Abstract
We propose a well-integrated, high-efficiency, high-precision, and non-destructive differential confocal measurement method for the multi-geometric parameters of the inner and outer spherical surfaces of laser fusion capsules. Based on the laser differential confocal measurement system with high tomography fixed-focus ability and high spatial resolution, the proposed method is used to perform the fixed-focus trigger measurement of the outer vertex, the inner vertex, and the spherical center of the capsule. From the rotation measurement around the Y-axis and the transposition measurement around the Z-axis, the inner and outer diameters, the three-dimensional inner and outer profiles, the shell thickness uniformity, and the shell non-concentricity of the capsule are measured with high precision and no damage. To the best of our knowledge, this is the first method to achieve the high-precision measurement for the multi-geometric parameters of the capsule inner and outer spherical surfaces with the same instrument.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser inertial confinement fusion (ICF) is a process that releases huge energy by using multi high-intensity lasers to focus ignition in space to rapidly compress fusion capsule filled with deuterium and tritium fuel through. It is an ideal technical means for human beings to obtain clean energy [1,2]. The hollow fusion capsule filled with thermonuclear fuel is a core component of the ICF experimental device. In the ICF experiments, the roughness of the inner surface of the capsule is the most important cause of hydrodynamic instability. The nano-scale fluctuations of the inner profile can be magnified by more than a hundred times, which causes experiment failure [3–6] and imposes demanding requirements on high-precision measurement.
In the previous studies on capsule measurement, [7,8] used the X-ray photography method to measure the inner surface, but the resolution was low and the high-frequency information of the inner surface may not be incorporated; [9] combined the atomic force microscope with the Wallmapper optical fiber probe to measure the outer surface profile and the shell thickness of the capsule, the inner surface profile data were then obtained by subtracting the thickness data from the outer surface profile data. However, it was difficult to accurately match the position of the atomic microscope probe with that of the Wallmapper optical probe; therefore, the improvement of the measurement accuracy for the inner profile was limited. [10,11] established a digital ray-tracing model, SHELL 3D, based on the backlight shadow imaging, and obtained the power spectrum curve of the inner profile. However, the method had certain requirements on the diameter-thickness ratio of the capsule. For example, when the thickness was smaller than a certain value, the bright ring of the shadow map can hardly be obtained, which caused almost indistinguishable inner surface profiles.
As shown in Fig. 1, the measurement of the inner surface profile of the capsule involves the outer and inner diameters, the outer surface profile, the refractive index, the shell thickness uniformity, etc. The above parameters are interrelated and coupled with each other. Therefore, accurate measurements for the inner surface profile requires precise and simultaneous measurements of the geometric parameters of the inner and outer spherical surfaces of the capsule to separate accurately the inner surface profile parameter from the aliased measurement information.
In the existing measurement methods for the multi-geometric parameters of the capsule, [12] used the white light interference microscope to measure the inner and outer diameters and the shell thickness of the capsule, whereas the inner surface profile was not measured; [13] used a scanning transmission ion microscopy to perform the tomographic measurement of the capsule, including the micrometer-level measurement of the three-dimensional (3D) outer and inner profiles, the shell thickness, and the shell non-concentricity. However, the resolution was low and therefore, the high-frequency information of the surface profile was mostly not considered; [14] used the spatial interpolation and edge recognition based on the X-ray tomography to reconstruct the capsule shell, the inner and outer profiles and shell thickness of the capsule were also obtained. However, the application of the method was restricted by the resolution of the CCD, which limited the improvement in the measurement accuracy for the geometric parameters of the capsule.
Capsules are normally sampled by the scanning electron microscope based on damage, and then the quality of the inner surface geometric mass is evaluated [15]. However, the capsules used for the ICF experiment are not measured by this method. Thus, an accurate measurement of the inner surface high-frequency information of the capsule is still a challenge.
