Experimental investigation of Gaussian random phase screen model for x-ray diffusers

The beam diffusing properties of stacked layers of diffuser material were evaluated experimentally and compared to a Gaussian random phase screen model. The model was found to give promising accuracy in combination with a Lorentzian auto-correlation model. The tail behaviour of the angular scattering distribution as a function of number of diffusing layers was particularly well described by the model, and in the case of an amorphous carbon diffuser, the model could describe the whole of the scattering distribution convincingly. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

incident beam and distributing it over the field of view, which is easily achieved in the new illumination by selecting an appropriate focal length for the condenser. If a smaller field of view is required, a longer focal length condenser should be used to allow the beam to condense more before arriving at the sample position. The second allegation on the other hand is a more challenging one, and may not be dismissed so easily. The benefits of aperture matching on the spatial resolution of a microscope can be linked to the coherence length of the illumination. To achieve optimal resolution in a transmission microscope, the coherence length should be shorter than the diffraction limited resolution so that there is no visible interference between adjacent features. Aperture matching, in the critical illumination scheme, produces a coherence length that is equal to the diffraction limited resolution. Here we propose that the diffuser placed between the sample and the condenser lens can provide the same effect. This would have the added benefit of providing field of view and coherence length that can be varied independently, which makes the microscope more flexible with regards to design constraints. The challenge, however, is to reliably engineer diffusers with the desired properties. The experiments and theory presented in this work will take steps towards this end.

