Comparison of monochromators with a bent parabolic mirror and a varied-spacing grating for the 2.0 GeV high-brilliance synchrotron radiation source (VSX)

A design study of monochromators for a 2.0 GeV electron/positron storage ring for high-brilliance synchrotron radiation in the vacuum ultraviolet (VUV) and the soft X-ray regions is described. Two types of VUV/soft X-ray grazing-incidence monochromators, one with a bent parabolic mirror and the other with a varied-spacing grating, are designed. Without any slope error, the expected resolving power of the former is much higher, but the latter is less affected by slope errors of the optical elements.

1. Introduction

In designing a grazing-incidence monochromator with high resolution and high purity it is important to correct ray aberrations. For this purpose, various ideas have been proposed, such as employing the SX-700 monochromator (Petersen, 1982), which uses an elliptical mirror, or using a monochromator with a varied-spacing grating (Koike & Namioka, 1995). While it is not easy to manufacture aspherical mirrors by polishing, an alternative approach is to use bent mirrors (Ishiguro et al., 1996). A simple wavelength-scanning mechanism is also important to achieve high performance.

Here, we report the design and performance of two types of monochromators: one using bent parabolic mirrors and equal-spacing plane gratings, and the other using spherical mirrors and varied-spacing plane gratings. Both monochromators have very simple mechanisms: fixed entrance and exit slits, and wavelength scanning only by grating rotation. Their design is described and a comparison of their performance is given.

2. Optics

2.1. Bent parabolic mirror monochromator

Fig. 1(a) is a schematic diagram of the monochromator, which consists of bent parabolic mirrors and equal-spacing plane gratings, where S₀ is an undulator source, and M₁ and M₂ are cylindrical or spherical mirrors. Rays with horizontal dispersion from S₀ are focused on the sample S₃ by M₁, and rays with vertical dispersion are focused on the entrance slit S₁ by M₂. The distances are S₀M₁ = 25 m, M₁M₂ = 1 m and M₂S₁ = 5.2 m. The deviation angles are ∠S₀M₁M₂/2 = ∠M₁M₂S₁/2 = 88°. The vertical magnification from S₀ to S₁ is 1/5.

Figure 1 (a) Schematic diagram of the bent parabolic mirror monochromator. (b) Schematic diagram of the varied-spacing grating monochromator.

M₃ is a bent parabolic mirror, collimating the rays with vertical dispersion from S₁ without aberration as S₁ is placed at the focal point of M₃. The deviation angle is ∠S₁M₃G/2 = 88°, and the distance S₁M₃ = 7 m. The equal-spacing plane gratings G₁–G₄ are interchangeable to cover the wavelength region. As they work with constant deviation-angle configuration, wavelength scanning is carried out only by grating rotation. The grating parameters are shown in Table 1. The bent parabolic mirrors M₄₁–M₄₃, which converge parallel rays on their foci without aberration, are also interchangeable to cover the wavelength region (see Table 1). The foci of these parabolic mirrors are fixed on the exit slit S₂, and consequently aberration-free monochromatic line images of S₁ are formed on S₂. The deviation angles are ∠GM₄₁S₂/2 = 88°, ∠GM₄₂S₂/2 = 86° and ∠GM₄₃S₂/2 = 81°, and the distances are M₄₁S₂ = 1.484 m, M₄₂S₂ = 1.844 m, M₄₃S₂ = 2.046 m, OS₂ = 2.2 m and OG = 50 mm. A spherical mirror, M₅, converges the rays with vertical dispersion from S₂ on the sample S₃. Thus, a monochromatic small beam is formed on the sample S₃.

Table 1
Grating parameters and wavelength coverage

The number of grooves is the value of the central position for the case of the varied-spacing gratings. K_g is a grating deviation angle: ∠M₃GM₄/2 for the bent parabolic mirror monochromator, or ∠M₃GS₂/2 for the varied-spacing grating monochromator.

