1. Introduction

Current developments in modern X-ray synchrotron radiation and free electron laser facilities enable a wide range of applications. X-ray beamlines in these facilities must deliver high-quality X-ray beams to their users. The effects of imperfect beamline optics have been widely studied both on and off the focal plane [1–11]. However, there is still a lack of a satisfactory physical explanation for the impacts on the far-field images. In this paper, “far-field” means far from the focal plane of the tested optics. For optics that have no focal plane, such as a plane mirror, “far-field” means far from the source.

For many X-ray imaging experiments, a clean and uniform beam of variable size in the far field is strongly desired. However, due to the limitations of the present manufacturing technology for modern X-ray optical elements, some fine structures always appear in such beams, especially after reflective optics [12–18]. An understanding of the origin of these structures is a necessary step towards their mitigation. Past works have shown them to be closely related to the curvature (second derivative) of the wavefront [19–27]. Recently, the authors of this work measured fine structures in X-ray beams reflected by flat mirrors [28] and confirmed the structures’ relationship to the curvature of the wavefront by using rigorous wave optics theory.

A new theoretical approach is presented in this work. First, with our new approach, the far-field beam structures can be quantitatively estimated from known optical imperfections. With such information, manufacturers of X-ray optics could be given clear specifications of permissible fabrication errors. Second, the theoretical approach used in this work extends to general X-ray optics, rather than only to reflective optics as in our previous work. Third, this approach can be applied to partially as well as fully coherent beams.

In this paper, we will show how our theory can be related to the transport of intensity equation (TIE), which is based on fundamental physical principles and is widely used in X-ray phase imaging [29–33]. Thus, they have the same loose assumptions about the source. The local variation of the intensity in the far-field images is shown to be simply proportional to the local variation of the wavefront curvature, which can be known from in-situ measurement using the speckle-based technique [28,34,35] or other in-situ wavefront sensing techniques [36–41]. It can also be calculated using the more routine ex-situ measurement [42]. The new theory has also been tested experimentally. Although some of the results have been shown in our previous paper [28], they were only used to verify the previous theory qualitatively. Here, using the new theory, we interpret them quantitatively.

2. Theory

2.1 Propagation of intensity of light

Our derivation starts from the conservation of energy for the electromagnetic field [43]. The conservation law tells us that the time-averaged Poynting vector of the field < P > in a homogeneous free space with no free charges or currents must obey the following equation:

Whereas in X-ray imaging TIE is mainly used to reconstruct the phase information, in this work, the phase information is always assumed to be known. We focus on the propagation of the intensity. By substituting Eqs. (3) and (4) into Eq. (2) and then solving Eq. (2), we can represent the light intensity at a point (x, y, z) in the space as [43] (see Appendix A.2 for detailed derivation):

The integration is along the light ray, which is parallel to the local normal vector of the wavefront. In a homogeneous medium, it is a straight line. The above equation provides us with an analytical expression for the propagation of the intensity of light. It indicates that the propagation of the light intensity is modulated by the second derivative of the wavefront. We use it to discuss the impacts of the imperfections of the optics on the far-field images. We only need to notice that the phase term ϕ(r) can be written as the addition of an ideal wavefront and a curvature error caused by the imperfections from the optics.

For the trivial case of a plane wave, Eq. (6) shows that the intensity distribution remains unchanged by propagation as expected. See Appendix A.3 for the details.

2.2 Application to an arbitrary wavefront

In a curved wavefront, the local propagation direction is no longer constant, but varies across the initial plane. For example, the local wavefront propagation directions on a perfect spherical wave all point away from the origin if the wave is divergent, or toward the focus if the wave is convergent.

It can be proved [43,44] that for any curved wavefront, the integral term in Eq. (6) can be represented as:

If the wavefront is a plane, the integrand in Eq. (8) is zero and the integration is trivial. If the wavefront is cylindrical, only one of the terms of the integrand in Eq. (8) is nonzero. Otherwise, to evaluate the integral term in the above equation, one notes that along the propagation direction, the wavefronts at two close positions are parallel and therefore [44]:

So, we have:

Using the above equation, Eq. (8) becomes [43]:

R₁ and R₂ are in the orthogonal directions.

Equation (11) obeys the energy conservation law. Figure 1 illustrates two small wavefront surface elements at two different positions along the propagation direction. The small surface elements are bounded by lines of curvature. A₀B₀C₀D₀ is at the position s₀, and ABCD is at the position s. R₁(s₀) and R₂(s₀) are the principal radii of curvature of the line segments A₀B₀ and B₀C₀, respectively. We have

 

Fig. 1. A₀B₀C₀D₀ and ABCD are two small wavefront surface elements at two different positions along the propagation direction. The mesh represents the orthogonal curvilinear coordinate system. A₀B₀, B₀C₀, AB and BC are lines of curvature.