Theory
The model presented here is an extension of the Gaussian random phase screen model [16]. The diffuser is modelled as consisting of n statistically independent diffuser sheets. It is assumed that the sheets are close enough for propagation to be negligible, i.e. the projection approximation is used [17]. The phase shift induced by the diffuser is the sum of contributions from individual layers.
φ(x, y) = φ 1 (x, y) + φ 2 (x, y) ... φ n (x, y). (1) As the phase shift of each individual layer is statistically identical, and statistically independent of the others, the root mean square value (rms) of φ(x, y), which is denoted by σ φ , is given by where σ layer is the rms of the phase shift distribution of a single layer. According to the central limit theorem, the phase distribution P φ (φ(x, y)) approaches a Gaussian as n increases regardless of the form of the underlying phase distribution of the individual layers. It can be shown [16,18] that if a plane wave was passed through such a diffuser, the resulting mutual coherence function would be Γ (∆x, ∆y) = e iφ(x,y) e −iφ(x+∆x,y+∆y) = e −σ 2 φ (1−γφ(∆x,∆y)) , where γ φ (∆x, ∆y) is the yet unspecified normalized correlation function of φ(x, y). Note that this result is valid for any value of σ φ , and does not rely on small phase approximation. The power spectral density, G q x , q y can be computed by invoking the Wiener-Khinchin theorem, which states that G q x , q y is the Fourier transform of Γ (∆x, ∆y).
The mutual coherence function and the power spectral density are convenient tools for analysing the effects the diffuser will have in different situations. Here in particular, the power spectral density is of interest because it can be directly measured experimentally. Suppose the diffuser is placed in a coherent and converging x-ray beam, as illustrated in Fig. 2, at a distance z upstream of the beam focal point, and an imaging detector is placed in the same plane to record the intensity of the resulting distribution. It is possible to calculate the intensity from the generalized Van Cittert-Zernike theorem [19], and the result is where x and y are the coordinates in the transversal plane where the beam comes to a focus, and I(x , y ) is the intensity distribution in that plane. Here it is assumed that the area of the beam at the diffuser is infinitely large, and would in the absence of a diffuser produce a truly infinitesimally small focal spot, at least within the small angle approximation. In practice, one will not produce such a beam. A real x-ray beam has finite area, partial coherence, both of which will result in a blurring of I(x , y ). As the number of layers is increased, σ φ behaves according to Eq. (2), whereas γ φ (∆x, ∆y) remains unchanged. Following from the central limit theorem, the mutual coherence function should asymptotically approach a Gaussian when n becomes sufficiently large, i.e.
where the value of σ q scales with σ φ in a way that depends on γ φ . This model has the advantage of not making any assumptions about phase-depth or phase gradient distributions, which allows it to capture an important feature of hard x-ray diffusers, namely the likely existence of a non-scattered fraction of the beam. This point is best illustrated by a demonstration. For this purpose, we use an example that will prove relevant for the experimental results, namely the case of a Lorentzian autocorrelation function, Here l is the half width at half maximum value of (∆x 2 + ∆y 2 ) and can be thought of as transversal coherence length. Figure 3 shows some power spectral densities and mutual coherence functions produced by this model. All the curves were produced using the same γ φ , the only difference between curves is the number of layers, which affects σ φ in accordance with Eq. (2). Figure 3(a) shows the evolution of the MCF. Initially, at n = 0 it is simply 1, as there is no randomness to destroy any correlation. As the number of layers start to increase, we see that a peak appears in the center, on top of a constant background. The value of the background, which is denoted by Γ ∞ , can be found by taking the limit in Eq. (4) with ∆x going towards infinity.
Note that this expression holds for any choice of γ φ that tends towards zero at infinity. As the number of layers increase further, Γ ∞ vanishes, and the MCF tends towards a Gaussian shape. With a Lorentzian γ φ , the value of σ q is given by Figure 3(b) shows a logarithmic plot of the power spectral densities calculated from the MCFs. In order to discuss the behaviour of these curves, it is convenient to separate them into two terms that may be thought of as the coherent and the diffuse part.
One can see from Eq. (12) that the power spectral densities contain a Dirac's delta function due to the non-zero background level, Γ ∞ . In the interest of making the figures more presentable, however, the shape has been convoluted by a Gaussian, representing a source with angular rms of 4 · 10 −6 at 17 keV photon energy. This results in an rms value of 3.46 · 10 5 in q-space. As the number of layers increase, initially, one can see that the central peak is attenuated and eventually disappears as the curve becomes more Gaussian. Figure 3(c) shows a semi logarithmic plot of how the peak attenuates with increasing number of layers. Initially the descent follows the exponential decay law of Eq. (12), but flattens out as the peak amplitude of G 0 becomes small enough to be comparable to the peak of G diffuse . Another property of interest is the width of G q x , q y . Here it is worth pointing out that G q x , q y for different number of layers are related to each other by successive convolutions, i.e.
This is significant because the second order moments of the result of a convolution is the sum of the second order moments of the arguments. This implies that the second order moments of G q x , q y ; n , which is denoted as rms q (n), are given by which holds, regardless of whether the G 0 peak is present or not. Figure 3(d) demonstrates this fact by showing that the log-log plot of n against rms q (n) gives a straight line with a slope of 0.5, indicating a power law relationship with 0.5 as exponent, i.e. a square root relationship. Consider now the case of a projective imaging system, where a parallel beam passes straight through a sample and the beam is recorded some distance downstream. A diffuser would affects an image from such an imaging system in two important ways. The first by smoothing out unevenness in the illumination, and the second by the blurring of image details. The former depends on how far upstream of the sample the diffuser is placed, while the latter primarily depends on the distance from the sample to the detector. A thorough theoretical description of this topic will be presented by Paganin et al [20] and is not discussed in detail here.
For imaging systems with objective lenses (stigmatic imaging systems), such as a full field microscope, the situation is different. With an objective lens, all rays emerging from a point on the sample plane are focused on a particular point in the image plane, provided the ray passes through the objective lens' aperture. The distance from sample to detector is therefore not directly relevant. What matters instead is the influence of the diffuser on the transmission cross-coefficients (TCC) [21,22]. The image intensity is given by where r is a two dimensional coordinate vector in the image plane,ψ is the Fourier transform of the wave front perturbation imparted by the sample, and q and q are two-dimensional reciprocal coordinate vectors. TCCs is given by where P is the pupil function of the imaging system. Unfortunately, computation of I (r) for arbitrary G and P is computationally heavy, as the TCC must be computed over a four-dimensional grid. The problem can be computationally simplified by singular value decomposing TCC(q , q ), known as the sum over coherent systems method [23,24], but this was not done here. Here we have made due with solving a simplified problem, where the sample is constant along the y-axis.
In this case, it follows from the Fourier slice theorem thatψ is non-zero only when q y = 0. One then only has to compute TCC in the two-dimensional sub-space where q y and q y is zero in order to evaluate I(r) from (16). Figure 4 gives a demonstration of how image resolution is affected by G. Figure 4(a) show the image resolution for a fixed value of l, with varying σ φ , and Fig. 4(b) show a similar case, but with a varying l and constant σ φ . This demonstrates that with Lorentzian γ φ (∆x, ∆t), both l and σ φ has an impact on resolution.

Experimental setup
The experiment was performed at ID06 at ESRF with a setup similar to the one illustrated in Fig.  2 [6,7]. While the theoretical effects of partial coherence on microscopy images are well understood, it is of interest to observe the real impact with actual diffusers in actual microscopes. In order to do so, a full field microscope was set up in the same beamline. A sketch of the Microscope can be seen in Fig. 1. The objective lens consisted of 89 Be lenslets with 50µm radius of curvature with an average spacing along the optical axis of T=1.7 mm, resulting in a focal length of 190 mm measured from the entrance of the lens stack. The condenser lens consisted of 32 Be lenslets of the same type, with a T =2 mm spacing along the optical axis, yielding a focal length of about 640 mm. The distance between sample and objective lens entrance was p = 210 mm, the length of the objective lens was L = 148 mm, and the distance between objective lens exit and detector was q = 4000 mm, yielding an x-ray magnification of 14. The detector was the same FReLoN camera used in the small angle scattering experiment. The effective pixel size, taking x-ray magnification into account, was measured to be 88 nm. The diffuser was placed approximately 170 mm from the sample. Figure 5 shows a semi-logarithmic plot of the collected scattering data. It is evident that curve broadens as the number of layers increase. It also appears that the tails of curves make straight lines in the logarithmic plot, which implies that the tails decay exponentially. We note that this phenomena appears to be present even at n = 1, where the phase-depth presumably was small. This is significant because at shallow phase-depth,

