High-precision laser differential confocal measurement method for multi-geometric parameters of inner and outer spherical surfaces of laser fusion capsules

Xianxian Ma; Lirong Qiu; Yun Wang; Mengshuang Lu; Weiqian Zhao

doi:10.1364/OE.387201

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.

Fig. 1. Parameters of the capsule.

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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.

Fig. 2. Light path of the high-precision laser differential confocal measurement for the multi-geometric parameters of the capsule.

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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(x_v, y_v) 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 I_D(u, u_M) obtained through the differential subtraction of curves I₁(u, +u_M) and I₂(u, -u_M) is as follows:

(1)$$\begin{aligned} &{I_D}(u,{u_M}) = {I_1}(u, + {u_M}) - {I_2}(u, - {u_M})\\ &= |\frac{1}{\pi }\int_0^{2\pi } {\int_0^1 {{p_1}(\rho ){p_2}(\rho )} {e^{j{\rho ^2}(2u + {u_M})/2}}} \rho d\rho d\theta {|^2}\\ & - \left|\frac{1}{\pi} \int_{0}^{2 \pi} \int_{0}^{1} p_{1}(\rho) p_{2}(\rho) e^{j \rho^{2}\left(2 \pi * k_{2}\right) / 2} \rho \mathrm{d} \rho \mathrm{d} \theta\right|^{2} \end{aligned}$$

where ρ₁(ρ, θ) and ρ₂(ρ, θ) are the pupil functions of the Ob and PL₁, respectively, ρ and θ are the radial normalized radius of the pupil and the polar angle, respectively, u is the axial normalization optical coordinate of the capsule, and u_M is the optical normalized axial offset of the detector.

As shown in Fig. 3, the differential confocal curves I_D_A(u, u_M), I_D_B(u, u_M), and I_D_C(u, u_M) of points A, B, and C, respectively, are obtained through the VPH. The zero-crossing points of I_D_A(u, u_M), I_D_B(u, u_M), and I_D_C(u, u_M) correspond to the physical position Z_A of point A, the optical position Z_B of point B, and the physical position Z_C of point C, respectively.

Fig. 3. Measurement principle of the outer and inner diameters of the capsule.

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According to the geometric relationship, the outer diameter D is given by

(2)$$D = 2 \times ({Z_A} - {Z_C})$$

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, Z_B is the optical position of point B. The optical shell thickness t of the capsule is

(3)$$t = {Z_A} - {Z_B}$$

Trace the light to get the physical shell thickness T of the capsule as follows:

(4)$$T = \frac{{\int_0^{\arcsin (NA)} {T(n,R,t,\beta )d\beta } }}{{\arcsin (NA)}}$$

where

(5)$$T(n,R,t,\beta ) = R + \frac{{\frac{{{n_0}}}{n} \times \sin \beta \times (t - R)}}{{\sin (\beta + \arcsin (\frac{{t - R}}{R} \times \sin \beta ) - \arcsin (\frac{{{n_0}}}{n} \times \frac{{t - R}}{R} \times \sin \beta ))}}$$

in which β is the half-value aperture angle of Ob, n₀ is the refractive index of air, n is the refraction index of the shell thickness, and R is the outer radius of the capsule.

The inner diameter d is given by

(6) $$d = D - 2T$$

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 I_D_Ai (u_i, u_M) (i = 1,2,3…) and I_D_Bi (u_i, u_M) (i = 1,2,3…) of the outer vertex A and the inner vertex B, respectively, are obtained by the PPH. The zero-crossing points Z_A of I_D_Ai (u_i, u_M) (i = 1,2,3…) and Z_B of I_D_Bi (u_i, u_M) (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 C_O(x_O, y_O) of the outer surface profile is given by

(7)$$\left\{ {\begin{array}{l} {{x_O} = \frac{2}{{nn}}\sum\limits_{i = 1}^{nn} {(\Delta {R_i}\cos {\theta_i})} }\\ {{y_O} = \frac{2}{{nn}}\sum\limits_{i = 1}^{nn} {(\Delta {R_i}\sin {\theta_i})} } \end{array}} \right.$$

where nn is the rotation measurement sampling points of the section, and ΔR_i is the height variation of the i-th sampling point on the outer surface.

Fig. 4. Measurement principle of the 2D (a) outer profile and (b) inner profile of the capsule.

