In order to achieve Inertial Confinement Fusion (ICF), high-power laser beams or particle beams are used to irradiate fusion fuels, which are compressed to high temperatures (above 5 000℃) with high density (600 g/cm³). X-ray microscopy which image the X-rays emanating from a laser-produced plasma provides an effective way to diagnose the spatial characteristics of laser plasma, which plays an important role in the study of laser radiation uniformity^[1], implosion compression symmetry^[2] and fluid instability^[3-4]. Since the refractive index of any materials to X-ray is lower than 1, and X-ray is strongly absorbed by materials, refraction imaging commonly used in the visible light band can not be used. When a glancing angle to a surface is small, the X-ray is totally reflected by the surface, the grazing incidence system is generally used according to the full external reflection. The structural forms of the grazing incidence reflection X-ray microscope system used for ICF experimental research predominantly come in two sorts: One is the Kirkpatrick-Baze (KB) structure firstly created by Kirkpatrick and Baze in 1948(two orthogonal cylindrical or spherical grazing-incidence mirrors reflecting the X-ray), AKB and KBA structure based on KB structure (two parallel spherical mirrors are applied to the meridian and sagittal surfaces)^[5-7]. The other one is the rotational axisymmetric grazing-incidence reflection system developed by Wolter in 1952 (confocal combination of spheroid and hyperboloid is used to realize the ideal imaging that decreasing the spherical aberration and coma)^[8].

The on-axis aberration of a grazing-incidence spheric-based Kirkpatrick-Baez compound microscope may be precisely corrected. However, for finite fields, the off-axis performance degrades too rapidly for high-spatial-resolution imaging of even the smallest objects of interest^[9-10]. The Wolter configuration is a coaxial confocal optical system, which can be used as one of the high resolution imaging to eliminate spherical aberration of the point on the axial and it can effectively reduce off-axial aberration such as coma aberration. However, due to its closed annular design, for high energy X-ray imaging, it is difficult to achieve the required smoothness and surface tolerance ^[11-12], which will cause serious X-ray scattering, resulting in the decline of resolution. In order to get high resolution in larger field of view in X-ray region, providing basis for ICF diagnosis, the system adopts two toroid mirrors which are nearly parallel to image the target for the compactness requirements of the structure of imaging system, and the image aberration can be decreased to acceptable level. An auxiliary aiming system and the bidirectional binocular intersection online aiming system ensure the precision of the mirrors' alignment.

The principle of this imaging system combines the characteristics of the KB mirrors and the Wolter microscope. In order to be compact and correct aberrations, toroid mirrors are used instead of two orthogonal cylindrical mirrors in the system. The toroid shapes also present some advantages of brightness compared to KB cylindrical mirrors because only a single toroid mirror can be used to achieve two-dimensional focusing imaging. Toroid surfaces have the same imaging qualities as quadric surfaces (i.e. elliptical and parabolic) and are easier to polish.

As shown in Fig. 1, A toroid is a surface of revolution generated by revolving a circle, whose radius is r, about the axis Z which is coplanar with the circle and a distance R from the center of the circle.

The toroid surface equation in Cartesian coordinates can be written as ^[13]

1 $ f(x, y, z) = {(\sqrt {{x^2} + {y^2}} - R)^2} + {z^2} - {r^2} $

where R is the major radius, and r is the minor radius. Toroid mirror is a part of the concave surface, and the base arc can be defined as

2 $ S = \frac{{r{x^2}}}{{1 + {{\left[ {1 - \left( {1 + {k_x}} \right){r^2}{x^2}} \right]}^{1/2}}}} + \sum\limits_{i = 1}^p {{A_i}} {x^{2(i + 1)}} $

where k_x is the conic constant, and A_i is polynomial aspheric coefficients. Usually the meridian radius of curvature R is typically quite large (10⁵ mm for our mirror) in comparison with sagittal radius of curvature r, so a toroid mirror becomes a feasible approximation to the ideal ellipsoidal figure. Its advantage is that a single mirror can be used to achieve two-dimensional focusing imaging instead of KB structure. It is of a simple structure and can constitute a multi-channel system. In addition, toroid mirrors could be more easily polished than quadratic surfaces.

It can be seen that the toroid mirror also has an axis of rotational symmetry, but it does not coincide with the axis of the optical system. Toroid mirror is a feasible approximation to the ideal ellipsoidal mirror when the meridian radius of curvature R is typically quite large in comparison with all the other dimensions of the microscope. Because the toroid mirror has different radius of curvature in meridian plane and sagittal plane, it can be considered separately when focusing. The focused imaging on the meridian plane is essentially the same as the imaging characteristics of a spherical mirror, which has been analyzed in many literatures^[14-15].