		No. of			Corres-
	Length	grooves	Kg	Wavelength	ponding
Element	(mm)	per mm	(°)	region λ (Å)	mirror
G1	±70	1200	87	12–30	M41 M31
G2	±70	1200	85	25–100	M42 M32
G3	±70	400	85	90–280	M42 M32
G4	±70	400	80	250–1000	M43 M33

2.2. Varied-spacing grating monochromator

Fig. 1(b) is a schematic diagram of the monochromator, which consists of spherical mirrors and varied-spacing plane gratings. The elements from S₀ to S₁ are the same as those in Fig. 1(a). The spherical mirrors M₃₁–M₃₃, which convert vertically diverging rays from S₁ into vertically converging rays, are interchangeable to cover the wavelength region (see Table 1). The deviation angles are ∠S₁M₃₁G/2 = 88°, ∠S₁M₃₂G/2 = 86° and ­∠S₁M₃₃G/2 = 81°, and the distances are S₁M₃₁ = 7 m, S₁M₃₂ = 7.359 m, S₁M₃₃ = 7.561 m, S₁O = 7.715 m, GS₂ = 2.2 m and OG = 50 mm. The varied-spacing plane gratings G₁–G₄ are interchangeable depending on the wavelength region (see Table 1). As they work with constant deviation-angle configuration, wavelength scanning is carried out only by grating rotation. Their spacing distributions were optimized with Code-V optical design software (Optical Research Associates) so that the vertical aberration between S₁ and S₂ is minimized. Consequently, monochromatic line images of S₁ are formed on the fixed-exit slit S₂. M₄ and S₃ are the same as M₅ and S₃, respectively, of Fig. 1(a).

3. Evaluation of the designed monochromators

The main purpose of this evaluation is to determine the resolving power of the designed monochromators with or without the slope errors of the optical elements. The resolving power was evaluated by means of ray tracing as described below.

3.1. Undulator source and the source for ray tracing

Table 2 gives the undulator source parameters of the 2.0 GeV high-brilliance synchrotron radiation that our calculation is based on. The vertical magnification from the source to the entrance slit is 1/5, so σ_y of Table 2 is reduced into σ_y/5 and σ_y′ is magnified to 5σ_y′ at the entrance slit S₁. As the angular spread by diffraction of the entrance slit is σ_diff = λ/(4πσ_slit), where σ_slit is the half width of the entrance slit, the angular spread after the entrance slit becomes Σ_y′ = [(5σ_y′)²+ σ_diff²]^1/2.

Table 2
Undulator source parameters

σ_y is the r.m.s. source size and σ_y′ is the r.m.s. angular spread. Both values are for the vertical plane.

λ (Å)	σy (µm)	σy′ (µrad)	Grating
12	74.25	16.90	G1
21	74.44	21.62	G1
30	74.63	25.47	G1
25	74.53	23.41	G2
60	75.26	35.40	G2
100	76.09	45.39	G2
90	75.90	42.94	G3
180	77.76	60.36	G3
280	79.77	75.13	G3
250	79.17	71.02	G4
600	85.88	109.74	G4
1000	92.96	141.58	G4

To achieve high resolving power we chose a 5 µm entrance-slit width (σ_slit = 2.5 µm), which is about 1/3 of σ_y/5. The rays intercepted by the entrance slit do not contribute to the performance of the monochromators. Consequently, for ray tracing we assume an equivalent source on S₁ with constant illumination intensity (whose vertical width is 5 µm) and with a Gaussian angular distribution (whose vertical half width is Σ_y′). Ray tracing was carried out between S₁ and S₂, where the resolving power is determined, with several thousand rays that are generated randomly from S₁ with the probability distribution of the above-mentioned equivalent source (namely, a flat intensity and a Gaussian angular distribution). We also assume that the horizontal distribution of the source is uniform, because there is no horizontal optical power between S₁ and S₂.

3.2. Ray tracing with slope errors

Here, a slope error means a local angular deviation of the surface normal from the designed surface of the optical element. As mentioned above, the resolving power of the monochromators designed here is determined by the optical elements placed between S₁ and S₂. So, for ray tracing, slope errors are added only to the following elements: M₃, G₁–G₄ and M₄₁–M₄₃ for the bent parabolic mirror monochromator, and M₃₁–M₃₃ and G₁–G₄ for the varied-spacing grating monochromator. We assume that the slope error distribution is Gaussian: f(δ) ∝ exp(−δ²/2σ_s²), where δ is a slope error. A ray, which is generated as described in §3.1, strikes some position of some element, and is then deflected by a slope error which is generated randomly with this probability distribution. Ray tracing with slope errors was carried out with σ_s = 0.5 µrad (0.1′′), a value that we consider to be possible (Miura et al., 1998). The slope errors were added to all elements at the same time.