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The least squares fitting roundness ɛ_O of the outer surface can be expressed as

(8)$$\begin{aligned} {\varepsilon _O} &= \max (\sqrt {(\Delta {R_i}\cos {{{\theta _i} - {x_o})}^2} - (\Delta {R_i}\sin {{{\theta _i} - {y_o})}^2}} ) \\ &- \min (\sqrt {(\Delta {R_i}\cos {{{\theta _i} - {x_o})}^2} - (\Delta {R_i}\sin {{{\theta _i} - {y_o})}^2}} ) \end{aligned}$$

According to Eqs. (3)–(5), the physical shell thickness T_i (i = 1,2,3…) of each sampling point of the section is obtained using the outer vertex physical position coordinates z_ai (i = 1,2,3…) and the inner vertex optical position coordinates z_bi (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

(9)$${z_{bi}}^{\prime} = {z_{ai}} - {T_i}(i = 1,2,3\ldots )$$

As shown in Fig. 4(b), according to Eqs. (7)–(8), the fitting spherical center C_I(x_I, y_I) and the fitting roundness ɛ_I of the inner surface are obtained.

The shell non-concentricity $\overrightarrow {\boldsymbol{nc}}$ is

(10)$$\overrightarrow {\boldsymbol{nc}} = ({x_O} - {x_I},{y_O} - {y_I})$$

The shell unevenness t_u of the capsule is

(11)$${t_{\boldsymbol{u}}} = {T_{i\max }} - {T_{i\min }}$$

where T_i_max is the maximum shell thickness and T_i_min 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

(12)$${\varphi _0} = \frac{\pi }{m}$$

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 (x_i, y_i, z_i) of the sampling point P_i on the measured surface Γ is given by

(13)$$\left\{ \begin{array}{l} {x_i} = {\rho_i}\cos ({\alpha_i})\cos ({\beta_i})\\ {y_i} = {\rho_i}\cos ({\alpha_i})\textrm{sin(}{\beta_i})\\ {z_i} = {\rho_i}\sin ({\alpha_i}) \end{array} \right.$$

where ρ_i is the polar coordinate radius of the point P_i, α_i is the angle between OP_i and the x-axis on the plane Γ, and β_i is the angle between the plane Γ and the plane xoz.

Fig. 5. Measurement principle of the 3D inner and outer profiles of the capsule.

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Assuming the equation of the ball is expressed as

(14)$${(x - {x_c})^2} + {(y - y{}_c)^2} + {(z - {z_c})^2} = {R_b}^2$$

According to the 3D measured data, the fitted spherical center C_O’ (x_O’, y_O’, z_O’) of the outer surface and the fitted spherical center C_I’ (x_I’, y_I’, z_I’) of the inner surface are obtained based on the least squares condition of the sphere.

Then, the least fitting sphericity ɛ_O’ of the outer surface is given by

(15)$$\begin{aligned} &{\varepsilon _O}^{\prime} = \max [\sqrt {{{({x_i} - {x_O}^{\prime})}^2} + {{({y_i} - {y_O}^{\prime})}^2} + {{({z_i} - {z_O}^{\prime})}^2}} ]\\ & - \min [\sqrt {{{({x_i} - {x_O}^{\prime})}^2} + {{({y_i} - {y_O}^{\prime})}^2} + {{({z_i} - {z_O}^{\prime})}^2}} ]\end{aligned}$$

According to Eqs. (13)–(15), the fitting sphericity ɛ_I’ 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

(16)$$\overrightarrow {\boldsymbol{n}{\boldsymbol{c}^{\prime}}} = ({x_O}^{\prime} - {x_I}^{\prime},{y_O}^{\prime} - {y_I}^{\prime},{z_O}^{\prime} - {z_I}^{\prime})$$

The shell uneven t_u’ 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, L_A and L_A’ 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, L_B and L_B’ are the ideal measured values and actual measured values of the sampling points on the inner surface, respectively.

Fig. 6. The analysis for the fixed-focus measurement error of (a) outer surface vertex and (b) inner surface vertex caused by the misalignment between the optical axis of the sensor and actuator scanning axis.