The best way to focus the beam with a single element is the use of a toroid mirror with unity magnification (Rowland configuration), which cancels the astigmatism and the coma. However, when a big magnification is required, coma aberration is unavoidable. Therefore, two toroid mirrors are used to improve the coma caused by the single toroid mirror which does not satisfy the Abbe sinusoidal condition, so as to improve the imaging quality of the off-axis point.

As an imaging optical system, the aim of the proposed configuration is to provide a high resolution image of the X-ray source with the following primary concerns.

The focal length and diameter of the mirrors determine not only the size of the system, but also the resolution of the optical system. Because the energy ICF is high, the environment surrounding the micro ball of Deuterium (D) and Tritium (T) nuclei is more complex. It's better to place the imaging system in a place far from the target, avoiding the damage of high energy beams and debris. Therefore, the object distance of the microscope is better more than 500 mm. At the same time, considering the limitation of the space for installing the microscope, assuming the magnification 15×.

When design the grazing incidence optics, the glancing angle must firstly be decided at the center points of the mirrors.The critical angle of reflection depends on the surface material and wave length of the X-ray. As the wavelength of X-rays becomes shorter and shorter, the critical angle of reflection is extremely small. However the mirrors should have sufficient reflectivity at a large grazing angle. And the design of the grazing angle should take into account the float of the incident angle because of the off-center edge of the mirrors. The grazing incident angle is set to θ=0.5°, achieved by using the multilayer.

The field of view of the microscope is determined by the size of the implosion compression region, which is only several microns, the ideal Field of View (FOV) restricted to 1 mm will be sufficient.

The structure of X-ray microscope is shown in Fig. 2. The system consists of three mirrors, two of them are toroid mirrors with the same radius of curvature, and the third one is a plane mirror. The first two mirrors M1 and M2 are arranged in a u-shape with an angle to converge the X-rays both on the meridian plane and sagittal plane.

Since the third mirror is plane who only deflects the optical axis and does not focus the light, only the first two toroid mirrors are considered in the calculation of the initial structural parameters. As shown in Fig. 3, AO₁O₂A^′ is the pseudo-axis of the optical system. The gracing incident angle of the first mirror M1 is θ₁=0.5° and the incident angle of the second mirror M2 is θ₂=0.5°, according to the geometrical relation of reflection imaging, a formula about angles can be obtained as follows

3 $ \angle A + \angle {A^\prime } = 4 \times {0.5° } = {2° } $

And the magnification can be approximately defined as M=∠A/∠A^′=15, combined with the Eq. (3), we got ∠A=1.875°, ∠A^′=0.125°. Assuming that the distance from the object A to the center of the first toroid mirror O₁ is p₁=AO₁=740 mm, and the distance between the two mirrors is d=O₁O₂=120 mm, then according to the triangle cosine relation, it can be calculated that the distance from the center of the second mirror O₂ to the image A^′ is q₂=A^′O₂=11 938 mm. The object distance p₂ of the second mirror and the image distance q₁ of the first mirror satisfy the following equation

4 $ p_{2}=q_{1}-d $

The total magnification M of the microscope and two toroid mirrors' respective magnification M₁ and M₂ meet the following requirements

5 $ M = {M_1}{M_2} = \frac{{{q_1}}}{{{p_1}}}\frac{{{q_2}}}{{{p_2}}} = \frac{{{q_1}}}{{740}}\frac{{11938}}{{{q_1} - d}} = 15 $

Then the following quantity can be figured out: q₁=－1 589 mm, M₁=2.14, M₂=6.985.

In the case of grazing incidence, the optical axis of the thin beam does not coincide with the normal line of the mirror on the incident point, and the light is no longer a concentric beam after reflection by the optical system, which is focused in the meridian and sagittal direction respectively, and its relationship satisfies the Young′s formula

6 $ \frac{{{n^\prime }{{\cos }^2}{i^\prime }}}{{{q_{\rm{m}}}}} - \frac{{n{{\cos }^2}i}}{{{p_{\rm{m}}}}} = \frac{{{n^\prime }\cos {i^\prime } - n\cos i}}{R} $

7 $ \frac{{{n^\prime }}}{{{q_{\rm{s}}}}} - \frac{n}{{{p_{\rm{s}}}}} = \frac{{{n^\prime }\cos {i^\prime } - n\cos i}}{r} $

where p_m and q_m are the object distance and image distance in the meridian plane respectively; p_s and q_s are in the sagittal plane; i is the incident angle and i′ is the exit angle; n and n′ are the refractive index of object space and image space. R is the radius of meridian curvature of the toroid mirror and r is the radius of sagittal curvature. Then the grazing incident angle θ=π/2－i, i^′=i, n^′=－n, p_m=p_s=p. According to Eq. (8) and Eq. (9), the meridian focal length f_m and sagittal focal length f_s of a toroid mirror can be obtained.