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Similarly, we have:

The energy conservation law requires [43]:

Combining Eq. (12)-(15), we get Eq. (11). This proves that Eq. (6) is also valid for any curved wavefront. Especially, for a perfect spherical wavefront, R₁= R₂= R and the Gaussian curvature is 1/R². The energy conservation law for a perfect spherical wavefront is I(s₀)R²(s₀)=I(s)R²(s).

2.3 Application to imperfect X-ray optics

In a practical optical system, the optical elements are not always perfect. The wavefront at the exit plane of the system is usually not uniform. This is particularly true in X-ray optical systems due to the extremely small wavelength and small grazing angle of X-ray reflective optics. Thus, very high-quality optics are needed to deliver high-quality X-ray beams. However, at present, the available X-ray optical elements are far from perfect. As a result, it is very common to see fine structures in the far-field images.

We only consider the wavefront error in this work. The optical element is assumed not to absorb or scatter any of the incident radiation. Thus, only the phase differs in the planes immediately before and after the optical element, while the intensity is the same. We use I₀(s₀) to represent the ideal intensity distribution at the plane immediately after the optical element, and I’(s₀) to represent the non-ideal intensity distribution at that plane. Here “ideal” means that the incident beam impinges on an ideal optic. Since the error of the optic is assumed to disturb only the phase of the beam, the intensity distribution for both ideal and non-ideal cases at the planes immediately after the optic is identical, I₀(s₀)=I’(s₀). We apply Eq. (11) to the non-ideal wavefront, using R’_{1, 2}(s₀) to represent the local wavefront curvature at the plane immediately after the optics, R’_{1, 2}(s) to represent the local wavefront curvature at the detector plane:

It is easy to find that the intensity distribution in the far field changes according to the local magnification factor R’_{1, 2}(s₀)/R’_{1, 2}(s), also see Fig. 2. As a result, the contrast of the fine structures that appeared in the far-field image is proportional to the variations of the local magnification factor. Letting d = s-s₀, the above equation can be written as:

 

Fig. 2. A typical experimental layout for X-ray optics.

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Equation (17) relates the contrast of the fine structures in the intensity distribution to the local change of the wavefront curvature at the detector plane. Similar results can be also found in Peterzol et al [45].

We can also rewrite Eq. (16) in terms of the local wavefront’s radii of curvature at the plane immediately after the optical element:

Under the assumption that the perturbation of the phase is small, we use the first terms of the Taylor expansion of the fractions in the above equation:

Unlike our previous work [28], the theory proposed in this work imposes no assumptions on the type of X-ray optical elements. It can be applied to refractive (such as CRL), reflective (such as mirror) as well as diffractive (such as Fresnel zone plate) optics, as long as the relationship between the error from the optical element and the wavefront has been set. For instance, because of the very small grazing angle of the X-ray mirror, the wavefront coordinates should be projected back to the mirror surface through division by the small grazing angle, usually several milliradians. For strongly focusing optics in the far field, such as the CRL stack, multilayer Laue lens or focusing mirror, for which d ≫ R_1,2(s₀) according to Eq. (19), the relative change of the wavefront curvature determines the contrast of the fine structures appearing in the far-field images. If the variation of the wavefront curvature is within 5%, then the expected intensity variation is also within 5%. For weakly or non-focusing optics, on the other hand, the contrast of the fine structures also depends on the distance from the optic to the detector as predicted by Eq. (17) or (19). We will give experimental evidence for both cases later in this paper. For 2D focusing optics such as most CRLs, one needs to note that the principal directions of the perturbed wavefront will be different from the principal directions of the individual components. This may add more difficulties in predicting the far-field intensity accurately. However, this does not arise for the 1D focusing optics used as examples in this paper.

It is easy to show that Eq. (6) can lead to the foundations of geometrical optics [43]. Under certain circumstances, the ray-optical approach is sufficient to explain the formation of the fine structures in the far-field intensity distribution. From our derivation of the TIE at the beginning of this section, the TIE can be applied to any type of light source, as long as the beam intensity and the phase are properly defined. For a partially coherent source, the phase can be defined statistically [33]. Peterzol et al. [45] have pointed out that with some restrictions on high-spatial-frequency features, the TIE can be derived from general diffraction theory. Our theory has the same assumptions as TIE. Thus, the same argument can also be applied to our approach. The change of the wavefront local curvature determines how much beam energy can be pulled out from the ideal intensity distribution. For a highly coherent beam, the energy removed from the ideal distribution can become concentrated in even finer structures in the far-field image. Equation (16), (17) and (19) still provide a good way to estimate the uniformity of the intensity distribution in the far-field image.

