Laser differential confocal inner-surface profile measurement method for an ICF capsule

Longxiao Wang; Lirong Qiu; Weiqian Zhao; Xianxian Ma; Shaobai Li; Dangzhong Gao; Jie Meng; Qi Wang

doi:10.1364/OE.25.028510

1. Introduction

Laser inertial confinement fusion (ICF) triggers nuclear fusion reaction by the bombardment of a high-energy laser beam on a capsule containing deuterium tritium fuel [1–3]. Owing to the direct contact between the deuterium tritium fuel and the inner surface, the effects of very small defects of the inner surface may be amplified during laser fusion ignition, leading to asymmetric compression or rupture of the ICF capsule, thereby resulting in laser fusion ignition failure [4]. Therefore, the shape precision of the inner profile of a laser fusion capsule is one of the key factors that determine the success or failure of an ICF experiment. To satisfy the explosion condition in the capsule, the inner surface profile needs to be 0.1~1.0 μm [5], which satisfies the high requirements for inner surface profile measurements of ICF capsules.

In US Nova Nuclear fusion devices, the inner surface profile of the capsule is measured using scanning electron microscopy (SEM) [6], but the capsule must be cut open before the measurement. Therefore, this method does not meet the screening inspection measurement requirement for ICF capsules.

In order to realize the nondestructive measurement of capsules, the Lawrence Livermore National Laboratory (LLNL) adds a fiber interferometer wall thickness measurement sensor in the diameter position of the Atomic Force Microscope (AFM) probe of an AFM SphereMapper [7,8], and this enables the inner surface profile of the capsule to be indirectly obtained by measuring the outer profile and shell thickness along the same contour section. However, during the actual measurement and alignment, it is difficult to ensure that the AFM probe and optical fiber interferometric wall thickness measurement sensor probe have a completely overlapping measurement trajectory, and the spot diameter of the optical fiber white light interferometer that is used is about 100 μm; therefore, its lateral resolution is low. Further, it is unable to satisfy the high-precision measurement requirements for the inner surface profile of the capsule.

In order to improve the lateral resolution of the capsule’s inner surface profile measurement, D. H. Edgell et al. employed backlit shadowgraph using the digital ray tracing simulation software (SHELL 3D) to simulate the shadow image of the capsule’s inner surface, and they used different directions of the target back light shadow graph to achieve the 3D reconstruction [9]. K. Wang et al. used the X-ray phase-contrast imaging method based on the strong penetration characteristics of X-rays to measure the inner surface profile of ICF targets [10]. However, the resolution was only 1.3 μm, and it was unable to measure the 3D inner surface profile of the capsule.

Therefore, it is challenging to obtain the non-contact inner surface sectioning measurement of the capsule with high spatial resolution during the screening inspection of ICF capsules.

Confocal microscopy has a lateral resolution that is 1.4 times better than ordinary optical measurement technology due to the imaging principle of its point illumination and point detection, and it has a unique longitudinal sectioning capability in the optical measurement field. It is therefore widely used in many sectioning measurement fields [11]. However, the existing confocal microscopy has the lowest sensitivity because it directly uses the extreme point position of its Sinc ‘bell-shaped’ intensity curve to do sectioning triggering. In order to optimize the axial resolution of confocal microscopy system, the differential confocal measurement method is proposed to improve the axial resolution of confocal microscopy systems to 1 nm [12–14]. However, when it is directly used to measure the inner and outer surface profiles of the capsule, the sensitivity of its differential confocal characteristic curve is susceptible to changes in the surface reflectance during the capsule rotation process, so it cannot be directly used for the high-precision measurement of the capsule’s inner surface profile.

Therefore, this paper proposes a new laser differential confocal inner surface profile measurement method for ICF capsules with high axial resolution and lateral resolution, which is insensitive to changes in the surface reflectivity of the capsule. The proposed method uses the cross-zero point of the differential confocal curve to precisely measure the inner and outer surface profiles on one section of the capsule, and rotates the capsule through the high-precision aerostatic rotary shaft and auxiliary rotary shaft, thereby achieving the high-precision and nondestructive 3D measurement of the capsule’s inner surface profile. The proposed method provides a new approach to the high-precision measurement of the inner surface profile of ICF capsules.

