2. University of Chinese Academy of Sciences, Beijing 100049, China
2. 中国科学院大学, 北京, 100049
For X-ray/soft X-ray imaging systems, surface roughness is an important factor that directly affects the reflectivity and the optical quality of the grazing incidence mirror. Therefore, it is impossible to fabricate high-quality grazing incidence mirrors without reliably testing surface roughness. Moreover, image degradation due to surface scattering cannot be evaluated without the surface roughness of grazing incidence mirror[1-6]. That is to say, the development of testing methods is necessary and part of the development of X-ray/soft X-ray imaging systems.
Presently, the methods for measuring surface roughness include the mechanical probe scanning method, optical interferometry, using an atomic force microscope(AFM), the optical scattering method etc. State-of-the-art profile instruments, such as optical profilometry and AFM, have a vertical resolution better than one angstrom[6-8]. However, they can only be used to characterize the surface roughness of flat samples[6-9]. In addition, the measuring area of AFMs is very small so the corresponding spatial frequency range is quite narrow. Moreover, the results of AFM and optical surface profiler measurements are very easily affected by surface defects and dust on the sample. The mechanical probe scanning method can be used to characterize the curved surface roughness but it is easy to scratch the sample and its measuring accuracy is limited.
The X-ray scattering method obtains the surface roughness by measuring the angular distribution of the radiation intensity(scattering diagram) scattered by the rough surface. As far as we know, the X-ray scattering method is usually adopted to measure the roughness of a super-smooth plane. For example, in 1988, Alexander Vinogradov first derived the theoretical basis to measure the roughness of a super-smooth plane using the X-ray scattering method. In 2004, V.E. Asadchikov, applied the X-ray scattering method to measure the roughness of a super-smooth plane mirror. In 2010, Wang Yong-gang, studied the X-ray scattering characteristics of a grazing incidence multilayer planar mirror. However, no work has been reported on how to apply the X-ray scattering method to measure the surface roughness of a curved surface. The difficulty in measuring super-smooth curved surface by X-ray scattering method lies in determining how the surface figure affects the scattering diagram of the curved rough mirror.
In section 2, the scattering diagram of a curved rough mirror is derived with consideration to the angular distribution of the radiation intensity due to its surface figure(herein named the surface figure effect) according to the general Harvey-Shack surface scatter theory(GHS) and image formation theory. In section 3, the characterizing scheme for the surface roughness of a secondary mirror is designed. The determination error of the one-dimensional power spectral density(PSD) function due to the width of the receiving slit is evaluated and the optimal width of the detector aperture is obtained. In addition, the scheme is simulated with Zemax by the non-sequence ray tracing method. In section 4, the extracted PSD data is compared with the exact data, which verifies that the surface figure only affects the measuring accuracy of curved surface roughness with low spatial frequency. Finally, some conclusions are given in section 5.2 Theory about surface figure effect
Consider the light-scattering configuration shown in Fig. 1, where θi is the grazing incidence angle, θ and φ are the scattering grazing and azimuth angles and Zf(
The effect of the surface figure is equivalent to an image forming system that is described by the transfer function according to image formation theory. Therefore, the surface transfer function H(
The figure transfer function Hf(
It is generally accepted that the inverse scattering problem of predicting the surface PSD from the scattering diagram is only possible when the optical surface satisfies the smooth-surface approximation[12-13]:
where θi is the grazing incidence angle shown in Fig. 1. In these conditions, the scatter transfer function is expressed as
where θi is the grazing incidence angle; θ and φ are the grazing and azimuth scattering angles as shown in Fig. 1; σrel and σs are the relevant root-mean-square of the surface roughness and the intrinsic root-mean-square of the surface roughness respectively; PSD(fx, fy) is 2-dimension PSD of surface roughness; fx and fy are the spatial frequencies.
