Since the first reported successful application of focusing X-rays using refractive lenses^[1], X-ray refractive optics has developed rapidly in the years following. X-ray refractive lenses usually have a strong curvature to focus the X-ray to a point, and they are often used in multiples, called compound refractive lenses (CRLs). X-ray refractive lenses are effective for focusing high-energy beams ($>5\mathrm{kev}$). They are usually used for focusing and collimating at synchrotron radiation sources^[2] or hard X-ray free electron lasers. They also play an important role in full-field microscopy and scanning microscopy^[3–5].

The index of refraction for the X-ray in matter can be written in the form of $n=1-\delta +i\beta$, where $\delta$ is the refractive decrement and $\beta$ describes the attenuation inside the material. For the X-ray, the real part of the refractive index is less than 1, and a single refractive lens must be concave in shape. The X-rays passing through a lens are deflected in the vertical grating line direction by an angle of $${\alpha }_l=\frac{\lambda }{2\pi }\frac{\partial {Ø}_l}{x_l},$$(1)where ${\alpha }_l$ is the beam deflection angle, $\lambda$ is the wavelength, and $\partial {Ø}_l/\partial x_l$ is the differential phase shift along the vertical grating line direction at the lens. For focusing the X-ray to a point, the ${\alpha }_l$ must be linearly proportional to $x_l$, so most X-ray refractive lenses are now in the shape of the paraboloid of rotation. The focal distance $f$ is determined by the refractive index decrement $\delta$, and the parabola vertex radius of the curvature $R$ of the lens is as follows: $$f=\frac{R}{2N\delta },$$(2)where $N$ is the number of doubly curved lenses in a stack.

X-ray refractive lenses are made of many different materials, such as Be, Al, SU-8, or diamond^[6–9]. Two groups have reported inspections of X-ray refractive lenses made by the mechanical pressing of Be and Al^[6,7]. The results showed that the edge part of the lens was relatively different from the ideal surface shape, which may be the result of mechanical pressing. Two-photon polymerization-induced lithography has unique advantages in the field of manufacturing complex three-dimensional micro-nano functional devices^[10]. Several examples of polymer refractive lenses manufactured by two-photon absorption lithography have been presented^[11–13]. Ptychographic characterization of polymer compound refractive lenses was presented several years ago^[14], but it did not directly show the defects of the lens surface profile. In the present paper, polymer compound refractive lenses were fabricated by two-photon polymerization lithography. The angular change and wavefront phase shift of the X-ray passing through the lens were reported, which was an ideal characterization for lens measurement. Further, the lens surface shape was presented. This research is valuable for improving manufacturing processes.

Two-photon polymerization is a unique three-dimensional micro-nanostructure processing technology that enables millimeter-sized structures to be fabricated with sub-hundred nanometer resolution, as presented in Fig. 1. When the laser is focused into the transparent photosensitive material through a high numerical aperture objective lens, the material can be induced to produce nonlinear absorption at the focused spot and a polymerization reaction occurs^[15–17]. CRLs were designed according to Fig. 2, and two sets of polymer refractive lenses were fabricated using photoresist (SZ2080). They comprised one and three individual lenses, respectively. The main component molecular formula is ${\mathrm{C}}_4{\mathrm{H}}_{12}\mathrm{SiZr}{\mathrm{O}}_2$^[18]. The density of the polymer material was estimated to be $1.2\mathrm{g}/{\mathrm{cm}}^3$^[19]. The resist was exposed from bottom to top in a cell containing uncrosslinked photoresist, and the height of the cell was approximately 81 µm. Each layer was printed with 400 nm slicing and hatching distances.

A fast steering mirror provided the beam-waist movement within the sample plane in the field of $120\mathrm{\mu m}\times 120\mathrm{\mu m}$. The average power of the incident radiation was 10 mW. The limited field of view and working distance of the focusing objective limited the lens geometry. The aperture of the individual lens was 50 µm, and the parabola vertex radius of curvature $R$ was 10 µm. The distance between the individual lenses was 120 µm.