To address the problem, we propose a high-precision differential confocal measurement method in the same coordinate system with common data for the multi-parameters of the capsule. The proposed method can measure the outer and inner diameters, outer surface profile, shell thickness, and shell non-concentricity of the capsule in one single installation and achieve high-precision measurements of the inner surface profile of the capsule.
2. High-precision laser differential confocal measurement method for the geometric parameters of the ICF capsule
As shown in Fig. 2, the measurement principle of the diameters and the profiles are integrated in the proposed method; based on that the focal point of the laser differential confocal system exactly corresponds to its zero-crossing point of the axial response curve [16]. The integrated capsule multi-parameters measurement system includes three parts: the large-field coarse aiming light path A, the large-range inner and outer diameters measurement light path B, and the micro-range inner and outer profiles measurement light path C.
In the large-filed coarse aiming light path A, the returned measurement beam is converged into a focused spot by the lens RL and enters the CCD, and the capsule position can be adjusted according to the Airy spot detected by the CCD.
The inner and outer diameters of the capsule are measured by the virtual pinholes detection light path (VPH) with a large-range with high-precision in part B. It is worth noting that, in the differential confocal optical path, the point detector is required to coincide with the optical axis of the measurement system to ensure the light beam reflected entering the point detector [17]. As the outer and inner diameters measurement requires a large-range of fixed-focus measurement on the outer surface vertex A, inner surface vertex B, and center position C of the capsule, the VPH was used instead of the traditional physical pinhole detection light path (PPH) in the laser differential confocal sensor (LDCS) to detect the light intensity. The VPH can implement the radial auto-tracking function of the point detector through a software algorithm. By the VPH, the measurement spot at the capsule spherical center successfully enters the point detector of the LDCS, which ensures the focus sensitivity of the optical path and reduces the adjustment difficulty of the focused spot at the spherical center in the light path. In the VPH, the returned measurement beam splits into two components via a 50:50 beam splitter BS1. After passing through, respectively, the microscope objectives Ob1 and Ob2, which are placed with an equal offset M before and after the focus of the lens PL1, the two components enter the CCD1 and CCD2, respectively. The Airy spot near the focal plane of PL1 is magnified by Ob1 and Ob2, and imaged on the CCD image plane in real-time. The sum of the gray values of the pixels ∑I(xv, yv) on the image plane is calculated, and then the differential confocal curve detected by the VPH detection light path is obtained.
The inner and outer profiles, shell uniformity, and shell non-concentricity are measured in a micro-range with nanometer accuracy by the PPH in part C. In the PPH, the returned measurement beam splits into two components via a 50:50 beam splitter BS2. The two components penetrate the pinholes P1 and P2, which are placed with an equal offset M before and after the focus of the lens PL2, and are then detected by the photomultiplier tube.
2.1 Measurement principle of the inner and outer diameters of the ICF capsule
In this study, the inner diameter d and outer diameter D are measured by the VPH. When the focus point of Ob scans near the outer vertex A, the inner vertex B, or the spherical center C of the capsule, the intensity response signal is obtained from the VPH. The differential confocal axial response curve ID(u, uM) obtained through the differential subtraction of curves I1(u, +uM) and I2(u, -uM) is as follows:
As shown in Fig. 3, the differential confocal curves IDA(u, uM), IDB(u, uM), and IDC(u, uM) of points A, B, and C, respectively, are obtained through the VPH. The zero-crossing points of IDA(u, uM), IDB(u, uM), and IDC(u, uM) correspond to the physical position ZA of point A, the optical position ZB of point B, and the physical position ZC of point C, respectively.