Since the Fourier transform of the surface transfer function, H(
Furthermore, the spatial wavelength of the surface figure is generally in the range of a few millimeters. The figure transfer function Hf(
In fact, the auto-correlation length l(a distance at which the auto-covariance function is reduced to 1/e, where e is the base of the natural logarithms) is generally in the range of several micrometers to several tens of micrometers, which is much longer than the wavelength of the incident X-ray. Therefore, the relevant root-square-mean surface roughness σrel is almost equal to the intrinsic root-square-mean surface roughness σs. So BRDF is expressed as the following:
where Q is the polar reflectivity
In Eq.(13), ε=ε2/ε1 is the relative dielectric constant of the interface where the subscripts 1 and 2 denote the incident and refracting side, respectively.
Since BRDF is a computed quantity calculated by dividing the measured radiant intensity by the cosine of the scattered angle, the scattering diagram (radiant intensity distribution) is expressed as the following:
The scattering diagram is characterized, in general, by two scattering angles (θ, φ). However, as the grazing incidence angle θi is typically smaller than the critical angle of the total external reflection θc at the wavelength, such a small value of θi leads to a large angular difference in the width of the angular distribution of reflected radiation intensity in the incidence (δθ~λ/(πlsinθi) and in the azimuth (δθ~λ/(πl) < < δθ) planes. In experiments, the scattering diagram is usually integrated over the azimuth angle as:
The integrated results could be divided into two parts:
where the surface figure Πf(θ) and the integrated scattering function Πs(θ) are respectively calculated as the followings:
For super smooth optical surfaces, Eq.(16) shows that the surface figure only affects the specular component, provided that the conditions described above are met. In these conditions, a superposition of the contributions of the figure and the roughness yields the integrated scattering diagram.3 Characterizing the surface roughness of the X-ray grazing incidence telescope
The space telescope has rigorous requirements on the thermal stability of the mirror body, especially for mirror material. Zerodur has excellent physical and chemical properties such as relatively high strength, a zero-expansion coefficient and high density. Therefore, the soft X-ray space telescope is fabricated with zerodur in our lab. After being polished, the relevant root-mean-square of its surface roughness is usually smaller than 1 nm, which is suitably characterized as being in the soft X-ray band. Tab. 1 lists the parameters of the Wolter-Ⅰ solar soft X-ray grazing incidence telescope in our laboratory.
|Nodal focal length/mm||659.885 495|
|Gap about joint/mm||5|
|Vertex radius/mm||-2.431 457 33||-2.440 466 51|
|Inner radius rinner/mm||80.075 946 99||75.414 332 72|
|Outer radius router/mm||81.505 495 11||79.771 451 35|
|Conic constant ε2||-1||-1.007 424 25|
|Location of focus/mm||1 364.867 410 92||705|
Since the primary and secondary mirrors are fabricated using the same process, their surface roughness are assumed to be identical. Therefore, only the surface roughness of the primary or secondary mirror need to be characterize. The radiation scattered from the primary mirror is blocked by the secondary mirror while the scattered radiation from the secondary mirror can arrive at the detector without being blocked. Therefore, it is more convenient and suitable to characterize the surface roughness of the secondary mirror.
The scheme shown in Fig. 2 is for measuring the surface roughness of the secondary mirror of the solar soft X-ray grazing telescope in our lab. Although, the grazing incidence mirror shown in Fig. 2 is non-nested, the scheme can also be applied to a nested X-ray grazing incidence mirror since each of its shell can be measured before integration. For the telescope in Tab. 1, the total reflection critical angle θc is equal to 5.94° when the incident wavelength λ is equal to 4.47 mm. When the incident radiation is parallel to the optical axis, the grazing incidence angle θo of 5.24° is smaller than the total reflection critical angle θc. That is to say, the wavelength and the grazing incidence angle are suitable for characterizing the surface roughness.
To condense the information contained in the PSD-function to a few parameters, the K-correlation model of the PSD-function shown in Eq.(19) is used.