An outline of the setup is shown in Fig. 3 on the basis of X-ray grating interferometer^[20]. It mainly consists of a microfocus X-ray source, a double grating shearing interferometer, and an X-ray camera. The phase grating $G_1$, with a phase shift of $\pi$, is used as a beam splitter to modulate the incoming wavefront^[21], and the absorption grating $G_2$ is performed as a transmission mask. In the downstream of $G_1$, the incoming beam is divided essentially into two first-order diffraction orders. They interfere and form periodic patterns at certain discrete distances. Based on the fractional Talbot effect^[22,23], this pattern exhibits a maximum modulation at the following distances: $$d_m=m\frac{p_1^2}{8\lambda }.$$(3)

The above equation is applied to a parallel beam setup, where $m$ is an odd integer that corresponds to the order of the fractional Talbot distance, $\lambda$ is the wavelength, and $p_1$ is the period of $G_1$. Considering a geometric magnification factor $l_1/(l_1-d_m)$ for a cone beam setup, the above formula can be rewritten as $$Z_m=\frac{l_1}{l_1-d_m}{d}_m,$$(4)where $l_1$ is the distance from the X-ray source to the phase grating. The lateral period $p_2=p_1/2$ of the interference pattern for a parallel beam setup can be rescaled to $$p_2^{*}=\frac{l_1}{l_1-d_m}\frac{p_1}{2}.$$(5)

In order to obtain a reasonable degree of modulation, the spatial coherence of the incident X-ray has to meet the condition that the transversal coherence length at the plane of $G_1$ is larger than the separation $h_m=m{p}_1/4$ of the interfering beams.

The incident wavefront is disturbed by the object in the beam path, and an angle change of the first-order diffraction angle at the phase grating can be detected as $$\mathrm{\Delta }{\alpha }_{\mathrm{PG}}=\frac{\lambda }{2\pi }\frac{\partial {Ø}_{\mathrm{PG}}}{\partial x_{\mathrm{PG}}},$$(6)where $\partial {Ø}_{\mathrm{PG}}/\partial x_{\mathrm{PG}}$ is the differential phase shift at the phase grating plane along the vertical grating line direction. The X-ray wavefront phase is not disturbed between the lens and the phase grating, so the wavefront phase shift ${Ø}_l(x_l)$ at the lens is equal to the wavefront phase shift ${Ø}_{\mathrm{PG}}(x_{\mathrm{PG}})$ at the phase grating. Due to the cone beam by the microfocus X-ray source, the $x_l$ is related to the $x_{\mathrm{PG}}$, with $x_l=x_{\mathrm{PG}}\times r/l_1$. Therefore, the differential phase shift follows the following basic relationship: $$\frac{\partial {Ø}_{\mathrm{PG}}}{\partial x_{\mathrm{PG}}}=\frac{r}{l_1}\frac{\partial {Ø}_l}{\partial x_l},$$(7)where $r$ is the distance from the X-ray source to the lens. Changes in the first-order diffraction angle cause the interference fringes to move. The position of the interference fringes is shifted by $s=\mathrm{\Delta }{\alpha }_{\mathrm{PG}}\times Z_m$. The phase information $\phi$ of the interference fringes can be reconstructed from either Fourier analysis based on the Moiré effect or the phase stepping method in grating interferometry. Consequently, the differential phase shift caused by lenses can be obtained by the phase information $\phi$ of the interference fringes with $$\phi =\frac{\lambda Z_m}{p_2^{*}}\frac{r}{l_1}\frac{\partial {Ø}_l}{\partial x_l}.$$(8)

The angle change transmitted through the lens is determined by the differential phase shift, which is an ideal measure for the focusing performance of the lens. The wavefront phase shift can be directly related to a material’s thickness. Therefore, the surface shape of the lens can be reconstructed based on the wavefront phase shift ${Ø}_l(x_l)$.