According to the geometric relationship, the outer diameter D is given by
When light passes through the capsule shell to the focus point B, the propagation direction changes due to the refraction of the light and thus, ZB is the optical position of point B. The optical shell thickness t of the capsule is
Trace the light to get the physical shell thickness T of the capsule as follows:
whereThe inner diameter d is given by
2.2 Measurement principle of the 2D inner and outer profiles, the shell thickness uniformity, and the shell non-concentricity of the capsule
The measurement principle is shown in Fig. 2, the two-dimensional (2D) section parameters, such as the outer and inner profiles, are measured by rotating around the Y-axis. As the piezoelectric transducer (PZT) scans rapidly along the radial direction of the capsule, the differential confocal signals IDAi (ui, uM) (i = 1,2,3…) and IDBi (ui, uM) (i = 1,2,3…) of the outer vertex A and the inner vertex B, respectively, are obtained by the PPH. The zero-crossing points ZA of IDAi (ui, uM) (i = 1,2,3…) and ZB of IDBi (ui, uM) (i = 1,2,3…) correspond to the physical positions of point A and the optical position of point B, respectively.
As shown in Fig. 4(a), according to the least squares fitting, the fitting center CO(xO, yO) of the outer surface profile is given by
The least squares fitting roundness ɛO of the outer surface can be expressed as
According to Eqs. (3)–(5), the physical shell thickness Ti (i = 1,2,3…) of each sampling point of the section is obtained using the outer vertex physical position coordinates zai (i = 1,2,3…) and the inner vertex optical position coordinates zbi (i = 1,2,3…).
Then, the physical position z’bi(i = 1,2,3…) of the sampling points of the inner surface can be obtained as
As shown in Fig. 4(b), according to Eqs. (7)–(8), the fitting spherical center CI(xI, yI) and the fitting roundness ɛI of the inner surface are obtained.
The shell non-concentricity $\overrightarrow {\boldsymbol{nc}}$ is
The shell unevenness tu of the capsule is
where Timax is the maximum shell thickness and Timin is the minimum shell thickness.2.3 Measurement principle of the 3D inner and outer profiles, the shell thickness uniformity, and the shell non-concentricity of the capsule
The measurement principle is shown in Fig. 2, according to the transposition measurement around the Z-axis, the capsule is driven to rotate φ0 degree to convert the measured section for achieving 3D measurements, where
in which m is the number of the measured sections.As shown in Fig. 5, Γ is a certain measured surface, according to the geometric relationship, the coordinate (xi, yi, zi) of the sampling point Pi on the measured surface Γ is given by
Assuming the equation of the ball is expressed as
According to the 3D measured data, the fitted spherical center CO’ (xO’, yO’, zO’) of the outer surface and the fitted spherical center CI’ (xI’, yI’, zI’) of the inner surface are obtained based on the least squares condition of the sphere.
Then, the least fitting sphericity ɛO’ of the outer surface is given by
According to Eqs. (13)–(15), the fitting sphericity ɛI’ of the inner surface is obtained.
Then, the 3D shell non-concentricity $\overrightarrow {\boldsymbol{n}{\boldsymbol{c}^{\bf ^{\prime}}}} $ of the capsule is obtained by
The shell uneven tu’ of the capsule is obtained based on Eq. (11).
3. Error analysis
3.1 Fixed-focus error caused by the misalignments between the measuring optical axis and the axis of the scanning actuator
The misalignment between the measuring optical axis of the sensor and the actuator scanning axis causes a fixed-focus measurement error on the outer surface vertex or inner surface vertex. As shown in Fig. 6(a), α is the angle between the optical axis and the actuator scanning axis for outer surface measurement, LA and LA’ are the ideal measured values and actual measured values of the sampling points on the outer surface, respectively. As shown in Fig. 6(b), γ is angle between the optical axis and the actuator scanning axis for inner surface measurement, LB and LB’ are the ideal measured values and actual measured values of the sampling points on the inner surface, respectively.
The outer surface vertex measurement error ΔLA and inner surface measurement error ΔLB derived from the geometric relationship shown in Figs. 6(a) and 6(b) are
As the angle α and angle γ can bd controlled within 1° by the careful adjustment, the measurement error ΔLA and ΔLB are negligible.