3. Experiments

X-ray mirrors are simple yet still suitable optical elements for the verification of the theory proposed in the previous section because the small grazing angle makes the effects of surface errors much less in the sagittal direction than in the tangential one. The fine structures that appear on the observation plane thus come mainly from the surface error along the mirror length, effectively reducing the problem from two dimensions to just one. The relationship between the local wavefront curvature and the far-field intensity distribution can be easily observed. The features would be much more difficult to match for 2D optics since contributions from both orthogonal directions will be added to the intensity distribution. No one-to-one correspondence can be easily found.

Two mirrors have been measured to verify the theory. The experiments using X-rays were conducted at the Diamond Light Source B16 Test beamline [46]. Two types of monochromatic beam with different energy bandwidths can be provided by either a Ru/B₄C double multilayer monochromator (DMM) or a Si (111) double crystal monochromator (DCM). Except for the monochromators, there are only windows between the tested mirror and the source. Thus, the ideal intensity image in the far field will have a relatively large area with gradual spatial variations in the intensity. The measurements using visible light were conducted at the Optical Metrology Laboratory [42], which is also at the Diamond Light Source.

3.1 Measurement of a plane mirror

A plane mirror with no coating was measured using the at-wavelength wavefront sensing technique. The rms slope error of this plane mirror is about 0.17 µrad. The slope error and the local wavefront curvature of this mirror were obtained using the speckle-based in-situ at-wavelength techniques [28,34,35]. A diffuser is used to generate the speckle pattern for X-rays. By comparing the local shifts of the speckle pattern with the reference pattern, the local wavefront curvature or slope can be reconstructed depending on the choice of references. The X-ray beam came from the double crystal monochromator. Except for the monochromator, there are only windows between the tested mirror and the source. Table 1 shows the basic experimental parameters for this mirror.

Table 1. Experimental parameters for the plane mirror

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Although this mirror has been shown in our previous work [28], there we only show the correspondence between the wavefront curvature and the fine structures in the far-field images qualitatively. Here, using the theory proposed in this paper, we can discuss the contrast of the fine structures quantitatively.

Figure 3(a) shows the reflected intensity image on the detector plane. Figure 3(b) shows the measured wavefront local curvature on the same plane. Firstly, although this mirror is quite smooth in terms of its slope error, those fine structures can still be seen from the intensity image. If we have a closer look at Fig. 3(a) and (b), we can find that fine structures shown in (b) can find their matches in (a). Secondly, some other features that appear on the intensity image are not caused by the wavefront curvature error in the measured direction, which is vertical in this case. Detailed inspection reveals that they are mainly from the incident beam. They are formed by other factors, such as the wavefront curvature error in the orthogonal direction or the absorption. Thirdly, for a plane mirror, we can use Eq. (17) to estimate the contrast of the fine structures that appeared in the far-field intensity image. Figure 3(c) shows a strip of the extracted and averaged intensity data and the intensity distribution estimated using Eq. (17) from the corresponding strip of wavefront curvature data. Observing the two curves, we can see that all the features match well with each other. It is known that the result from the speckle-based technique is smeared a little bit by the convolution with several detector pixels due to the choice of finite window size. That can explain the discrepancies between the intensity curve and the theoretical results, especially on the sharp jumps. We define the contrast of the fine structures here as σ_I/I, where σ_I is the standard deviation of the beam intensity. According to Eq. (17), the standard deviation of the wavefront curvature at the detector plane multiplied by d can also represent the contrast of the fine structures, where d is the distance between the mirror center and the detector plane (1.705 m). From Fig. 3(c), according to our definition, the standard deviation of the extracted intensity is around 138. The mean value is around 3300. Thus, σ_I/I ≈ 0.042. The standard deviation of the extracted wavefront curvature is 0.0278 m^-1. Multiplying it by the distance d also represents the contrast of the fine structures on the far-field image. The value is around 0.0474, which is close to σ_I/I. These results verify Eq. (17).

 

Fig. 3. (a) is the far-field intensity image after the reflective mirror. (b) is the wavefront curvature on the detector plane. A strip of intensity data is extracted, averaged, and showed in (c) in a blue curve. The average is along the horizontal axis. The corresponding strip of wavefront curvature data is also extracted and averaged. It is then used for Eq. (17). The result is shown in a red curve in (c).