2. Laser differential confocal inner surface profile measurement principle of ICF capsule

2.1 Laser differential confocal measurement principle

The laser differential confocal inner surface profile measurement principle of the ICF capsule is shown in Fig. 1. The principle is based on the property that the zero point of the normalized differential confocal intensity curve I_D(u,u_M) in a laser differential confocal system precisely corresponds to the focus of the objective, which uses the cross-zero point of the normalized axial differential confocal curve I_D(u,u_M) precisely to focus and measure the inner and outer surface profiles of the capsule at the center of the rotary spindle equatorial section point-by-point at equal intervals. In addition, it obtains the inner surface profile through the corrections and compensation by the inner and outer diameters as well as the refractive index of the capsules. Then, it rotates the auxiliary rotary shaft to drive the capsule at angle of transposition, θ, along the z-axis. At this position of angle θ, it measures the inner and outer surface profiles of the capsule at the center of the rotary spindle equatorial section point-by-point at equal intervals. Finally, it reconstructs the 3D contour of the capsule’s inner surface using the inner-profile data obtained at different rotation angle positions, and obtains the precise 3D inner surface profile of the ICF capsule.

Fig. 1 Laser differential confocal inner surface profile measurement principle of ICF capsule.

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2.2 Radial focusing property of laser differential confocal ICF capsule measurement

The focusing process at one point of the inner and outer surfaces of the capsule obtained using the laser differential confocal system is shown in Fig. 1. The light emitted from the laser is expanded through the expander, and is then focused on the measured capsule by the objective after passing through the polarized beam splitter (PBS) and a quarter-wave plate. When either the inner or outer surface of the capsule is near the focus of the objective, the measurement beam containing the height information of the capsule’s inner or outer surface is reflected to again pass through the objective and quarter-wave plate, and it is sent as a focusing beam by emerging lens L after being reflected by PBS. The focusing beam is divided into two beams by the beam splitter (BS), and they enter the respective pinholes P₁ and P₂ placed with an equal offset before and after the focus of the converging lens. Then they are detected by detectors D₁ and D₂ to obtain the corresponding axial response curves I₁(u, + u_M) and I₂(u,-u_M). The differential confocal axial response curve I(u,u_M) obtained through the differential subtraction of curves I₁(u, + u_M) and I₂(u,-u_M) is [12]:

\begin{array}{l} I (u, u_{M}) = {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u + u_{M} / 2)} ρ d ρ d θ |}^{2} - {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u - u_{M} / 2)} ρ d ρ d θ |}^{2}, \\ = {[\frac{\sin ((2 u + u_{M}) / 4)}{(2 u + u_{M}) / 4}]}^{2} - {[\frac{\sin ((2 u - u_{M}) / 4)}{(2 u - u_{M}) / 4}]}^{2} \end{array}

where,

u = \frac{2π}{λ} \cdot N A^{2} \cdot z, and u_{M} = \frac{2π}{λ} \cdot N A^{2} \cdot M .

Here u is the axial normalized optical coordinate, M is the offset of detector from the focus of lens L, u_M is the axial normalized offset M of the detector, z is the axial optical coordinate, λ is the laser wavelength, N.A. is the numerical aperture of the objective.

In order to eliminate the influence of the changes in the reflectivity of the capsule surface, the normalized process is performed for two intensity signals I₁(u, + u_M) and I₂(u,-u_M), and the spherical differential confocal axial response I_D(u,u_M) that is obtained with anti-reflectivity is [13]:

I_{D} (u, u_{M}) = \frac{I_{B} (u, + u_{M}) - I_{B} (u, - u_{M})}{I_{B} (u, + u_{M}) + I_{B} (u, - u_{M})} = \frac{{[\frac{\sin ((2 u + u_{M}) / 4)}{(2 u + u_{M}) / 4}]}^{2} - {[\frac{\sin ((2 u - u_{M}) / 4)}{(2 u - u_{M}) / 4}]}^{2}}{{[\frac{\sin ((2 u + u_{M}) / 4)}{(2 u + u_{M}) / 4}]}^{2} + {[\frac{\sin ((2 u - u_{M}) / 4)}{(2 u - u_{M}) / 4}]}^{2}} .

The slope at the zero point is obtained by differentiating Eq. (3) with respect to u, and the calculated axial resolution Δu is:

Δ u = \frac{λ}{2 π \cdot N A^{2} \cdot S N R \cdot | \frac{\partial I_{D} (u, u_{M})}{\partial u} |_{u = 0}} .

Here SNR is the signal-to-noise ratio of the detector. When λ = 405 nm, N.A. = 0.80, SNR = 200:1, the axial differential curve I_D(u,u_M) and axial resolution curve obtained by Eqs. (3) and (4) are as shown in Fig. 2 for different offset u_M.

Fig. 2 The property curves for different offset u_M. (a) Laser differential confocal axial response curve I_D(u,u_M). (b) Axial resolution curve Δu.