3.2 Error caused by the offset inconsistency of the two pinholes in the VPH or PPH
The offsets of the two pinholes in the VPH or PPH should be consistent and the optimal normalized value is uM=5.5 [16]. However, in this study, the offsets of the two pinholes are not completely identical after multiple adjustments, which can affect the zero-crossing point position and the fixed-focus resolution of the axial respond curve of the system. Assuming that the normalized deviation of the pinhole before the focus point is uM=5.5, the normalized deviation of the pinhole after the focus point is uMδ. According to Eq. (1), the respond curves and the axial fixed-focus resolution curve within different uMδ are simulated and the results are shown in Fig. 7.
From Fig. 7, inconsistent deviations of the two pinholes can cause the offset of the zero-crossing point of the differential confocal curve and the reduction of the axial fixed-focus resolution. According to Eq. (1), the normalized fixed-focus error of the fixed-focus points of the capsule is
According to Eq. (18), the fixed-focus errors of the fixed-focus points are the same. Because of the relative position measurement of the geometric parameters of the capsule, the inconsistent deviations of the two pinholes can hardly cause the measurement error. However, in order to maximize the axial resolution, the deviation from the focus points of the two pinholes should be made consistent.
3.3 Influence of the outer radius error on the inner surface profile fixed-focus measurement
The outer radius R of all sampling points of the capsule is uneven because of the defect of the outer surface. As it is almost impossible to measure the R of each sampling point, it can cause a small error in the physical shell thickness T. Assuming n0 = 1, the relationship between the outer radius error ΔR = 1 µm and the thickness error ΔT is simulated according to Eqs. (4)–(5).
As shown in Fig. 8, when the optical shell thickness of the capsule is small, the physical shell thickness error ΔT caused by the uneven outer radius ΔR is in the nanometer level, which is negligible.
3.4 Radius measurement error caused by the misalignment between the actual measuring optical axis and the ideal measuring optical axis
Even with the precision adjustment, a residual error still exists between the actual measuring optical axis and the ideal measuring optical axis. As shown in Fig. 9(a), ω is the angle between the actual measuring optical axis and the ideal measuring optical axis. Point A, point B, and point C are the ideal measured points, and point B’ and point C’ are the actual points. According to the geometric relationship, the outer radius measurement error δRω and the inner radius measurement error δrω, caused by ω, can be expressed as
As shown in Fig. 9(b), h is the offset between the actual measuring optical axis and the ideal measuring optical axis. According to the geometric relationship, the outer radius measurement error δRh and the inner radius measurement error δrh, caused by h, are given by
Assuming that R = 400 µm and r = 390 µm, the errors δRω and δrω caused by angle ω are simulated according to Eq. (19). As shown in Fig. 10(a), when ω = 1°, the outer radius error is δRω = 0.061 µm and the inner radius error is δrω = 0.059 µm.
According to Eq. (20), the error δRh and δrh caused by offset h are simulated. As shown in Fig. 10(b), when h = 2 µm, the outer radius error is δRh = 0.005 µm and the inner radius error is δrh = 0.005 µm.
3.5 Rotation measurement error of profiles caused by the misalignment between the actual measuring optical axis and the ideal measuring optical axis
The horizontal offset error h’ between the actual measuring optical axis and ideal measuring optical axis causes a fixed-focus error in the rotational measurement for the inner and outer profiles of the capsule. For the rotational fixed-focus measurement of the outer surface, as shown in Fig. 11(a), points A and A’ are the ideal measured point and the actual measured point, respectively; e is the eccentricity error of the capsule; and ΔA is the fixed-focus error of point A. For the rotational fixed-focus measurement of the inner surface, as shown in Fig. 11(b), points B and B’ are the ideal measured point and the actual measured point, respectively; nc is the shell non-concentricity; and ΔB is the fixed-focus error of point B. According to the geometric relationship, ΔA and ΔB are given by
Assuming that R = 400 µm, r = 390 µm, and nc = 1 µm, the eccentricity error e is set to less than 2 µm after adjustment [18]. According to Eq. (21), when θ =π/2, ΔA reaches its maximum; and when θ =π/2 and φ =π/2, ΔB reaches its maximum. The fixed-focus error caused by h’ is simulated with θ =π/2 and φ =π/2, the results are shown in Figs. 12(a) and 12(b). In general, h’ can be controlled to less than 2 µm after precision adjustment, the fixed-focus errors of point A and point B are ΔA = 0.005 µm and ΔB = 0.010 µm, respectively.
