3.1

Bimorph mirror

The bimorph mirror is an uncoated 150 mm long, deformable piezo bimorph mirror with a silica substrate and eight electrodes. The central 115 mm of the substrate was “super-polished” with elastic emission machining¹² by JTEC (Japan) to an elliptical cylinder of source-mirror distance 41.5 m, mirror-image distance 0.4 m, and grazing incidence angle 3 mrad. Thales-SESO (France) attached the piezoelectric elements to the substrate. Over more than five years of operation, the mirror’s surface profile and voltage responses have remained stable¹³.

3.2

Mirror figure measurements

As even figure errors of a few nanometers disturb the profile of the beam reflected by the mirror, metrological techniques of correspondingly high accuracy are required to detect them. Ex situ metrology was performed on the bimorph mirror using the Diamond-NOM, a non-contact slope-measuring profiler that measures angular surface deviations with an accuracy better than 50 nrad¹⁴. This device is inside the Diamond Light Source Optics and Metrology laboratory, a Class 10,000 (ISO7) cleanroom. For in situ metrology, performed while exposing the mirror to X-rays, X-ray speckle tracking was applied¹⁵. A sandpaper diffuser was placed upstream from the mirror in order to introduce a speckle pattern into the reflected beam. As the diffuser was scanned transversely to the incident beam, cross-correlations between the speckle patterns at different diffuser positions were used to determine the mirror’s slope error (first derivative of its figure error).

3.3

X-ray beam profile measurements

The mirror was examined with X-rays at the Diamond Light Source Versatile Test Beamline B16¹⁶ using the setup shown in Fig. 1. X-rays with a photon energy of 9200 eV were selected by a Si (1 1 1) double crystal monochromator (DCM). Slits downstream of the DCM selected a 0.03 mm (horizontal) × 0.3 mm (vertical) section of the X-ray beam produced by the bending magnet source. The small vertical slit size justifies the assumption of uniform illumination of the mirror. The EEM bimorph mirror was oriented for upward deflection inside a closed chamber that was continuously flushed with N2 gas at a rate of 1 L/min to prevent the generation of reactive hydrocarbons which can contaminate the optical surface. Kapton windows allowed the X-rays to enter and exit the sealed chamber.

Figure 1:

Schematic of the experimental setup used to measure the reflected X-ray beam profiles in the vicinity of the focus of the bimorph mirror at the Versatile Test Beamline B16 of Diamond Light Source. The figure is not to scale. From Sutter, Alcock, Kashyap et al¹⁰.

The transverse profile of the reflected X-ray beam in the vicinity of the mirror’s focus were measured by knife-edge scans using a horizontal gold wire 200 μm in diameter. The narrow horizontal width of the slits ensures that any striations are not blurred by sagittal slope errors on the mirror or by accidental inclination of the gold wire. The X-ray intensity passing by the gold wire was measured by a PIPS detector with a current amplifier.

The knife-edge scans taken while the re-entrant modifications were applied to the mirror were compared with a scan taken when the mirror was simply defocused. The latter scan was taken using X-rays of 8000 eV, but with all other experimental conditions the same except for possible slight variations in temperature and mounting forces. Despite the difference in photon energy between the older and the newer scans, the main features in the beam profiles can be explained to a good approximation by geometrical theory, in which the photon energy is not important¹⁰.

4.1

Selection of re-entrant surface modifications for bimorph mirror

The re-entrant surface modifications applied to the bimorph mirror were chosen by inspection of its piezo response functions (PRF) and of its figure when all voltages were set to 0 V. Each electrode of the bimorph mirror has a PRF, which shows how a specific voltage increment applied to that electrode changes the mirror’s overall figure. The PRFs may be expressed in terms of height or slope, the latter being the derivative of the former. They were determined by using the Diamond-NOM according to the procedure described by Hignette et al¹⁷ and by Sutter, Alcock & Sawhney¹⁸, beginning with all voltages set to 0 V and using a +200 V increment. Experience has shown that the mirror figure’s response to voltage changes on the electrodes is highly linear; therefore, voltage optimization was performed by the linear algebra method of least squares minimization. The PRFs are shown in Fig. 2 in terms of slope because the region of the mirror influenced by each electrode is then obvious. The mirror’s slope error when all voltages are at 0 V is also displayed there. This is equal to the slope measured by the Diamond-NOM minus the ideal focusing ellipse.