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The outer surface vertex measurement error ΔL_A and inner surface measurement error ΔL_B derived from the geometric relationship shown in Figs. 6(a) and 6(b) are

(17)$$\left\{ \begin{array}{l} \Delta {L_A} = {L_A}^{\prime} - {L_A} = {L_A}(1 - \cos \alpha )\\ \Delta {L_B} = {L_B}^{\prime} - {L_B} = {L_B}(1 - \cos \gamma ) \end{array} \right.$$

As the angle α and angle γ can bd controlled within 1° by the careful adjustment, the measurement error ΔL_A and ΔL_B 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 u_M=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 u_M=5.5, the normalized deviation of the pinhole after the focus point is u_Mδ. According to Eq. (1), the respond curves and the axial fixed-focus resolution curve within different u_Mδ are simulated and the results are shown in Fig. 7.

Fig. 7. Effects of the offset inconsistency of the two pinholes on (a) the response curves and (b) the axial resolution.

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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

(18)$${\delta _A} = {\delta _B} = {\delta _C} = ({u_M} - {u_{M\delta }})/4$$

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.

Fig. 8. Simulated results of different shell refractive index n of the capsule with (a) R = 400 µm and (b) R = 800 µm.

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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 δ_Rω and the inner radius measurement error δ_rω, caused by ω, can be expressed as

(19)$$\left\{ \begin{array}{l} {\delta_{R\omega }} = {R^{\prime}} - R = \frac{R}{{\cos \omega }} - R\\ {\delta_{r\omega }} = {r^{\prime}} - r = \frac{R}{{\cos \omega }} - R\cos \omega + \sqrt {{r^2} - {R^2}{{\sin }^2}\omega } - r \end{array} \right.$$

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 δ_Rh and the inner radius measurement error δ_rh, caused by h, are given by

(20)$$\left\{ \begin{array}{l} {\delta_{Rh}} = \sqrt {{R^2} - {h^2}} - R\\ {\delta_{rh}} = \sqrt {{r^2} - {h^2}} - r \end{array} \right.$$

Fig. 9. Radius measurement error caused by (a) the angle ω and (b) the offset h.

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Assuming that R = 400 µm and r = 390 µm, the errors δ_Rω and δ_rω caused by angle ω are simulated according to Eq. (19). As shown in Fig. 10(a), when ω = 1°, the outer radius error is δ_Rω = 0.061 µm and the inner radius error is δ_rω = 0.059 µm.

Fig. 10. Simulated results of the radius measurement error caused by (a) the angle ω and (b) the offset h.

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According to Eq. (20), the error δ_Rh and δ_rh caused by offset h are simulated. As shown in Fig. 10(b), when h = 2 µm, the outer radius error is δ_Rh = 0.005 µm and the inner radius error is δ_rh = 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 Δ_A 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 Δ_B is the fixed-focus error of point B. According to the geometric relationship, Δ_A and Δ_B are given by

(21)$$\left\{ \begin{array}{l} {\Delta _A} = \sqrt {{R^2} - {{(e\sin \theta - h)}^2}} - \sqrt {{R^2} - {{(e\sin \theta )}^2}} \\ {\Delta _B} = \sqrt {{r^2} - {{(e\sin \theta - h + nc\sin \varphi )}^2}} - \sqrt {{r^2} - {{(e\sin \theta + nc\sin \varphi )}^2}} \end{array} \right.$$

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, Δ_A reaches its maximum; and when θ =π/2 and φ =π/2, Δ_B 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 Δ_A = 0.005 µm and Δ_B = 0.010 µm, respectively.

Fig. 11. Effects of the misalignment between the actual measuring optical axis and the ideal measuring optical axis on the rotational fixed-focus measurement of (a) the outer surface and (b) the inner surface.

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Fig. 12. Fixed-focus error simulation results of (a) the outer surface and (b) the inner surface.

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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.

Fig. 13. Schematic diagram of capsule multi-geometric parameters measurement system.

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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.

Fig. 14. High-precision and high-integration differential confocal measurement system for multi-geometric parameters of the ICF capsule.

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Here, the axial resolution Δu and lateral resolution r_ρ of LDCS can be calculated by Eq. (22).

(22)$$\left\{ \begin{array}{l} \Delta u = \frac{\lambda }{{2\pi \cdot N{A^2} \cdot SNR \cdot {{\left|{\frac{{\partial {I_D}(u,{u_M})}}{{\partial u}}} \right|}_{u = 0}}}}\\ {r_\rho } = \frac{{0.436\lambda }}{{NA}} \end{array} \right.$$

where, SNR is the signal-to noise ratio of the detector. When λ = 632.8 nm, N. A. = 0.8, SNR = 200:1, the axial resolution is Δu = 1.48 nm, the lateral resolution r_ρ = 0.345 µm.