8 $ \frac{1}{{{f_{\rm{s}}}}} = \frac{1}{p} + \frac{1}{{{q_{\rm{s}}}}} = \frac{{2\sin \theta }}{r} $

9 $ \frac{1}{{{f_{\rm{m}}}}} = \frac{1}{p} + \frac{1}{{{q_{\rm{m}}}}} = \frac{2}{{R\sin \theta }} $

During the optical design of the microscope, associated with the initial structural parameters (p, M, θ, d), a unique set of toroid mirrors' parameters could be established. In order to make the object be imaged on the same position both on meridian plane and sagittal plane, the focus of both direction should be equal:

10 $ {f_{1{\rm{m}}}} = {f_{1{\rm{s}}}} = {f_1} = 1385\;{\rm{mm}} $

which is substituted into Eq.(8) and Eq.(9), then we get the meridian radius of the first mirror R₁ and sagittal radius r₁ respectively.

11 $ {R_1} = \frac{{2{f_{1{\rm{t}}}}}}{{\sin \theta }} = \frac{{2 \times 1385\;{\rm{mm}}}}{{\sin {{0.5}° }}} = 317422\;{\rm{mm}} $

12 $ {r_1} = 2{f_{1{\rm{s}}}}\sin \theta = 2 \times 1385\;{\rm{mm}} \times \sin {0.5° } = 24.17\;{\rm{mm}} $

In the same way, we get the radius of the meridian plan R₂ and radius of sagittal plane r₂ of the second mirror.

13 $ {f_{2{\rm{t}}}} = {f_{2{\rm{s}}}} = {f_2} = 1495\;{\rm{mm}} $

14 $ {R_2} = \frac{{2{f_{2{\rm{t}}}}}}{{\sin \theta }} = \frac{{2 \times 1495\;{\rm{mm}}}}{{\sin {{0.5}° }}} = 342633\;{\rm{mm}} $

15 $ {r_2} = 2{f_{2{\rm{s}}}}\sin \theta = 2 \times 1495\;{\rm{mm}} \times \sin {0.5° } = 26.09\;{\rm{mm}} $

This is not the only solution for the microscope structure. A set of toroid mirrors with appropriate radius of curvature can be calculated with different grazing angles, object distances, magnification and spacing of mirrors.

The optical system can be optimized after the determination of the initial structure. Since the optical system of the grazing incidence structure is not coaxial, and the chief optical axis rotates at the break point during modeling, the grazing incidence reflection belongs to the large off-axis of the optical element, and the toroid mirror belongs to aspherical mirror, so the default evaluation function and constraint conditions cannot be used for the optimization by ZEMAX.

The mirror length l₁ of the first toroid mirror is an adjustable quantity whose length is determined by the expected spatial resolution and the geometric light gathering angle. To some extent, increasing l₁ can improve the light collection efficiency of the system, but it will damage the off-axis image quality. The second toroid mirror length l₂ can be adapted according to the first mirror length l₁, so that all (or part) of the incident light is received and reflected. A proper shortening of l₂ can act as an aperture and further improve off-axis aberration.

The radius of meridian curvature, radius of sagittal curvature of the two toroid mirrors, the distance between toroid mirrors, and object distance, image distance are all variables.The optimal evaluation function measures the imaging quality of the system by minimizing the spot radius of the image plane. The on-axis performance is modestly degraded and uniformly aberrations distributed over a chosen object field. The Spot diagram of the optimized system is shown in Fig. 4.

Obviously, the spot size is sufficiently bigger than the Rayleigh limit and the microscope belongs to the large aberration system. Spot diagram is still a feasible method to evaluate the imaging quality of large aberration system, which can show the concentration and dispersion degree of point source on the image plane. According to the design parameters, ZEMAX is used to simulate the performance of the microscope, and the spot diagrams of different field of view are calculated. As shown in Fig. 4, the maximum RMS diameter of the spot in each field of view is 52 μm, which meets the requirements of system resolution in the image field of view. The results show that as the field of the object is increased, the spot on the image surface is gradually increasing.