Figure 2:

Piezo response functions displayed as slope changes scaled linearly to a voltage change of 500 V for comparison with the thick black curve that shows the mirror’s slope error when all electrodes are at 0 V. From Sutter, Alcock, Kashyap et al¹⁰.

By placing the breakpoints on the electrode junctions, one can approach as closely as possible the ideal discontinuous change in the sign of the curvature there, thus limiting intensity spikes in the beam at the focus. Experimentally, the positions of the electrode junctions were defined as the midpoints between the centers of adjacent PRFs in Fig. 2. The first breakpoints are placed at those junctions where the derivative of the slope error at 0 V most abruptly switches sign, in this case between electrodes 2 and 3 and between electrodes 6 and 7. This by itself produces a three-segment modification, which is displayed in Fig. 3(a) along with one five-segment modification generated by adding two more breakpoints and one seven-segment modification generated by adding four more breakpoints. Figure 3(b) shows a simple defocusing of the mirror. All modifications are designed to expand the beam size at the mirror’s focus to 4 μm.

Figure 3:

(a) Bottom to top: mirror slope error at 0 V, then theoretical three-, five-, and seven-segment surface modifications approximating the slope error at 0 V. Each plot has been shifted upward by 5 μrad from the one before for visibility. (b) Theoretical simple concave surface modification with expected best approximation achieved by mirror as calculated from its slope error at 0 V and its measured PRFs. Intended beam size at focus is 4 μm for all cases. From Sutter, Alcock, Kashyap et al¹⁰.

4.2

Verification of surface modifications with the Diamond-NOM

Figure 4 shows the three-segment modification. The list of voltages in Fig. 4(a) confirms that all voltage differences between adjacent electrodes remain within the 500 V limit. Figure 4(b) shows the slope error, the mismatch between the ideal re-entrant modification and the measured one. Figure 5 shows the same information for the five-segment modification with some small voltage changes made to keep the voltage difference within the 500 V limit. Figure 6 shows the seven-segment modification, with somewhat larger voltage changes made to keep the voltage difference within the design limit. Agreement of the final measured surface modifications is good throughout: 0.19 μrad for three segments, 0.29 μrad for five segments, and 0.44 μrad for seven segments. As expected, the curvature around the breakpoints is not in fact truly discontinuous; some rounding of the slope is always observed. A flattening of the measured slope modification is visible in the seven-segment modification at the junction of electrodes 4 and 5, and to a lesser extent at the center of the mirror when the three-segment modification is applied. This “imprint” effect, caused by the electrodes’ finite width, was observed by Alcock, Sutter et al¹⁹ and discussed more fully in Sutter, Alcock, Rust et al⁶.

Figure 4:

(a) Theoretical, predicted, and measured three-segment surface modification. (b) Slope error (deviation of measured surface modification from theoretical one). From Sutter, Alcock, Kashyap et al¹⁰.

Figure 5:

(a) Theoretical and measured five-segment surface modification. (b) Slope error. Arrows in the voltage list in (a) show where ideal voltages had to be compromised to keep voltage differentials below the 500 V limit. From Sutter, Alcock, Kashyap et al¹⁰.

Figure 6:

(a) Theoretical, predicted, and measured seven-segment modification. (b) Slope error. Arrows in the voltage list in (a) show where the voltages had to be shifted away from their theoretically optimal values to keep the voltage differentials below the 500 V limit. From Sutter, Alcock, Kashyap et al¹⁰.