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3.2 Measurement of an elliptical mirror

An elliptical mirror with multiple coated stripes was also measured. These stripes are designed to have parabolic sections with different amplitudes to modulate the exit beam [47]. The stripe measured here has the highest modulated amplitude. The parameters of the mirror are shown in Table 2. The slope of the parabolic modulation was measured offline using the Diamond-NOM [48] and is shown in Fig. 4(a).

 

Fig. 4. (a) Mirror slope modulation measured using Diamond-NOM. (b) The intensity distribution in the far field. The mean intensity curve (averaged along the vertical axis) within the red rectangular box is extracted and shown in the red curve in (c). The blue curve in (c) is the estimated intensity distribution from the wavefront curvature error using Eq. (19). The horizontal coordinate of the blue curve is along the mirror surface.

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Table 2. Main parameters of the elliptical mirror

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Since the distance from the mirror centre to the detector is much larger than to the mirror focus, Eq. (19) is appropriate for the estimation of the far-field intensity variation. To do that, we need to convert the mirror surface slope modulation to the wavefront curvature modulation, which is the derivative of the wavefront slope modulation. Note that the wavefront slope modulation is twice the mirror surface slope modulation. Considering the elliptical shape of the mirror surface, we cannot just project the mirror surface coordinates to the wavefront coordinates by simply multiplying the angular factor of sin(θ) [49]. To be accurate, one needs to do ray tracing. We use the xrt python package [50] for our calculation. The converted wavefront curvature can be used to estimate the intensity distribution. This is shown in Fig. 4(c) with a blue curve. This curve is calculated using Eq, (19).

The measured far-field X-ray intensity distribution after this elliptical mirror is shown in Fig. 4(b). The horizontal stripes shown in Fig. 4(b) are introduced by the double multilayer monochromator. The mirror was placed facing sideways to decouple the impact of the incident beam. The intensity data within a red rectangular box is extracted and averaged. This is shown in Fig. 4(c) with a red curve. The two curves in Fig. 4(c) show clearly the correlation between the wavefront curvature modulation and the intensity distribution. They both have step-like jumps. For the measured intensity distribution, we define the contrast as (I_max-I_min)/(I_max+ I_min). The contrast of the first jump (at around pixel 100) is (3650-2750)/(3650 + 2750)≈0.14, the contrast of the second jump (at around pixel 390) is (3850-3000)/(3850 + 3000) ≈0.124, and the contrast from the lowest step to the highest is (3850-2750)/(3850 + 2750)≈0.17. The corresponding contrast of the estimated intensity distribution using Eq. (19) is around 0.24 for the first jump, 0.19 for the second jump and 0.27 from the lowest step to the highest. These values are compatible with those from the measured intensity distribution. The discrepancies come from several reasons. First, the mirror pitch angle may not be at the exact nominal one due to the error coming from the installation. Second, at the position with a large jump, our theory may not be effective because the nonzero window size chosen for the speckle technique creates a convolution effect on the data processing procedure. This can further impact the correspondence between the theory and the experiment. In any case, we can still use the theory proposed in Section 2 to estimate the variation of the intensity image in the far field.

4. Conclusions

In this paper, we have confirmed with theoretical derivations the correlation between the wavefront curvature and the fine structures that appear in far-field images produced by imperfect X-ray optics. We have also provided equations that allow quantitative estimation of the contrast of the fine structures if the imperfections of the optic are known. Our new theoretical approach has a wider range of applications than our previous work. It can be used for general X-ray optics and for all sources, including partially coherent ones, whose radiation has a defined intensity and phase. The X-ray intensity variations in the far field after a plane mirror and an elliptical mirror have been investigated experimentally. The measured contrast of the fine structures is in good agreement with the theory proposed here. Thus, the proposed theoretical approach can help us to set manufacturing tolerances on the optical imperfections when smooth, structure-free beams in the far field are required.

