In order to accurately evaluate the influence of the surface errors on the scattering, it is necessary to measure the surface errors in all relevant spatial frequencies. The power spectral density (PSD) of surface errors as a function of the spatial frequencies is accepted as an all-inclusive way to characterize optical surfaces, which can quantitatively describe the distribution of super-polished surface topography in spatial frequency ranges, and provides abundant data for the analysis of surface errors. The PSD [17] curve over the entire spatial frequency ranges can be obtained by Fourier transform of the surface profile errors as follows: P S D ( f ) = 1 L | ∫ 0 L z ( x ) exp ( − j 2 π f x ) d x | 2 . (1)

Here, L is the sampling length that can represent its geometric characteristics. It is the diameter of the mirror if sampling using an interferometer, and it is the field-of-view corresponding to different objective magnifications if sampling using a white-light optical profiler. z (x) is the surface profile error function, and “x” is the 1D coordinates of the sample along the surface. f is the spatial frequency. Equation (1) can be used to evaluate whether the surface errors of M1 meet the stray-light requirement for SCI. It is of great guidance to the surface polishing improvement.

According to the Harvey–Shack scattering theory, surface errors in different spatial frequency ranges have different effects on imaging quality [18–20]. Low-spatial frequency surface errors produce conventional aberrations, which affect the convergence of the Sun at the position of M2 and result in the stray-light of the inner FOV for SCI; mid-spatial frequency surface errors produce small-angle scattering, which affects the stray-light level of the inner FOV for SCI; and high-spatial frequency surface errors produce large-angle scattering, which affects the stray-light level of the outer FOV for SCI. Therefore, it is necessary to specify and measure the surface errors of M1 over the entire spatial frequency ranges to ensure that M1 meets the requirement, thereby suppressing the stray-light over the full FOV for SCI.

To specify the contribution of surface errors of M1 to the stray-light level, the Harvey–Shack ABg bidirectional scattering distribution function (BSDF) model is applied to characterize the surface errors of M1. This model is suitable for the scattering mainly caused by roughness, and the RMS roughness is much less than the wavelength. The BSDF function of an optical surface with an RMS roughness of “σ” [21, 22] is expressed in Eq. (2): B S D F = A B + ( sin ⁡ θ ) g . (2)

Here, A = π 2 ⋅ Δ n 2 σ 2 λ 2 , B = ( λ 2 π l ) 2 , and g is the logarithmic slope of the tail of BSDF. It is a dimensionless constant and is usually taken as 2, △n is the change in the refractive index (△n = 2 for mirrors), and λ is the wavelength. l is the auto-correlation length related to the scattering properties. When the RMS roughness of M1 is 0.3 nm and the wavelength is 700.0 nm, then A = 1.15 × 10⁻⁶ andB = 1.24 × 10⁻⁶; the BSDF resulting from surface errors of M1 is shown in Figure 5A. A simulated solar disk source with a divergence angle of 32′ is established in non-sequential ray-tracing software to illuminate the SCI, and the stray-light distribution at the focal plane is simulated as shown in Figure 5B. Under this condition, the irradiance of the stray-light and the solar disk source at the focal plane can be obtained, and the ratio of the two represents the stray-light suppression level. Simulation results show that the stray-light suppression level can be as low as 10⁻⁷ B_⊙ at 1.1 R_⊙ and 10⁻⁸ B_⊙ at 2.5 R_⊙.

When the wavelength is 121.6 nm, then A = 3.82 × 10⁻⁵ andB = 3.75 × 10⁻⁸; the BSDF resulting from the surface errors of M1 is shown in Figure 6A, and the stray-light distribution at the focal plane is shown in Figure 6B. The stray-light suppression level can be as low as 10⁻⁵ B_⊙ at 1.1 R_⊙ and 10⁻⁶ B_⊙ at 2.5 R_⊙.

After specifying surface errors of M1, different metrology instruments (laser interferometer for low-spatial frequency and white-light optical profiler for mid-spatial and high-spatial frequencies) are required to measure them. The spatial frequency range of different metrology instruments for measuring surface errors is determined by the sampling length L and the resolution of CMOS [23]. The minimum spatial frequency that can be measured is 1/L, where L is the FOV corresponding to different objective magnifications for the white-light optical profiler or the measurable aperture diameter for the laser interferometer. The maximum spatial frequency that can be measured is 1/(2d), which is the Nyquist cut-off frequency, where d is the sampling interval corresponding to the pixel size of CMOS [24, 25].

Since surface scatter phenomenon is a diffraction process, according to the grating equation (26), components with a spatial frequency greater than 1/λ will produce a small amount of optical scattering for SCI, which can be ignored. Therefore, the upper limit of the spatial frequency of the PSD curve is 1/λ, and the lower limit of the frequency is 1/D, where D is the aperture diameter of M1. According to the metrology data for different measuring instruments [27] in their specified spatial frequencies, the PSD [28] curve over the entire spatial frequencies is shown in Figure 7, which is obtained by Eq. 1. While measuring the surface errors, random errors such as mechanical vibration and electronic noise are removed by multiple phase averaging, and the system error is removed by subtracting the surface errors of the standard sample. The metrology data after removing random errors and system errors are under the PSD limit line of the polishing requirement; otherwise, another polishing and testing cycle is necessary.