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 Z_A = 0.170 µm, the optical position of point B is Z_B = 9.357 µm, and the physical position of point C is Z_C = 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.

Fig. 15. The differential confocal fixed-focus curves of the capsule.

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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 R_avg = 384.387 µm within a standard deviation of 10 nm, the average value of 10 inner radiuses of the capsule is r_avg = 366.841 µm within a standard deviation of 20 nm.

Fig. 16. Repeated measurement results of (a) the outer radius and (b) the inner radius.

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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 r_ρ is approximately the same as its lateral resolution [16]. Then, the number of the sampling point nn for the capsule rotation measurement is

(23)$$nn \ge \pi r/{r_\rho }$$

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 ɛ_O = 0.942 µm and the inner roundness is ɛ_I = 0.967 µm after the 1-50 upr Gaussian filtering of the data.

Fig. 17. Outer and inner circular traces of the equatorial section of the capsule.

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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.

Fig. 18. Shell thickness of the equatorial section of the capsule.

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Table 1 shows the results of 10 repeated measurements on the ɛ_O, ɛ_I, and t_u of the section of the capsule.

Table 1. Results of 10 repeated measurement of the section of the capsule.

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From Table 1, the average of ɛ_O is 0.949 µm with a standard deviation of 24 nm; the average of ɛ_I is 0.941 µm with a standard deviation of 16 nm; and the average of t_u is 1.512 µm with a standard deviation of 27 nm.

According to Eq. (7), the fitting center of the outer surface is C_O(−0.377, 0.424) and the inner fitting center of the inner surface is C_I(−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.

Fig. 19. Repeated measurement results of (a) the center fitting results of the outer and inner surfaces and (b) the scatter plots of the shell non-concentricity.

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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.

Fig. 20. 3D measurement results of (a) the outer surface, (b) the inner surface, (c) the vertical 3D section profile and (d) the horizontal 3D section profile.

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According to Eqs. (12)–(15), the fitting spherical center coordinate of the outer surface is C_O’(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 C_I’(0.070, −0.086, 0.665) and the sphericity of the inner surface is 2.306 µm.

The 3D shell thickness unevenness is t_u’ = 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.

References

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Results	1	2	3	4	5	6	7	8	9	10
ɛ_o (µm)	0.942	0.952	0.923	0.957	0.919	0.964	0.957	0.966	0.995	0.912
ɛ_I (µm)	0.967	0.920	0.948	0.958	0.951	0.944	0.934	0.933	0.942	0.913
t_u (µm)	1.511	1.480	1.505	1.536	1.493	1.534	1.466	1.508	1.518	1.564

High-precision laser differential confocal measurement method for multi-geometric parameters of inner and outer spherical surfaces of laser fusion capsules

Abstract

1. Introduction

2. High-precision laser differential confocal measurement method for the geometric parameters of the ICF capsule

2.1 Measurement principle of the inner and outer diameters of the ICF capsule

2.2 Measurement principle of the 2D inner and outer profiles, the shell thickness uniformity, and the shell non-concentricity of the capsule

2.3 Measurement principle of the 3D inner and outer profiles, the shell thickness uniformity, and the shell non-concentricity of the capsule

3. Error analysis

3.1 Fixed-focus error caused by the misalignments between the measuring optical axis and the axis of the scanning actuator

3.2 Error caused by the offset inconsistency of the two pinholes in the VPH or PPH

3.3 Influence of the outer radius error on the inner surface profile fixed-focus measurement

3.4 Radius measurement error caused by the misalignment between the actual measuring optical axis and the ideal measuring optical axis

3.5 Rotation measurement error of profiles caused by the misalignment between the actual measuring optical axis and the ideal measuring optical axis

4. Experiments and results

4.1 Measurement system

4.2 Measurement of the outer and inner radiuses of the ICF capsule

4.3 2D outer and inner profiles, shell thickness uniformity, and shell non-concentricity measurements of the capsule

4.4 3D outer and inner profiles, shell uniformity, and the shell non-concentricity measurements of the capsule

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (20)

Tables (1)

Equations (23)

Optics Express