2.1

Measurement of coherence length

The measurement of the transverse coherence of X-ray beams is extremely important for many experiments. The intrinsic coherence properties of radiation from a synchrotron source may also be modified due to imperfections of beamline optics. The complex coherence function (CCF) can be measured from the decaying visibility of fractional Talbot distances by using two gratings to generate a Moiré pattern[19]. As described in reference [19], the CCF can be expressed with a Gaussian of the form

The visibility of fringes is equal to the degree of coherence of the beam at the phase grating position. Here, the transverse coherence of the X-ray beam from the B16 beamline was measured using the grating interferometer. The grating pitches are 4μm and 2μm for the phase and absorption gratings respectively. The gratings were oriented with the grating line in the horizontal directions, and the vertical transverse coherence was measured in this case. The first grating was mounted on the stage at R=47.5m from the source, and the absorption grating was located a few millimeters in front (upstream) of the detector. An X-ray energy of 18keV was selected using a double multilayer monochromator (DMM)[20]. The distance between the two gratings was scanned, taking an image every two millimeters. As shown in Fig. 1a, the visibility of a Moiré fringe pattern will reach a maximum at the odd order fractional Talbot distances, and the visibility will drop to a minimum at the even order as shown in Fig 1c. By fitting the measured visibility as a function of the distance from the phase grating to the detector, the transverse coherence length can be derived. The measured coherence length is 20 μm at 18keV, slightly less than the theoretical value of 22 μm, as calculated usingthe DLS machine parameters. This result indicates that imperfection of the multilayer only moderately deteriorate the beamline’s coherence.

Figure 1.

(a-c) Moiré fringe pattern recorded at three distances of phase grating to detector: (a) odd-order fractional Talbot distance, (b) intermediate distance, (c) even-order fractional Talbot distance, (d) the experimental visibility (red curve) as a function of the distance from the phase grating to the detector, and the fitting curve (blue) using equation 1.

2.2

Characterization of Compound refractive lenses

A one dimensional (1D) compound refractive lens (CRL) was characterized using a grating interferometer[21]. The 1D concave parabolic lens was made of beryllium, with an aperture of 1 mm vertically and 3 mm horizontally. The radius of curvature R at the apex of the parabolas is 500μm. For a single lens, the focal length f can be calculated from the radius of curvature R of the lens at the apex of the parabolic lens by

Here δ_Be is equal to 1.52×10^-6 at the energy of 15keV, and the corresponding focal length f is 164.5 m. The 1D CRL and grating shearing interferometer were mounted on three motorized towers on a versatile optics test bench in the experimental hutch of the B16 beamline. The distance between the source and lens was 46.6m, and the grating G₁ placed 0.5m downstream from the lens. The energy was set to 15 keV by a silicon double-crystal monochromator (DCM). An X-ray 2D CCD detector with indirect illumination and an effective pixel size of 6.4μm was used. The lens was mounted such that it focused in the vertical plane. No other focusing optical elements were used upstream of the CRL. The gratings were oriented with the grating line in the horizontal directions, giving access to the vertical wavefront gradient. A single 1D CRL was tested at different tilt angles θ between the axis of the lens and the incident X-ray beam. Phase stepping mode was used with 96 images recorded during a piezo scan over two periods of the grating, with an exposure time of 400ms.

Fig.2 summarizes the amplitude image (top), dark-field image (middle) and phase gradient image (bottom) for single 1D CRL at 0 and 30 degrees. The field of view decreased as the CRL rotates from 0° to 30° and scattering is more visible at the edge of the CRL from the dark-field image. In addition, the effective radius of curvature of the lens at the apex was calculated from the phase gradient map to be 503μm and 432μm at 0° to 30°, respectively. The experimental data proves that the focal length changes as the CRL rotates according to equation 2. Therefore, it provides valuable input for the proof of the principle for maintaining the focal position by rotating the 1D CRL as the photon energy is varied.

Figure 2.

The amplitude image (top), dark-field image (middle) and phase gradient image for a single CRL at 0° (a, c, e) and 30° (b, d, f)

2.3

Optimization of a bimorph mirror

An X-ray grating interferometer was employed for in-situ optimization of an X-ray bimorph mirror. Moiré fringe analysis was used to calculate the wavefront slope error [9, 16]. Only a single interferogram image captured outside the focal plane is required, enabling the optical surface to be quickly measured and optimized. The 8 piezo electrodes enable the substrate to be bent to a large range of cylinders or ellipses. The mirror’s figure was pre-polished as a tangential ellipse whose foci are located at distances p = 41.5 m (source) and q = 0.4 m (image) from the bimorph mirror when no voltages are applied to its electrodes [22]. Phase grating G₁, with a pitch of d₁ = 3.99 μm, was mounted on a rotation stage. Absorption grating G₂, with a pitch of d₂ = 2.00 μm, was fabricated by depositing electroplated gold onto a silicon grating. Silicon substrates for both gratings were etched to a 50 μm thickness to minimize X-ray absorption. The grating G₁ was placed at a distance 720 mm downstream of the mirror’s focal plane, and the inter-grating distance, G₁ to G₂, was set to 33 mm.

To determine the optimum voltages which minimize slope errors, an initial interferogram was collected with all piezo voltages set to 0 V as shown in Fig.3(a). Fig. 3(b) shows an interferogram after the first three electrodes’ voltages had been incremented to 400 V. As highlighted in the region marked by a line, the fringes were distorted after applying the voltages on the piezo actuators. Wavefront slopes were calculated using the method described in reference [8]. As shown in Fig. 3(c), the response of the j^th electrode is calculated by subtracting the slope extracted from the j^th and (j+1)^th image. It can be seen that the slope change induced by each piezo actuator is below 0.01μrad/V. An interaction matrix, which defines the response of each electrode per unit voltage change (so called piezo response functions), can then be constructed and used for the mirror optimization [23]. The detailed optimization procedure and the results can be found in reference [16]. Since only single interferograms are required for each voltage set, the optical surface can be quickly optimized. In this case, the slope error of a bimorph mirror was reduced to < 200 nrad (rms) in only two iterations for a range of ellipses.