Appendix

A.1 Derivation of the TIE

We derive the TIE from Eq. (2). Replacing the wavefront propagation direction s in Eq. (3) with Eq. (4), we obtain:

(20)$$< {\mathbf P} > = I({\mathbf r})\frac{1}{k}\nabla \phi ({\mathbf r})\,.$$

We now write r as (x, z). x = (x,y) is a 2D vector of transverse coordinates that is perpendicular to the longitudinal direction z. The operator ∇ can be decomposed as:

(21)$$\nabla = {\nabla _x} + \frac{\partial }{{\partial z}}{{\mathbf e}_z}\,,$$

where

(22)$${\nabla _x} = \frac{\partial }{{\partial x}}{{\mathbf e}_x} + \frac{\partial }{{\partial y}}{{\mathbf e}_y}\,.$$

e_x, e_yare the unit vectors along the x and y direction, respectively. Under the paraxial assumption, the gradient of the phase term can be represented as:

(23)$$\nabla \phi ({\mathbf r}) = {\nabla _{\boldsymbol x}}\phi ({\boldsymbol x},z = 0) + k{{\mathbf e}_z}\,,$$

where e_zis the unit vector along the longitudinal direction z. Substituting Eq. (20) and Eq. (23) into Eq. (2), we have:

(24)$$- k\frac{{\partial I({\boldsymbol x},z)}}{{\partial z}} = {\nabla _{\boldsymbol x}} \cdot [I({\boldsymbol x},z){\nabla _{\boldsymbol x}}\phi ({\boldsymbol x},z = 0)]\,.$$

Equation (24) is Eq. (5) in the main text. It is the well-known transport of intensity equation (TIE).

A.2 Derivation of Eq. (6) [43]

To do this, we introduce the derivative along the light path:

(25)$$\frac{d}{{ds}} = {\mathbf s} \cdot \nabla \,,$$

where s is the length of the arc along the ray. The propagation direction of the light s is defined in Eq. (4). We have |ds/ds|=1.

Substituting for < P > from Eq. (20) into Eq. (2), and using the identities ∇·uv = u(∇·v)+v·∇u and ∇·∇=∇², Eq. (2) yields:

(26)$$\frac{1}{k}I({\mathbf r}){\nabla ^2}\phi ({\mathbf r}) + \frac{1}{k}\nabla \phi ({\mathbf r}) \cdot \nabla I({\mathbf r}) = 0\,.$$

The above equation can also be written as:

(27)$$\frac{1}{k}{\nabla ^2}\phi ({\mathbf r}) + \frac{1}{k}\nabla \phi ({\mathbf r}) \cdot \nabla \ln I({\mathbf r}) = 0\,.$$

Using Eq. (4) and the directional derivative defined in Eq. (25), the above equation can be further expressed as:

(28)$$\frac{d}{{ds}}\ln I({\mathbf s}) ={-} \frac{1}{k}{\nabla ^2}\phi ({\mathbf r})\,.$$

The solution of Eq. (28) is:

(29)$$I({\mathbf s}) = I({x_0},{y_0},{z_0}){e^{ - \int_{{s_0}}^s {\frac{1}{k}} {\nabla ^2}\phi ({\mathbf r})ds}}\,.$$

Equation (29) provides us with an analytical expression for the propagation of the intensity of light. It is Eq. (6) in the main text.

A.3 Application of Eq. (6) to a plane wave

Equation (6) or (29) indicates that the propagation of the light intensity is modulated by the second derivative of the wavefront. For the trivial case of a plane wave, the phase term ϕ(r) is:

(30)$$\phi ({\mathbf r}) = {\mathbf k} \cdot {\mathbf r} = {k_x}x + {k_y}y + {k_z}z\,.$$

For any point (x₀, y₀, z₀) in the initial plane, the propagation direction of the light is a constant unit vector, s = (k_x/k, k_y/k, k_z/k). As a result, the integration path for Eq. (6) or (29) is a straight line along the direction of s, and the equation becomes:

(31)$$I({\mathbf s}) = I[{x(s),y(s),\,z(s)} ]= I({x_0},{y_0},{z_0})\,,$$

thus showing that as expected the intensity distribution remains unchanged by propagation.

A.4 Eq. (17), (18), (19) in different geometries

Figure 2 shows one case when the focus sits between the optical element and the detector. There are other cases. Figure 5 shows the corresponding sign changes for Eq. (17), (18) and (19) in different geometries. The radii of curvature in these cases are always assumed to be positive.

 

Fig. 5. Different forms of Eq. (17), (18) and (19) in different geometries. s₀ represents the position of the tested optic, s represents the detector. d always represents the distance between the optic and the detector. The radii of curvature in the diagram are always positive.

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Funding

Diamond Light Source (beamtime NR26501 , NT31201).

Acknowledgments

We thank Dr Oliver Fox, Dr Vishal Dhamgaye and Andrew Malandain for their support on the X-ray experiments. We also thank Dr Simon Alcock and Dr Ioana Nistea for providing the visible light measurement results.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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