The experimental device is shown in Fig. 4. The setup is designed for 8 keV photon energy, and it consists of a Hamamatsu L9421-02 microfocus X-ray source with a tungsten target operating at 40 kV and 200 µA. The focal spot is approximately 7 µm at 8 W. A source-detector distance of $l_2=1178\mathrm{mm}$ was chosen. The phase grating $G_1$ is made of epoxy on a polyimide substrate and has a pitch of $p_1=5.366\mathrm{\mu m}$ and a thickness of 20 µm, yielding a phase shift of $\pi$. The absorption grating $G_2$ has a pitch of $p_2=3\mathrm{\mu m}$, with electroplated gold on a polyimide substrate. Experiments were performed at the third fractional Talbot distance ($Z_m=77.89\mathrm{mm}$ for 8 keV photon energy). An in-line setup consisting of a 100 µm-thick CsI scintillator and a Dhyana 400D CCD camera was used to record the interferograms. The CCD has a pixel of 11 µm and a $4\times$ optical magnification objective lens, which allows for high spatial resolution imaging.

A single polymer lens and a set of three polymer lenses, which were both known to be imperfectly manufactured, were tested. The lenses were placed as close to the source as possible to facilitate detection. The projection magnification of $M=r/l_2=6.12$ was chosen. With the assistance of the laser device, a three-dimensional precision translation stage was used to move the lens to the center of the detection range. To avoid aberrations, the alignment of the lens optical axis with the system optical axis was conducted by a high-resolution X-ray CCD camera. A single polymer lens recorded with a microfocus X-ray source is shown in Fig. 5(a). In this experiment, the interference fringes are discontinuous because the phase grating used is discontinuous. We selected the area where the fringes were continuous for analysis. Figures 5(b) and 5(c) show the reference image and the inspection image of the lens recorded at the BL09B beamline of the Shanghai Synchrotron Radiation Facility.

The phase can be extracted from the interference pattern using the Fourier analysis method. Further, the angle change along the vertical grating line direction at the lens can be derived, which is shown as a function of $x$ and $y$ in Fig. 6(a). For focusing the X-ray to a point, the beam deflection angle ${\alpha }_l$ must be proportional to $x_l$. Figure 6(c) displays the measured beam deflection angle ${\alpha }_{l,m}$ and its target value ${\alpha }_{l,t}$ for a line along the $x$-axis. The target value was derived by linearly fitting the measured data for $|x|\le 20\mathrm{\mu m}$. The deviation between ${\alpha }_{l,m}$ and ${\alpha }_{l,t}$ for a line along the $x$-axis is shown in Fig. 6(d). The beam deflection angle of a set of three lenses of ${\alpha }_{l,\text{thr}}$ shown in Fig. 6(b) was obtained by the same method, which is about three times that of a single lens.

The interference pattern directly recorded by X-ray camera contained much noise, which could directly affect the inspection quality. Noise levels can be characterized on the basis of images without the lens in the beam path. Each image was acquired with an exposure time of 16 s, and the standard deviation of the measured beam deflection angle ${\delta }_{l,m}$ for many pixels was approximately 0.06 µrad. Averaging multiple acquired images considerably reduces noise. Exposure for 320 s and averaging of 20 frames of images can reduce the standard deviation to 0.04 µrad.

For the setup applied, a single lens showed an X-ray transmission of 91.17%. According to the data in Ref. [24], an effective mean photon energy of 14.97 keV was deduced. Using Eq. (2) and the target lens curvature of $R=10\mathrm{\mu m}$, the focal distance of $f=4.54\mathrm{m}$ was calculated. The focal distance is suitable for practical applications where multiple lenses are used in combination. The test data can be interpreted by simple geometric optics, ignoring the diffraction effects of the lens aperture. For $|x|\le 15\mathrm{\mu m}$, the maximum deviation was 0.05 µrad, which is approximately equivalent to the level of noise. However, for the edge part, the maximum deviation reached 0.22 µrad, which is not acceptable for imaging and focusing applications.