Results
The curves, according to Eq. (6), are measurements of the Fourier transform of Γ (∆x, ∆y).
As the constant term in Eq. (18) does not contribute to the tails of the Fourier transform, the exponential tails stem from γ φ , i.e. the Fourier transform of γ φ have exponential tails. A simple function that behaves in this way is the Lorentzian function. This can be seen in Fig. 3(b), particularly in the curve for n = 1. The Fourier transform of a Lorentzian is a symmetrical, two-sided sided, exponential function (Laplacian distribution). We therefore attempted to model the data using a Lorentzian phase-correlation function. Simulated curves were produced by first choosing a value of l and phase depth for a single layer, σ layer . Then, for every n, σ φ was set to σ layer √ n. From this, Γ (∆x, ∆y) could be determined according to Eq. (4), allowing G q x , q y to be calculated from Eq. (5). To simulate scattering patterns, an image of the focal spot, recorded without any layers, was convoluted by G q x , q y . The displayed curves are cross-sections of the resulting scattering patterns. The results can be seen in the dashed lines in Fig. 5. The tail behaviour seem to scale appropriately with n. Note that only two parameters, namely l = 2.3µm and the phase depth for a single layer, taken to be 0.65 , was used to produce all of the curves. Near the origin, however, there are notable discrepancies suggesting that γ φ was not truly a Lorentzian. We speculate that scattering from the 3 mm thick polyester film which the Al 2 O 3 particles were attached to might play a role in this. Figure 6 shows the result of applying the same procedure to the scattering data from the carbon diffuser disc. The model here produced a convincing match to the experimental data. The Lorentzian model appears to be an excellent fit for this diffuser. Fig. 5. Semi-logarithmic plot of recorded small angle scattering pattern. The intensity was rescaled so as to be equal to 1 at the origin. Note the tendency for constant slope away from the origin, suggesting that the tails decay exponentially. The modelling parametres used were l = 2.3µm, and σ φ = 0.65 √ n. Next we wished to investigate the validity of Eq. (15). Figure 7 shows a log-log plot of the measured rms q against n. The slope of the curve appears to be approximately correct, however, it should be mentioned that the resulting rms values are highly sensitive to the treatment of background noise. In order to investigate the effects of diffuser on actual CRL microscopy images, a grating with 0.3 µm pitch was placed in the object plane of the microscopy setup and recorded with different number of diffuser layers. The result can be seen in Fig. 8. The first thing that can be observed is that there is the smoothing of the illumination. The large intensity oscillations that are present in most of the field of view in Fig. 8(a) are almost completely removed, even with only 6 diffuser sheets, as seen in Fig. 8(b).
Regarding effects on resolution, the numerical aperture of the objective lens is 3 × 10 −4 . The angular distribution of the scattering patterns produced by the diffusers ranged up to approximately 2 × 10 −5 rad. Theoretically, based on Eq. (16) and Eq. (17), this angle is much too small to produce an appreciable effect on the resolution. Nevertheless, it is clear that the diffusers are having a significant impact, as can be seen in Fig. 9. It is evident that adding more sheets the shoulder fringes near the absorbing line are attenuated, but so is the depth of the lowest point on the graphs. The observed behaviour is not consistent with the simulation based on Eq. (16) and Eq. (17) as there should not be any shoulder fringes under ideal condition. Conditions are of course not ideal, however, and while it is difficult to identify the cause with high certainty, it seems plausible that the observed effect is a result of interaction between aberration, defocus and partially coherent illumination. We note that significant aberration is known to be present in the type of lenses that were used, but good progress have been made towards corrective optics, which have already been applied in nanofocusing [27]. It is clear that the diffusers have an effect, but it is not obvious which diffuser configuration is better. Assuming lens aberration is in fact the culprit, it is reasonable to expect significant quality improvements with corrective optics, but so far, experiments on full field imaging with corrective optics have not been reported.
It should be noted that in order to bring the angular spread from the diffuser layers to a level closer to the N.A. of the lens, we may extrapolate from Fig. 7 that about 2000 to 3000 layers would be required, suggesting that this particular diffuser material would not be suitable for the purpose of filling the aperture of this objective lens at 17 keV.

Conclusions
The Gaussian random phase screen model has been identified as a useful theoretical frame work for the design of x-ray diffusers, and can likely be used to evaluate the suitability of a diffuser material quantitatively. The relevant parameters of the model are the phase depth per layer and the auto-correlation function for a single layer. The parameters of the auto-correlation model is independent of number of layers. The model can therefore prescribe how many layers are necessary to achieve the desired quality of illumination. It was shown that a Lorentzian auto-correlation model correctly predicts tail profiles and their behaviour in response to changing number of layers, and in the case of the carbon disc diffuser, it accurately models the full scattering pattern, including the central peak.