4.3

X-ray measurements of surface modifications and beam profiles at focus

Figures 7(a) and 7(b) compare, respectively, the theoretical five-segment and seven-segment surface modifications with measurements on the Diamond-NOM and with the results of X-ray speckle tracking at B16. They are not exactly in agreement and indeed were not expected to be, partly because X-ray speckle tracking is still a technique under development, and partly because the mounting forces and temperature variations to which the mirror was subjected were less well controlled at B16 than inside the cleanroom. Nonetheless, both the Diamond-NOM and the X-ray speckle tracking agree exactly in the number of segments and approximately in the positions of the breakpoints. The main disagreements among the three plots in each graph are observed close to the breakpoints. The measured curvature there is significantly rounded while the theoretical curvature changes sign abruptly, and the level of the slope at the breakpoints varies significantly. The imprint effect observed in the central segment of the seven-segment modification on the Diamond-NOM is also visible, though less pronounced, in the X-ray speckle tracking data.

Figure 7:

(a) 5-segment and (b) 7-segment surface modifications: comparisons of theory with measurements by Diamond- NOM and by X-ray speckle tracking at B16. The theoretical modifications have sharp changes in curvature at their breakpoints. The data from B16 are represented by the smooth curve with the rounded extrema. From Sutter, Alcock, Kashyap et al¹⁰.

Figure 8 compares typical beam profiles obtained by knife-edge scans at the mirror’s focus when the seven-segment modification, the five-segment modification, and a simple concave modification were applied to the mirror. The seven- segment modification produces a series of spikes that are dispersed throughout the beam profile by the quite different levels of the slope at different breakpoints. However, these spikes appear at evenly spaced intervals, and the two central spikes are of similar strength and lie on a similar background level. The five-segment modification, by contrast, is dominated by the intense spikes at both edges. These hot edges are typical of slope modifications of continuous curvature such as sinusoids, because near the extrema of the slope, the position of the reflected ray on the detector varies minimally with the position of the incident ray. However, the center of the beam has considerably less striation. The profile produced by a simple concave modification has a strong side shoulder and shows a main peak that is breaking up into two equal parts even at this small (5 μm) level of beam expansion. These arise from regions of zero curvature at random positions on the mirror. The positions of these regions depend on the polishing errors and would therefore be different for every mirror.

Figure 8:

Beam profiles at mirror focus. (a) Seven-segment for 4 μm beam (b) Five-segment for 4 μm beam (c) Simple concave for 5 μm beam: Maximum experimental intensity normalized to unity for easy comparison of plots. From Sutter, Alcock, Kashyap et al¹⁰.

5.1

Examination of mirrors with different piezo response functions

Each bimorph mirror used at Diamond Light Source has piezo response functions of different characteristic shape. Their differences are most obvious when one examines them in terms of their slope. As stated before, the mirror used in the experiments described above has the piezo response functions of Fig. 2. The strong overshoot and undershoot on the edges of each PRF’s central region distinguish this mirror from two others that will be examined here. Also significant is the lack of overlap between the central regions of the PRFs of adjacent electrodes. Note that each PRF reaches its minimum slope at approximately the position where the next PRF reaches its maximum slope. In the following section, the bimorph mirrors were all assembled by Thales-SESO (France).

One of the older bimorph mirrors at Diamond Light Source, the vertical focusing mirror at beamline I02, is 50 mm wide and has 8 electrodes over a length of 600 mm. Figure 9(a) shows two typical PRFs of this mirror as measured with X- rays on the beamline. Note the absence of overshoot or undershoot and also the clear separation between them. Figure 9(b) shows the calculated result of adding the first PRF and subtracting the second PRF, thus simulating a real re-entrant surface modification. The breakpoint for the mirror at I02 has a sharper change in curvature than is seen at the breakpoints in Figs. 5(a) and 6(a), thus indicating that the overshoot and undershoot around the PRFs of this article’s test mirror are causing some of the observed rounding.