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From Fig. 2, it can be seen that the slope of curve I_D(u,u_M) at the zero point first increases and then decreases with the axial offset u_M of the detector. When u_M = 12.57, the axial resolution calculated using Eq. (4) is 4.57 × 10⁻⁷ μm, which is the highest; however, the linear range of the differential confocal curve is very small. Considering the linear range and axial resolution, the axial offset used is u_M = 5.5, and the anti-reflectivity differential confocal curve obtained using Eq. (3) is shown in Fig. 3. So, the normalized linear range is u ≈ ± 1.4, and the linear range calculated by Eq. (2) is z ≈ ± 0.14 μm.

Fig. 3 Anti-reflectivity differential confocal curve when u_M = 5.5.

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The axial resolution calculated using Eq. (4) is [14]:

Δ u = \frac{λ}{3.32 \times N A^{2} \times S N R} = \frac{0.405}{3.32 \times {0.80}^{2} \times 200} μm \approx 9.52 \times 10^{- 4} μm .

The lateral resolution of the differential confocal system is [14]:

Δ v_{1} = \frac{0.436 λ}{N A} = \frac{0.436 \times 0.405}{0.80} μm \approx 0.221 μm .

Therefore, the axial resolution of the laser differential confocal system is 1 nm, and the lateral resolution is 0.22 μm, and they satisfy the measurement requirements of the ICF capsule.

2.3 Sectioning measurement of radial position for inner and outer surfaces of ICF capsule

As shown in Fig. 1, when the objective moves with the piezoelectric ceramic micro-displacement actuator along the radial direction of the capsule, the differential confocal focusing curves I_DA(u,u_M) and I_DB(u,u_M) near point A of the inner surface and point B of the outer surface are obtained, and the optical shell thickness is the difference between positions Z_A and Z_B, which correspond to the zeroes of the curves I_DA(u,u_M) and I_DB(u,u_M).

T_{0} = Z_{B} - Z_{A} .

As shown in Fig. 4, when the focusing beam is used to identify point B at the inner surface, the measurement ray is deflected in the shell owing to the influence of the refractivity n of the capsule shell. Then, the geometrical shell thickness T_avg of the capsule is calculated by ray tracing as follows [15].

T_{a v g} = \frac{\int_{0}^{\arcsin (N A)} T (n, R, T_{0}, β) d β}{arc \sin (N A)},

where,

Fig. 4 Calculation of shell thickness using the ray-tracing technique.

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T (n, R, T_{0}, β) = R + \frac{1}{n} \times \frac{\sin β \times (T_{0} - R)}{\sin [β + arc \sin (\frac{T_{0} - R}{R} \times \sin β) - arc \sin (\frac{1}{n} \times \frac{T_{0} - R}{R} \times \sin β)]} .

Here n is the refractive index of the capsule shell, R is the radius of the capsule’s outer surface, T₀ is the optical thickness, β is the numerical aperture angle of the measuring beam.

The position Z_C corresponding to point B on the capsule’s inner surface is calculated using position Z_A on the capsule’s outer surface and the calculated shell thickness T_avg.

Z_{C} = Z_{A} + T_{a v g} .

2.4 Laser differential confocal 3D profile measurement of inner surface of ICF capsule

As shown in Fig. 1, when the ICF capsule rotates with the air shaft around the rotating shaft, the objective of the laser differential confocal sensor (LDCS) rapidly scans using a piezoelectric ceramic micro-displacement actuator along the radial direction of the capsule. Let the sampling points of the sensor on the first circle profile be N, and the data D₁(i) at the ith sampling point is obtained using Eqs. (7)–(10). Then, the data array of sample points at the first circle is {D₁(1),⋅⋅⋅⋅⋅⋅⋅, D₁(i),⋅⋅⋅⋅⋅⋅⋅, D₁(N)}.

The Fourier expansion of the circular contour sampling data {D₁(i)} is shown in Eq. (11) [16].

D_{1} (i) = D_{10} + \sum_{k = 1}^{N - 1} [p_{k} \cos (2 k i π / N) + q_{k} \sin (2 k i π / N)] .

where i = 1, 2,…N, and N is the number of sampling points of the inner profile, k is the harmonic frequency, and D₁₀, p_k and q_k are the coefficients of the Fourier series.