The biggest advantage of the reflective imaging system is with no chromatic aberration. Compared with axisymmetric optical systems (without considering high order aberrations), The aberrations cannot be analyzed by the software ZEMAX exactly, the primary aberration of off-axis reflection optical system is still spherical aberration, coma astigmatism, etc. However, coma includes linear coma and constant coma, while astigmatism contains quadratic, linear and constant terms. The pseudo axis is regarded as the optical axis, actually, incident rays are all off axis.

For a single toroid mirror, since the radius of meridian curvature is much larger than the radius of sagittal curvature, the spherical aberration is mainly sensitive to the meridian surface. The aperture longitudinal aberration is

16 $ {\mathit{\Delta }_{{\rm{sperical }}}} = - \frac{{3\alpha {q^2}(\theta R - q)}}{{{\theta ^3}{R^2}}} $

where α is the aberrant angle of the reflection ray to the center of the mirror.

As shown in Fig. 5, for two toroid mirrors, with the correct choice of parameters, the spherical aberration can be corrected when

17 $ {\mathit{\Delta }_1}(\alpha ) + {\Delta _2}( - \alpha ) = 0 $

with the grazing incidence angle same and mirrors 1 and 2 facing.

Due to the different magnification M of each point on the image plane, off-axis focusing is degraded, especially when the object deviates from its principal axis, and when the grazing incidence angle is small. The coma influence cannot be ignored. For a toroid mirror, the change of the scaling ratio of the beam across the length of the mirror is

18 $ {\Delta _{{\rm{coma}}}} = (M + 1)\frac{l}{{4p}} $

As can be seen from the coma aberration formulas above, the aberrations of the toroid mirror focusing image depends on the length l of the mirror illuminated by the beam, the grazing incidence angle θ, the distance p from the mirror to the source, and the magnification M. The coma aberration can be reduced with small mirror size or large object distance.

The first mirror introduces coma, which is completely compensated by the second one, so the coma is mutually compensated with two successive mirrors configuration.

As shown in Fig. 6, the angle between the two mirrors is η, then the angle between the incident light and the reflective light is 2η, and the angle between them is unchanged regardless of how the incident angle changes. This is the intrinsic characteristic of the double mirror system. So, two toroid mirrors can correct the field obliquity because it satisfy the Abbe sine condition.

The only condition for overcoming astigmatism in the use of a single toroid mirror is r/R=sin²θ, which is completely satisfied for the point on the axis, but with the increase of the field of view, the off-axis point will cause the change of the grazing incident angle, so that the sagittal and meridian image distances are not equal, and the astigmatism will appear again.

The optical structure is non-axial symmetrical, so off-axis aberration exists. When off-axis aberration has the same order of magnitude as spherical aberration, only correcting spherical aberration of points on the axis cannot guarantee the best performance of the system in the whole field of view.

In the actual assembly process, there will be errors in the parameters of each mirrors and their positions in the system, which will reduce the imaging quality of the system. Therefore, machining and assembly tolerances should be determined by the allowable quality reduction limit.

To determine the effect of each misalignment on the image quality, spot diagram in the image plane were calculated for various misalignment conditions. There are mainly four misalignment parameters: the grazing incident angle, the angle between the two toroid mirrors, the distance between the toroid mirrors, the distance from the object to the first toroid mirror.

It can be seen from the Fig. 7 that the optimal resolution position is not at the center where the chief ray intersects the image plane, but is about 0.2 mm offset to the center of the image in the positive direction of the field of view. Such imaging characteristics are related to the asymmetric structure caused by grazing incidence. Spatial resolution is more sensitive to the change of the grazing angle, and the larger the grazing angle is, the higher the imaging quality will be.

Besides the parameters analyzed above, other parameters change independently, and we evaluate the range of each tolerance, the central field resolution should be less than 2 μm under the tolerance. Then, we adjusted the tolerance so that 80% of the microscope resolution remained below 5 μm in the whole field. This calculation was obtained by randomly selecting the optical location of 60 000 previously calculated tolerances of different configurations. The final tolerances for positioning are summarized in Table 1.

The image resolution is highly sensitive to the mirror figure. Slope errors is the error between the rugged reflecting surface and the ideal smooth surface.The coating material of the mirror is Pt, and the RMS surface roughness of the mirrors was assumed to be 0.5 nm.