4. Results

4.1. Line profiles

Fig. 2 shows examples of line profiles that were obtained from the vertical (dispersion direction) cross sections of the spot diagrams on the exit slit S₂ by the above-mentioned ray tracing, line profiles of the bent parabolic mirror monochromator without (a) and with (b) slope error, and of the varied-spacing grating monochromator without (c) and with (d) slope error. These profiles are at the same wavelength (λ = 180 Å) as the grating G₃ (see Table 1). The width of profile (a) is much smaller than that of (c). The long-tailed profile of (c) results from the remaining coma aberration of the varied-spacing grating monochromator. (The coma aberration is corrected at λ = 280 Å with grating G₃.) With σ_s = 0.5 µrad slope error, the profiles of both monochromators are spread out to almost the same width (Figs. 2b and 2d), but the long tails still remain in the profiles of the varied-spacing grating monochromator (Fig. 2d).

Figure 2 Line profiles of the bent parabolic mirror monochromator at λ = 180 Å with G3, (a) without slope error and (b) with slope error σs = 0.5 µrad (0.1′′). Line profiles of the varied-spacing grating monochromator at λ = 180 Å with G3, (c) without slope error and (d) with slope error σs = 0.5 µrad (0.1′′).

4.2. Resolving power

Some definitions of wavelength resolution Δλ have already been given in Koike & Namioka (1995); here we simply define Δλ as full width at the 1/e-maximum point of the line profile. Namely, Δλ = W_ePsinβ/F, where W_e is the full width at the 1/e-maximum point of the line profile, P, the pitch of the grating grooves, β, the diffraction angle from the grating surface, and F, the distance M₄S₂ for the bent parabolic mirror monochromator or the distance GS₂ for the varied-spacing grating monochromator. In both cases, the exit-slit width is regarded as zero.

Fig. 3(a) shows the resolving power λ/Δλ of the bent parabolic mirror monochromator and Fig. 3(b) shows that of the varied-spacing grating; each depends on the wavelength region. Without the slope error, the resolving power of the former is much higher than that of the latter. However, with σ_s = 0.5 µrad slope error the resolving power of both monochromators is reduced to almost the same value.

Figure 3 Resolving power λ/Δλ, with and without slope error, (a) of the bent parabolic mirror monochromator and (b) of the varied-spacing grating monochromator.

Fig. 4 shows the reduction rate of the resolving power, which is defined by [(λ/Δλ) with slope errors]/[(λ/Δλ) without slope errors]. This figure indicates that the degree of resolving power reduction of the varied-spacing grating monochromator is smaller than that of the bent parabolic mirror monochromator.

Figure 4 Reduction rate of the resolving power. This value is defined by [(λ/Δλ) with slope errors]/[(λ/Δλ) without slope errors].

5. Discussion and conclusions

In Figs. 1(a) and 1(b), the incidence angle of G is α and the diffraction angle of G is β (both angles are from the grating surface), with the grating equation where P is a pitch of the grating grooves. When an incidence ray deviates by dα, which is generated by a slope error dθ of M₃, the diffraction angle deviates dβ, which is obtained by differentiating the grating equation In the case of α < β (this design), the ray deviation by slope errors of M₃ is reduced through the grating. On the other hand, the ray deviation by slope errors of M₄ is 2dθ. The slope error effect of a mirror placed before the grating is smaller than that of a mirror placed after the grating. Consequently, in this design, the varied-spacing grating monochromator is less affected by slope errors of the optical elements, as shown in Fig. 4.

The bent parabolic mirror monochromator, which employs aberration-free optics, has higher resolving power, but is more affected by slope errors. As a result, the slope error reduces the resolving power of both monochromators to almost the same value. However, the wavelength purity of the varied-spacing grating monochromator should be worse because of the long tails of the line profiles.

The bent parabolic mirror monochromator has the potential to achieve much higher resolving power as the manufacturing precision of the optical elements is improved in the future.