The width of the incident beam is set to 0.2 mm to ensure that Eq.(11) holds true according to the analysis in Sec.2. The factors causing systematic errors in the determination of the PSD-function include the finite width of the receiving slit before the detector(simply called "receiving slit"), the divergence and spectral width of the incident beam, the noise of the detector etc. In this paper, we will only address the dominant factor and the finite width of the receiving slit in detail. Fig. 3 illustrates the effect of the finite width of the receiving slit.
For simplicity, the width of the incident beam is ignored and the error of determination of the PSD-function is written as follows:
Dependency of the error ηs on the receiving slit S is shown in Fig. 4. In calculations, we used the following parameters: radiation wavelength λ=4.47 nm; grazing incidence angle for the incident beam θ0=5.24°; sample-detector distance D=336 mm. Besides, the PSD-function is supposed to behave in accordance with the fractal-like law as shown in Eq.(19), which is typical for the samples.
Since the error ηs decreases rapidly with a decrease in the width of the receiving slit as shown in Fig. 4, it is important to minimize the width of the receiving slit. However, when the slit is too narrow, the detected scattered radiation falls sharply and the measured spatial frequency becomes more narrow. As shown in Fig. 5, ηs is less than 0.2% at spatial frequencies larger than 0.02 μm-1 when the receiving slit is less than 0.1 mm. According to the above analysis, the error of the PSD function is small when the width of the receiving slit is less than 0.1 mm.3.2 Simulation
In the simulation, the PSD is modeled by Eq.(19), where A=1 000.322×10-5 μm3, B=12 000 μm and C=1.089. The smooth-surface approximation is satisfied on the simulation. BRDF is calculated from the PSD according to Eq.(12). The characterizing scheme in Sec.3.1 is designed with Zemax and the simulation is created using the following steps. First, the surface figure effect Πf(θ), is obtained using the ray tracing method when the secondary mirror surface is perfect. Second, the integrated scattering diagram Π(θ) is obtained using the non-sequential ray-tracing method when the BRDF is added to the secondary mirror. Finally, the PSD function is obtained from Eq.(16).4 Results and discussion
The simulated results are shown in Fig. 5. Fig. 5(a) shows the scattering diagram of the rough secondary mirror П(θ), whereas Fig. 5(b) shows the surface figure effect Пf(θ). The scattered component of the scattering diagram Пs(θ) is shown in Fig. 5(c), which is obtained by subtracting the surface figure effect Пf(θ) from the scattering diagram П(θ). Furthermore, the surface figure effect Пf(θ), the scattering diagram П(θ), and the scattered component Пs(θ) are shown in Fig. 5(d) to demonstrate the surface figure effect on the scattering diagram. As shown in Fig. 5(d), the scattering diagram П(θ) agrees well with its scattered component Пs(θ) when the scattering angle is outside the angular range of the surface figure effect Пf(θ), which verifies that surface figure only affects the specular component of the scattering diagram.
Fig. 6 shows the PSD data extracted from the scattering diagram П(θ) and the exact PSD data. The PSD data extracted from the scattering diagram П(θ) agrees well with the real value(exact PSD data) when the spatial frequency is more than 28/mm, which confirms that surface figure only affects the measuring accuracy of surface roughness with low spatial frequency. On the other hand, surface roughness with high spatial frequency can be characterized accurately using the X-ray scattering method when smooth-surface approximation is met and the width of the incident beam is about a tenth of spatial wavelength of the surface figure.5 Conclusions
In this paper, a theoretical analysis was performed to prove that surface figure only affects the measuring accuracy of the roughness of a curved surface with low frequency when smooth-surface approximation is met and when the width of the incident beam is about a tenth of the spatial wavelength of the surface figure. Based on this analysis, the X-ray scattering method was applied to characterize surface roughness of the secondary mirror of the X-ray grazing incidence telescope. The scheme was simulated with Zemax and the simulated results show that the surface figure only affects the measuring accuracy of the roughness of a curved surface with spatial frequency less than 28/mm, which verifies the accuracy of the theoretical analysis. This method will be valuable and helpful for efficiently developing high quality X-ray imaging systems. It can also be used in other fields, such as modern optics, micro-electronics and micro electro-mechanical systems. The factors causing systematic errors will be analyzed comprehensively in the future.