The microscopy system is a grazing incident imaging system which is non-axisymmetric.The accuracy of grazing angle is of great influence on imaging quality. It is difficult to achieve high accuracy by the common method used in the rotational axisymmetric structure under the condition of limited space.

Since the mirrors of microscope (including M1~M3) is to be installed in the chamber of laser fusion, the space of the chamber is quite limited, and the system requires a highly small grazing incident angle, so the installation and adjustment of the mirrors are typical difficult. Off-line adjustment is mainly to precisely align three mirrors. The fine alignment of the optical system was carried out by especially adjusting the glancing angle to mirror M1 and M2 respectively.

As shown in Fig. 8, The adjustment of the toroid mirror resort to an auxiliary laser imaging, an optical lens with a large numerical aperture is adopted to be equivalent to the conjugate relationship between the object and image of the toroid mirror, the error and depth of field of the equivalent lens are designed considering the positioning accuracy requirements of the X-ray microscope. The optical structure of the auxiliary adjustment system is shown in Fig 8. BS1 and BS2 are the beam splitters. The He-Ne laser irradiates the target and forms a cross-wire image at the image plane through the auxiliary lens. The BS1 and a mirror deflects another beam of light to the main optical axis, so that the target is focused through the toroid mirror. Then the angle of the toroid mirror is adjusted, when the two images formed by the two paths of light coincide, the grazing angle is adjusted. The relative positions of the mirrors are adjusted by the manipulator which can be used to adjust all the degrees of freedom of the mirrors according to the alignment tolerances.

Off-line mounting ensures that the internal optical elements of the microscope be installed in place, and then online alignment is required to realize that the microscope aim at the center of the object. The microscopic system is different from ordinary optical instruments in that its object and image are not in the same horizontal plane. In addition, the microscopy is installed in the chamber of laser fusion whose space is limited, causing a lot of difficulties in setting and adjusting. The binocular intersection aiming lasers is designed to realize the optical system aiming through the spatial intersection of two laser beams. For the space of the target chamber is limited, the opening angle of the two laser beams can not be too large.

The aiming structure of object and image planeis shown in Fig. 9. There are two pairs of tiny telephoto lenses. The lens image the target on the CCD, the microscope can be adjusted by judging the direction needs to move through the quadrant position of the center cross on the image plane. The object field of view is 6 mm×6 mm.

Image quality evaluation of binocular vision aiming system is shown in Fig. 8. Figure 11 (a) is the spot diagram of the system image plane. The size of the spot is similar to the diffraction airy spot diameter, so the resolution of the object plane is 20 μm. Figure 11 (b) is the Modulation Transfer Function (MTF), The MTF values at 50 lp/mm are more than 0.3 in object field of view, which satisfies the imaging requirements.

The source is an X-ray tube open Mo target with metal materials as the anode and filament as the cathode, and its maximum voltage is 50 kV and current is 1 mA. The imaging detector is a PI1 300×1 340 X-ray CCD camera with pixel size of 20 μm. Figure 12(a) is an image of a Ni grid, whose line width is 6 μm and the period is 20 μm. Figure 12(b) describes the light intensity distribution along the horizontal direction, the abscissa is the field of view in the object space and the ordinate is the normalized light intensity. The effective field of view can reached about 500 μm. The intensity distribution curve is essentially to image the mesh edge with Edge Response Function (ERF) in different field of view, so the (20% ~ 80%) peak-valley value of the ERF is correspond to the resolution of the image plane, which can be used to calculate the resolution of the object plane according to the magnification M, the pixels can be converted into microns which is defined as the resolution of the microscope. As shown in Fig 12(c), the spatial resolution in the center of the object plane is up to 2 μm, better than 5 μm in the 500 μm field of view. The best resolution achieved and the obtained field of view was worse than the simulation results. We consider the most dominant factor is the limitation of toroid surface processing accuracy and the influence of grazing angle scattering.

According to the spatial resolution requirements of ICF optical diagnosis and the limitation of experimental space, the X-ray microscope based on the toroid mirrors is designed and optimized. The imaging simulation shows that the spatial resolution of the optimized system is less than 5 μm in the range of 1 mm field of view. Backlight experiments show that the resolution of the system is about 5 μm at 500 μm. The errors between experimental results and simulation results are mainly due to the influence of mirror shape accuracy and diffraction effect. The microscope basically meets the requirements of large field of view and high resolution in laser plasma X-ray imaging. The following work direction is multi-light path structure design to gather more information at the same time, and the design of the multilayer, for the increase of the grazing incidence angle would be useful for image-forming in the hard X-ray region.