4. Experiments and results
4.1 Measurement system
Based on the proposed method shown in Fig. 2, the schematic diagram of the measurement system is shown in Fig. 13. The system mainly includes the laser interference length measurement system A, laser differential confocal measurement sensor B, rotary shaft system C, rail motion system D, and computer & controller system E.
Before the measurement, the intersection property and coplanar property of the optical axis of LDCS should be ensured by the adjustment.
The intersection property is adjusted with the aid of a calibrated standard oval object. Firstly, a precision inductive sensor (TALYMIN4 Taylor Hobson) is used to compensate the eccentricity of the standard oval object located on the high-precision vertical air-slewing shaft A. When the eccentricity of the oval object is less than 50 nm, the center of the oval object is considered to coincide with the centerline of the axis of the high-precision vertical air-slewing shaft A. Secondly, LDCS was adjusted in the X direction by using the 3D adjustment table A, and used to scan the surface of the oval object until the difference between the maximum value and minimum value of the measured profile data matches the difference between the major axis and the minor axis of the oval object. In this case, the optical axis of LDCS is considered to coincide with the centerline axis of the high-precision vertical air-slewing shaft A.
The coplanar property is adjusted using the coarse aiming CCD. The LDCS moves in the Z direction with the precision air bearing guide, the focus spot of the objective lens Ob at the outer surface vertex and center position of the capsule are observed. Adjusting the position of the LDCS in the Y direction and the position of the capsule in the X direction until both spots are at the center of the CCD, the optical axis of LDCS is considered to coincide with the equatorial plane of the capsule.
Based on the schematic diagram shown in Fig. 13, the measurement system was built as shown in Fig. 14. In the experimental system, the laser interference length measurement system A uses the Renishaw-XL80 laser interferometer. The laser differential confocal measurement sensor B uses a He-Ne laser with a wavelength λ of 632.8 nm as the light source, an objective lens with numerical aperture of 0.8 as the measuring objective, and a PI-P726.1CD as the scan driver. Rotary shaft system C mainly includes the high-precision vertical air-slewing shaft A and high-precision horizontal slewing shaft B. The rail motion system D uses a precision air-bearing guide made of granite. The precision 3D adjustment tables A and B can achieve adjustments of the LDCS and horizontal slewing shaft B, respectively.
Here, the axial resolution Δu and lateral resolution rρ of LDCS can be calculated by Eq. (22).
4.2 Measurement of the outer and inner radiuses of the ICF capsule
Based on the system shown in Fig. 14, the geometric parameters of the capsule can be measured. The experimental conditions are the pressure of 102540 ± 60 Pa, the temperature of 21.0 ± 0.5 °C, and the relative humidity of 44 ± 5%.
As shown in Fig. 15, the differential confocal curves of points A, B, and C are obtained. Based on the zero-crossing points of the curves, the physical position of point A is ZA = 0.170 µm, the optical position of point B is ZB = 9.357 µm, and the physical position of point C is ZC = 384.565 µm. According to Eqs. (2)–(6), the outer diameter is D = 768.790 µm, the physical shell thickness is T = 17.549 µm, and the inner diameter is d = 733.692 µm.
Figure 16 shows the repeated measurements of the inner and outer radiuses of the capsule for 10 times. The average of 10 outer radiuses of the capsule is Ravg = 384.387 µm within a standard deviation of 10 nm, the average value of 10 inner radiuses of the capsule is ravg = 366.841 µm within a standard deviation of 20 nm.