Scratches and digs on the optical surface are cosmetic defects, which will generate unnecessary light scattering and contribute to stray-light for SCI. In order to quantify the stray-light level from cosmetic defects, a dark-field microscope is used to measure the number and size of scratches and digs on M1. The sub-field testing results are shown in Figure 8, in which mark 1 is a scratch, marks 2–5 are digs, and 6–9 are dust.

According to the Gary L. Peterson’s cosmetic defects scattering theory, stray-light due to scratches or digs includes geometric refraction inside the scratches or digs and diffraction of light that passes around scratches or digs can be quantitatively analyzed by BSDF.

BSDF from scratches [29] is given by the expression: B S D F s = N s w l π [ 1 + π w l λ 2 ( 1 + sin 2 ⁡ θ l s 2 ) − 3 2 ] . (3)

Here, N s = 1 π l D , l s = ( 1 2 π 2 ) 2 3 λ ( w 2 l ) 1 3 , N _s is the density of the scratches (number of scratches per unit area), D is the diameter of the optical element, w and l are their typical width and length from microscopy, respectively, θ is the scattering angle with respect to the surface normal of the optical component, and l _s is the roll-off angle of the BSDF curve. We note that the factor l in the numerator of Eq. (3) will cancel with the factor of l in N _s , so the width has a dominant effect on the BSDF from scratches. According to the statistical testing results of scratches for M1 over the full FOV, the typical width w is less than 0.002 mm.

The Harvey–Shack scattering model based on the scalar diffraction theory can predict the scattering distribution of a super-polished optical surface, which is given by the expression: B S D F = b 0 ( 1 + ( sin ⁡ θ L ) 2 ) s 2 . (4)

Here, b ₀ is the intercept of the BSDF at the specular angle, θ is the scattering angle, and L is the shoulder of the BSDF curve and equals the sine of the shoulder angle. According to the calculation result of BSDF from scratches, a fitting Harvey–Shack scattering model is established with b ₀ = 3 × 10⁻², L = 2 × 10⁻³, and s = -2. The fitting BSDF curve is shown in Figure 9A, and the simulated stray-light distribution resulting from scratches is shown in Figure 9B.

BSDF from digs is given by the expression: B S D F d = 1 4 N d d 2 [ 1 + π 2 d 2 4 λ 2 ( 1 + sin 2 θ l d 2 ) − 3 2 ] . (5)

Here, N d = 1 5 π D , l d = ( 4 π 4 ) 1 3 λ d , N _d is the density of the digs (number of digs per unit area), d is the diameter of the dig from microscopy, θ is the scattering angle, l _d is the roll-off angle, and D is the diameter of the optical element. Combined with the statistical testing result of digs for M1 over the full FOV, the diameter of the dig d is less than 0.01 mm. According to the calculation result of BSDF from digs, a fitting Harvey–Shack scattering model is established with b ₀ = 1 × 10⁻⁵, L = 2 × 10⁻², and s = −2. The fitting BSDF curve is shown in Figure 10A, and the simulated stray-light distribution resulting from digs at 700.0 nm is shown in Figure 10B.

The surface BSDF scattering resulting from particulate contamination is a function of the particle density function f(D) [30]. According to the Mie scattering theory, BSDF can be estimated when the MIL-STD-1264B Cleanliness Level (CL) is used to describe f(D). Spyak and Wolfe’s work shows that particulate contamination scattering is also shift invariant, and the Harvey–Shack model can be used to predict particle scattering in accordance with Spyak and Wolfe’s BSDF curve. For particulate contamination scattering [31], the parameter b ₀ in Eq. (4) can be taken by the total integrated scattering (TIS) [32] corresponding to the cleanliness level of M1 as follows: b 0 = ( T I S ) L s ( s + 2 ) 2 π [ ( 1 + L 2 ) ( s + 2 ) 2 − ( L 2 ) ( s + 2 ) 2 ] . (6)

According to the statistical testing results of dust for M1 over the full FOV, the cleanliness level for M1 is better than class 100, and the fitting parameters are inserted into the Harvey–Shack scattering model with L = 2 × 10⁻², TIS = 1.3 × 10⁻⁵, b ₀ = 2 × 10⁻³, and s = −2.2. The BSDF curve for cleanliness level 100 at 700.0 nm is shown in Figure 11A, and the simulated stray-light distribution resulting from particulate contamination is shown in Figure 11B.

According to the aforementioned BSDF Eqs. 2–5, combined with the testing results of the white-light optical profiler and the dark-field microscope for M1, the BSDF scattering curves resulting from roughness (red line), scratches (violet line), digs (green line), and particulate contamination with cleanliness level 100 (blue line) at 700.0 nm are shown in Figure 12. Their contributions to the stray-light level for SCI are shown in Table 1. After using the aforementioned testing results for stray-light simulation, the overall stray-light level for SCI can be suppressed to 10⁻⁸ B_⊙ at 700.0 nm.