The wavefront phase shift caused by the lens can be deduced from the measured beam deflection angle data, which is cumulative data. The measured wavefront phase shift of ${Ø}_{l,m}$ and the target wavefront phase shift of ${Ø}_{l,t}$ for a line along the $x$-axis are shown in Fig. 7(a) as solid and dashed lines, respectively. The target wavefront phase shift was deduced from the target beam deflection angle. The deviation between the measured wavefront phase shift and its target value is shown in Fig. 7(b). The maximum deviation reached 0.012$\pi$ at the edge part, which corresponded to a thickness deviation of 0.23 µm.

The wavefront phase shift is directly related to the thickness of the lens. Further, the surface shape of the lens can be reconstructed by taking half of the thickness variation. The grating was rotated to 90 deg and the lenses were re-measured under the same experimental conditions. The surface shape of the lens can be reconstructed by a linear combination of the results of these two experiments^[25]. The reconstructed result is shown in Fig. 8. According to the reconstructed surface shape, the edge height of the lens was lower than the design parameters, which was the reason why the refraction change angle of the edge part was smaller than the ideal value.

X-ray speckle tracking (XST) is a basic and intuitive method for wavefront detection using X-ray near-field speckle characteristics. Similar to the principle of differential phase contrast imaging (DPCI), XST uses speckles generated by a speckle generator (such as sandpaper) to detect the angular changes in the X-ray transmitted through the object, thereby deriving the wavefront phase shift. The surface shape can be reconstructed using the wavefront phase shift. The detailed principle of XST was reported in Refs. [26–28]. The lens was inspected using XST for an X-ray energy of 15 keV with an exposure time of 1 s at the BL09B beamline of the Shanghai Synchrotron Radiation Facility to confirm the accuracy of DPCI. The surface shapes for a line plot along the $x$-axis obtained through DPCI and XST are shown in Fig. 9(b), and their deviation is displayed in Fig. 9(c). The results were in good agreement with a maximum deviation of 0.4 µm.

The measured beam deflection angle results showed good linearity, except at the edge of the lens, which meant the central part of the lens could be used in practical applications. The reconstructed lens surface shape showed that the thickness of the edge part was lower than the design parameters, which may be caused by the shrinking of the polymer material or improper handling during the development process. In order to improve the focusing performance of the lens, we can modify the original design or printing process. An error in the assumed effective energy causes a cumulative error for the measured wavefront phase shift ${Ø}_{l,m}$, but it cannot affect the directly obtained beam deflection angle ${\alpha }_{l,m}$. When the phase is extracted from the interference fringes using the Fourier analysis method, the accuracy depends on the size of the Fourier plane window used to select the order, which reduces the spatial resolution. Some improved Fourier analysis methods can be used in subsequent experiments to weaken this effect, such as the technique of Fourier analysis using two complementary interferograms^[29]. When the X-ray passes through an object, it can interact with the object in several manners, such as changes in intensity, phase, refraction angle, scattering angle variance, and second-order derivative of the phase. Some of these signal changes were not taken into account in this study. While image filtering filters out noise, it inevitably removes some useful information, which could affect the experimental results.

We demonstrated an experimental setup capable of measuring the refractive properties of polymer lenses with high accuracy. The DPCI results on the polymer lenses and the XST experimental results showed the feasibility of the device and its accuracy is greater than 0.4 µm. The lens error is primarily characterized by the deviation between the measured beam deflection angle and its target value. The polymer lenses have less X-ray absorption loss and could be the development trend of X-ray refractive lenses in the future. Under actual working conditions, the refractive lens could be interfered with by radiation damage, mechanical vibration, and other factors. The proposed device can perform non-destructive, in situ, and high-resolution inspection of the lenses, which is important for improving the fabrication process and focusing performance of the lens. The effective aperture of the lens we tested was only 50 µm and the CCD that we used had a pixel of 11 µm with a $4\times$ optical magnification objective lens. The use of a detector with higher resolution could optimize experiments in the future.