4.3 2D outer and inner profiles, shell thickness uniformity, and shell non-concentricity measurements of the capsule
In general, the LDCS convergence spot radius rρ is approximately the same as its lateral resolution [16]. Then, the number of the sampling point nn for the capsule rotation measurement is
In order to facilitate data processing such as the Gaussian filter on data, we take nn = 4096. Figure 17 shows the results of the outer and inner circular traces of the capsule section. According to Eqs. (7)–(9), the outer roundness is ɛO = 0.942 µm and the inner roundness is ɛI = 0.967 µm after the 1-50 upr Gaussian filtering of the data.
Figure 18 shows the shell thickness of the equatorial section of the capsule. According to Eq. (11), the shell thickness unevenness of the section is 1.511 µm.
Table 1 shows the results of 10 repeated measurements on the ɛO, ɛI, and tu of the section of the capsule.
From Table 1, the average of ɛO is 0.949 µm with a standard deviation of 24 nm; the average of ɛI is 0.941 µm with a standard deviation of 16 nm; and the average of tu is 1.512 µm with a standard deviation of 27 nm.
According to Eq. (7), the fitting center of the outer surface is CO(−0.377, 0.424) and the inner fitting center of the inner surface is CI(−0.063, 0.697). According to Eq. (10), the shell non-concentricity is $\overrightarrow {\boldsymbol{nc}}$(−0.314, −0.273).
Figure 19 shows the repeated measurement results of the shell non-concentricity.
4.4 3D outer and inner profiles, shell uniformity, and the shell non-concentricity measurements of the capsule
Based on the system shown in Fig. 14, the high-precision horizontal shaft B is used to drive the capsule to rotate 10° along the radial direction to measure different sections. Figure 20 shows the measurement results of the inner and outer surface profiles of 18 sections.
According to Eqs. (12)–(15), the fitting spherical center coordinate of the outer surface is CO’(0.063, −0.066, 0.731) and the sphericity of the outer surface is 2.061 µm; the fitting spherical center coordinate of the inner surface is CI’(0.070, −0.086, 0.665) and the sphericity of the inner surface is 2.306 µm.
The 3D shell thickness unevenness is tu’ = 4.895 µm according to Eq. (11) and the 3D shell non-concentricity is $\overrightarrow {\boldsymbol{n}{\boldsymbol{c}^{{\prime}}}} ( - 0.007,0.02,0.066)$ according to Eq. (16).
In the measurement, the movement speed of the precision air bearing guide is set to 500µm/min, the vibration frequency of PZT is 125 Hz, the rotation speed of the high-precision vertical air-slewing shaft A is 1r/min, and the high-precision horizontal slewing shaft B is 1r/min. It takes about 40 minutes to complete the measurement of the outer and inner diameters, 2D inner and outer spherical parameters, and 3D outer and inner spherical parameters.
5. Conclusions
In this study, we proposed a novel, high-precision, and high-efficiency differential confocal method for the multi-geometric parameters of the capsule. Compared with the existing methods, the proposed method demonstrates the following significant advantages:
- (1) High-precision measurements of the outer and inner diameters, outer surface profile, and shell thickness are achieved on the same instrument with data in the same coordinate system for the first time. Based on the measurement results, the inner surface profile is separated by ray tracing, and then the inner surface profile, the shell thickness uniformity, and the shell non-concentricity are simultaneously measured with high precision in the same coordinate system.
- (2) Based on the high spatial resolution of the LDCS and the 4096 high-density sampling, the outer and inner vertexes of the capsule are measured simultaneously, which can improve the measurement accuracy of the inner profile and the shell thickness;
- (3) The multi-geometric parameters of the inner and outer spherical surfaces of the capsule are measured on the same instrument, which improves the measurement efficiency and provides an effective means to select the capsules with a qualified geometric quality from a large number of capsules.
In general, the proposed method provides an effective and novel approach for the high-precision measurement and characterization of the geometric parameters of the capsule.
Funding
National Key Research and Development Program of China Stem Cell and Translational Research (2016YFF0201005); National Natural Science Foundation of China (51535002); China National Funds for Distinguished Young Scientists (51825501).
Disclosures
The authors declare no conflicts of interest.
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