Figure 9:

(a) Two typical adjacent PRFs E5 and E6 of the vertical focusing mirror of the Diamond Light Source beamline I02 (electrodes 5 and 6 of 8) given per volt. Measured using X-rays at the beamline. (b) 200 x (E6-E5), simulating the application of 200 V on electrode 5 and -200 V on electrode 6 to produce a re-entrant surface modification.

At the other extreme is one of the newest bimorph mirrors to be installed at Diamond Light Source, the vertical microfocusing mirror of beamline I24. This mirror is 34 mm wide and has 16 electrodes along its 240 mm length. The PRFs of this mirror were measured both on the Diamond-NOM and with X-rays on the beamline. Because the measurement noise was considerable, an idealized model was developed in which each PRF was replaced by an error function of approximately the same width, strength, and position (Fig. 10(a)). The strong overlap between adjacent PRFs has been preserved. Only 14 electrodes are modeled here because the PRFs of the last two electrodes of the real mirror were cut off at the end. Nevertheless, this is sufficient to show that the strong overlap of the PRFs makes the achievement of a re-entrant surface modification impossible, at least by the expected method of adding the PRFs together with alternating sign (Fig. 10(b)). Figure 10(c) shows that the central region of this bimorph has only a small oscillating slope variation probably due to the imprint effect.

Figure 10:

(a) Simplified theoretical model of PRFs of the vertical microfocusing mirror installed on the Diamond Light Source beamline I24. Based on measurements with the Diamond-NOM but showing only the first 14 of the 16 electrodes. Voltage increment is 200 V. (b) Addition of PRFs in (a) with alternating signs. (c) Enlargement of vertical scale in (b) to show the imprint effect.

The reasons for the great variation of the PRFs’ shape and overlap among these three mirrors are not fully understood. However, it should be mentioned that the test mirror of this article and the mirror used at I02 both belong to the first generation of bimorphs, in which the piezoelectric elements are sandwiched between an upper and a lower substrate². On the other hand, the mirror installed at I24 is one of the first examples of the second generation of bimorphs, in which the piezoelectric elements are attached to the sides of the substrate²⁰. The dependence of the PRFs on the construction of a bimorph will be an important topic for future studies.

5.2

Re-entrant modifications of continuous curvature

Because in practice a deformable bimorph mirror cannot have a discontinuity in its curvature at any point along its length, it would be helpful to know how the features of surface modifications of continuous curvature might be chosen to minimize the striation in the reflected beam at the focus. One hint comes from the intense hot edges produced by the five-segment modification, which are shown in Fig. 8(b). The X-ray speckle tracking data in Fig. 7(a) show that the slope maxima are all at nearly the same level, as are the slope minima. Therefore, one could consider breaking up the hot edges by purposefully giving randomly varying values to the slopes at the breakpoints. To keep the beam’s center clear, one could limit the random values to an upper band and a lower band, as shown in Fig. 11. This is in fact what occurred accidentally in the measured seven-segment modifications of Fig. 7(b).

Figure 11:

Schematic of re-entrant surface modification of continuous curvature set up to break up hot edges and ensure even spread of illumination across center of beam at focus. Shown with segments of equal length for simplicity but can be generalized to segments of unequal length.

A second unacceptable feature of beam profiles produced by sinusoidal modifications is that the hot edges have tails extending a long distance toward the center of the beam. These arise from the gradual reduction in the curvature of the sinusoidal modification as the slope extrema are approached. To avoid this problem, the slope of the modification’s segments should follow the form given in Eq. (4) as closely as possible over as great a range as possible. This ensures an even spread of illumination across the reflected beam profile at the focus. For short mirrors such as that of this paper, for which Eq. (4) can be well approximated by a linear function, this condition can be met reasonably well by the PRFs of Fig. 2.