HARVEY J E, MORAN E C, ZMEK W P. Transfer function characterization of grazing incidence optical systems[J]. Applied Optics, 1988, 27(8): 1527-1533. DOI:10.1364/AO.27.001527
HARVEY J E. Scattering effects in X-ray imaging system[J]. Proceedings of SPIE, 1995, 2515: 246-272. DOI:10.1117/12.212595
PETERSON G L. Analytic expression for in-field scattered light distributions[J]. Proceedings of SPIE, 2004, 5178: 184-193. DOI:10.1117/12.509120
HARVEY J E, CHOI N, KRYWONOS A, et al. Image degradation due to scattering effects in two-mirror telescopes[J]. Opt. Eng., 2010, 49(6): 063202. DOI:10.1117/1.3454382
KOZHEVNIKOV I V, ASADCHIKOV V E, ALAUDINOV, et al. X-ray investigations of super smooth surfaces[J]. Proceedings of SPIE, 1995, 2453: 141-153. DOI:10.1117/12.200271
DE KORTE P A J, LAINE R. Assessment of surface roughness by X-ray scattering and differential interference contrast microscopy[J]. Applied Optics, 1979, 18(2): 236-242. DOI:10.1364/AO.18.000236
ZANAVESKIN M L, GRISHCHENKO Y V, TOLSTIKHINA A L, et al. The surface roughness investigation by the atomic force microscopy, X-ray scattering, and light scattering[J]. Proceedings of SPIE, 2006, 6260: 62601A-1-62601A-10.
RUPPE C, DUPARRE A. Roughness analysis of optical films and substrates by atomic force microscopy[J]. Thin Solid Films, 1996, 288(1-2): 8-13. DOI:10.1016/S0040-6090(96)08807-4
VINOGRADOV A V, ZOREV N N, KOZHEVNIKOV I V, et al. X-ray scattering by highly polished surfaces[J]. Journal of Experimental and Theoretical Physics, 1988, 67(8): 1631-1638.
ASADCHIKOV V E, KOZHEVNIKOV I V, KRIVONOSOV Y S, et al. Application of X-ray scattering technique to the study of super smooth surfaces[J]. Nucl. Instr& Meth A, 2004, 530(3): 575-595.
王永刚, 孟艳丽, 马文生, 等. 掠入射X射线散射法测量超光滑表面[J]. 光学 精密工程, 2010, 18(1): 60-68.
WANG Y G, MENG Y L, MA W SH, et al. Measurement of super-smooth surface by grazing X-ray scattering method[J]. Opt. Precision Eng., 2010, 18(1): 60-68. (in Chinese)
ANDREY KROYWONOS. Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction[D]. Central Florida: University of Central Florida, 2006: 178-180. https://www.researchgate.net/publication/47715393_PREDICTING_SURFACE_SCATTER_USING_A_LINEAR_SYSTEMS_FORMULATION_OF_NON-PARAXIAL_SCALAR_DIFFRACTION
KRYWONOS A, HARVEY J E, CHOI N. Linear systems formulation of scattering theory for roughsurfaces with arbitrary incident and scattering angles[J]. J. Opt. Soc. Am. A, 2011, 28(6): 1121-1138. DOI:10.1364/JOSAA.28.001121
DITTMAN M G. K-correlation power spectral density and surface scatter model[J]. Proceedings of SPIE, 2006, 6291: 6290R.
ZEMAX Development Corp. ZEMAX Manual:Optical Design Program User's Guide[M]. Sandiego: ZEMAX Development Corp, 2009, 331-446.