Figure 3.

Interferograms with (a) 0 V applied to all electrodes (b) 400V applied to first three electrodes only. (c) Piezo response functions of the bimorph mirror showing the slope change induced by applying a fixed voltage to each piezo actuator in sequence. Here line 1-0 represents the slope response of the 1^st electrode found by subtracting the 0^th image (no voltages applied) from the 1^st image (voltage only applied to 1^st electrode).

2.4

In-situ metrology of X-ray KB mirror

Apart from the X-ray wavefront characterization, the grating was also used as a deflectometer to measure the mirror’s slope error [24]. In this case, only one absorption grating was mounted on a piezo translation stage located upstream of the test mirror. A pair of X-ray KB mirrors was characterized using the grating deflectometer. For the vertical focusing mirror (VFM), the mirror’s figure was polished to a tangential ellipse with: p = 47.11m and q = 0.125m; and a grazing incident angle θ of 3mrad. The pitch of the grating is 10μm such that the fringes can be resolved by the CCD camera thanks to the magnification effect. The effective pixel size of camera was 6μm, and the camera was placed outside the focal plane. The distance between the mirror and detector is 3125mm. This large distance increases both angular sensitivity and spatial resolution through the magnification offered by the focusing mirrors.

The fringes for the VFM are shown in Fig. 4(a). Note that the fringe period is much larger for the top region than the bottom region. This indicates that the wavefront gradient varied gradually due to the change in the radius of curvature along the mirror’s elliptical surface. The grating was scanned, using the piezo actuator, in the direction perpendicular to the grating lines, and an image recorded for each position of the grating. A stack of 64 images was recorded by scanning the piezo over 20μm, which is equivalent to two periods of the grating pitch. The phase of the projected fringe was then recovered for each detector position. In order to retrieve the mirror’s slope error, an iterative procedure was employed and described in reference [24]. The recovered mirror slope error was then compared with that measured by the Diamond-NOM. Fig.4(b) shows that the ex situ measurement and the in situ measurements are in good agreement.

Figure 4.

(a) The X-ray fringe recorded out of the focal plane, (b) the calculated slope error from grating deflectometry is compared with from the ex-situ measurement using the Diamond-NOM

3.1

Wavefront characterization

As a simple illustration [10], the X-ray wavefront from the bending magnet source of B16, after passing through a Si(111) DCM, was measured using absolute mode. The membrane was placed at 47.5m from the source, and the X-ray energy was set at 14.5keV. The beam size of 2mm × 2mm was defined using slits. The first speckle image was collected with the detector 600 mm from the membrane, and a second image acquired by moving the detector 100mm further downstream. Fig. 5 (a) shows the reconstruction of the beam wavefront, which was then fitted with an ellipsoidal shape. The best-fit radii of curvature of the wavefront are 49.7m and 48.8m for the vertical and the horizontal direction, respectively. The difference between the two orthogonal directions might be due to the heat load imposed on the DCM. The corresponding wavefront error shown in Fig. 5(b) might have been introduced by imperfections of the monochromator crystals or the beryllium windows.

Figure 5.

(a) Reconstruction of the B16 wavefront after double reflection on Si (111) monochromator (b) wavefront error after removing the best-fit ellipsoidal wavefront

3.2

Measurement of compound refractive lenses

As a demonstration, a 2D CRL was characterized using the speckle differential mode [10]. The CRL was made of beryllium and had a parabolic shape with a design radius of curvature 200μm at the apex. The beam energy was set to 15keV using a Si(111) DCM of the beamline B16. As illustrated in Fig.6, two images were recorded: (a) one without the CRL in the X-ray beam and (b) the other with the lens. A sub-pixel accuracy, zero-normalized, cross-correlation algorithm was employed to calculate the displacement vector for each pixel from the two images. By dividing the displacement by the distance between the lens and the detector, both (d) the horizontal and (e) the vertical wavefront gradient can be calculated. The reconstructed wavefront, illustrated in Fig. 6 (f), shows dramatically enhanced contrast compared to the absorption image (c). The corresponding aberrations of the parabolic, rotationally symmetric lens can be retrieved from the wavefront profile. Such aberrations are difficult to measure using other techniques. Precise measurement of the refractive properties is essential for further improvement of the lens fabrication process, especially in the context of efforts to achieve ever smaller X-ray focal spot sizes.

Figure 6.

(a) Imaging with the membrane but without sample, (b) imaging with the sample and the membrane, (c) absorption image, (d) horizontal phase gradient, (e) vertical phase gradient, and (f) integrated phase from (d) and (e)

3.3

Characterization and optimization of an X-ray mirror

As described in reference [13], the grating technique is a special case of the general speckle scheme in which the membrane is scanned across the incoming beam. Absolute wavefront curvature, which is the second derivative of the wavefront, can be accurately retrieved in scanning mode. Hence, for grating deflectomety, the grating can be replaced with a membrane to measure the X-ray mirror’s slope error. Reference [15] demonstrates that the surface profile of the same pair of X-ray KB mirrors has been recovered based on near-field speckle with nanometer height precision.

Apart from the mirror profile characterization, the speckle based technique has also been used for optimizing a bimorph mirror. The differential mode was used to measure the piezo response functions, and scanning mode was then employed to measure the absolute wavefront error. The X-ray mirror can be optimized to focus or defocus beams by applying the derived voltage on each piezo actuator [22].