Hybrid height and slope figuring method for grazing-incidence reflective optics

Abstract

Grazing-incidence reflective optics are commonly used in synchrotron radiation and free-electron laser facilities to transport and focus the emitted X-ray beams. To preserve the imaging capability at the diffraction limit, the fabrication of these optics requires precise control of both the residual height and slope errors. However, all the surface figuring methods are height based, lacking the explicit control of surface slopes. Although our preliminary work demonstrated a one-dimensional (1D) slope-based figuring model, its 2D extension is not straightforward. In this study, a novel 2D slope-based figuring method is proposed, which employs an alternating objective optimization on the slopes in the x- and y-directions directly. An analytical simulation revealed that the slope-based method achieved smaller residual slope errors than the height-based method, while the height-based method achieved smaller residual height errors than the slope-based method. Therefore, a hybrid height and slope figuring method was proposed to further enable explicit control of both the height and slopes according to the final mirror specifications. An experiment to finish an elliptical-cylindrical mirror using the hybrid method with ion beam figuring was then performed. Both the residual height and slope errors converged below the specified threshold values, which verified the feasibility and effectiveness of the proposed ideas.

Keywords: ion beam figuring; surface slopes; synchrotron optics; two-dimensional.

Figure 1

The height errors in (a) and (c) are generated using equation (1) with (f _x , f _y ) = (150, 0) mm⁻¹ and (f _x , f _y ) = (10, 0) mm⁻¹, respectively. Although they have the same height error of 50 nm RMS, the corresponding slope errors shown in (b) and (d) are very different from each other.

Figure 2

Height (a), slope in x (b) and slope in y (c) of the analytical Gaussian TIF with FWHM = 5 mm used in the simulation.

Figure 3

Dwell time, residual height errors and residual slope errors estimated from the height-based method (a–c) and slope-based method (d–f), respectively, for the sinusoidal surface shown in Figs. 1 ▸(a) and 1 ▸(b) with f _x = 150 mm⁻¹.

Figure 4

Dwell time, residual height errors and residual slope errors estimated from the height-based method (a–c) and slope-based method (d–f), respectively, for the sinusoidal surface shown in Figs. 1 ▸(c) and 1 ▸(d) with f _x = 10 mm⁻¹.

Figure 5

Height (a), slope in x (b) and slope in y (c) of an analytical surface generated with a 260-order Chebyshev polynomial.

Figure 6

Dwell time optimized from the height-based method (a), slope-based method (b) and hybrid method (c).

Figure 7

Residual height and slope errors estimated from the height-based method (a–c), slope-based method (d–f) and hybrid method (g–i).

Figure 8

Integrated PSD distributions in the x-direction calculated from the residual height errors estimated with the height-based, slope-based and hybrid methods.

Figure 9

Target elliptical-cylindrical mirror with the object distance p = 14254.7 mm, image distance q = 2448.8 mm and grazing angle θ = 1.25°.

Figure 10

Initial spherical mirror with ROC = 199 m.

Figure 11

Desired removals of height (a), slope in x (b) and slope in y (c) from the sphere to the elliptical cylinder, where the slope maps are generated from the height map with the 2.5 mm sliding window.

Figure 12

Height (a), slope in x (b) and slope in y (c) of the IBF TIF generated with a 5 mm diaphragm.

Figure 13

The initial figuring was guided by the estimation obtained from the height-based method (a). The figuring result after 50 IBF runs was measured with the SI system (b).

Figure 14

The finishing was guided by the estimation obtained from the hybrid method (a). The finishing result after two IBF runs was measured with the SI system (b).

Figure 15

Cross-validation between the SI and NSP measurements of the residual slope errors along the x-direction.