is unachievable.
The position of the diffuser along the beamline may therefore be chosen to produce the required degree of coherence for the imaging required. More importantly, the diffuser will spatially even out the beam and reduce phase effects in the final image.
The authors thank the Japan Synchrotron Radiation Research Institute (Proposal number 2008B1985) for the privilege of using the SPring-8 facility to conduct these experiments. We acknowledge funding for the trips to SPring-8 for the Australian co-authors from the Access to Major Research Facilities Program, which is supported by the Commonwealth of Australia under the International Science Linkages program. The computational component of our project was supported by the Victorian Partnership for Advanced Computing HPC Facility and Support Services. Kaye Morgan and Sally Irvine acknowledge the support of an Australian Postgraduate Awards. Kaye Morgan also acknowledges a J. L. William Scholarship from Monash University. David Paganin acknowledges the Australian Research Council. We thank Andreas Fouras for the use of his PCO camera during the experiment, as well as Stephen Dubsky and Simon Higgins for their assistance and the use of their optical microscope for Fig. 4.
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