Removing the DC component D₁₀ and the first harmonic component that resulted from the capsule eccentricity adjustment, i.e., k ≥ 2, then

D_{1}^{'} (i) = D_{1} (i) - D_{10} - [p_{1} \cos (2 i π / N) + q_{1} \sin (2 i π / N)],

where,

{p_{1} = \frac{2}{N} \sum_{i = 1}^{N} D_{1} (i) \cos (2 i π / N), q_{1} = \frac{2}{N} \sum_{i = 1}^{N} D_{1} (i) \sin (2 i π / N), D_{10} = \frac{1}{N} \sum_{i = 1}^{N} D_{1} (i)} .

D₁₀ is the DC component in the capsule’s inner surface measurement data, p₁ and q₁ are the coefficients of the first harmonic component, and D₁′(i) is the inner surface profile data of the capsule for roundness evaluation using the minimum zone circle (MZC) method [17,18].

As shown in Fig. 1, when the inner surface profile measurement is completed for the circumferential capsule equatorial position, the capsule turns by rotation angle γ = π/w to the second capsule’s circle section of the equatorial position using a rotating system with an air-assisted orthogonal rotary spindle, after which the inner profile is measured. As shown in Fig. 5, after the second section profile of the capsule is measured, in turn, the capsule turns to the next measured cross-section profile in the equatorial position, and its inner surface profile is measured until the w circle section surface profile data are obtained, which are required for the sphericity evaluation of the capsule’s inner surface profile.

Fig. 5 3D profile measurement scheme of the capsule’s inner surface.

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To obtain the 3D profile of the capsule’s inner surface, data processing and 3D reconstruction are then performed using the w capsule circumferential cross section measured in the surface profile data. In addition, the sphericity of the inner profile in the capsule is evaluated by the MZC.

3. Analysis of measurement error of the capsule’s inner profile

In the inner profile measurement of an ICF capsule, the main influence factors include LDCS error, spherical aberration and astigmatism, capsule eccentricity adjustment error, precision spindle error, misalignment error between the optical axis and the capsule center, misalignment error between the optical axis and the axis of the scanning actuator, and the inclination angle of the working table.

However, in the assembly of the measurement system, the inclination angle of the working table and the misalignment errors between the axes can be controlled within a very small value by the careful adjustment, so their effects on the measurements can be negligible.

3.1 LDCS error

The LDCS system uses the zero-crossing point of the differential confocal curve to identify the position of the capsule’s inner and outer surfaces in the capsule radial direction. The LDCS resolution at the focus is 1 nm in the axial direction and 0.22 μm in the lateral direction, so the measurement range and linearity of the LDCS system are mainly determined using the scanning system of the nanometer-precision objective lens.

3.2 Effect of spherical aberration and astigmatism on focusing on inner surface

As shown in Fig. 4, when the measurement beam is focused on point B at the capsule’s inner surface, the outer surface and the refractive index of the shell affect the focusing accuracy at point B, and the resulting aberration of the laser differential confocal focusing system is expressed according to Seidel formula as [19]:

ϕ (ρ, θ) \approx A_{040} ρ^{4} + A_{022} ρ^{2} \cos^{2} θ + A_{120} ρ^{2} \dots

Where A₀₄₀ is the primary spherical aberration coefficient, A₀₂₂ is the astigmatism coefficient and A₁₂₀ is the curvature of the field coefficient.

The primary spherical aberration A₀₄₀ρ⁴ and astigmatism A₀₂₂ρ²cos²θ affect the focusing property of the laser differential focusing system, and the analyses are as follows.

1) Effect of the primary spherical aberration A₀₄₀ρ⁴ at point B on the inner surface

The axial differential confocal response with primary spherical aberration A₀₄₀ρ⁴ is obtained using Eq. (3), and

\begin{array}{l} I_{D B} (u, u_{M}, 0, ϕ) \\ = \frac{{| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u + u_{M} / 2)} \cdot e^{- 4 π j A_{040} ρ^{4} / λ} ρ d ρ d θ |}^{2} - {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u - u_{M} / 2)} \cdot e^{- 4 π j A_{040} ρ^{4} / λ} ρ d ρ d θ |}^{2}}{{| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u + u_{M} / 2)} \cdot e^{- 4 π j A_{040} ρ^{4} / λ} ρ d ρ d θ |}^{2} + {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u - u_{M} / 2)} \cdot e^{- 4 π j A_{040} ρ^{4} / λ} ρ d ρ d θ |}^{2}} . \end{array}

Using Eqs. (4) and (15), Fig. 6 shows the axial differential confocal intensity curves and axial resolution curve at point B of the capsule’s inner surface for different primary spherical aberration coefficient values A₀₄₀.

Fig. 6 Effect of primary spherical aberration. (a) Focusing curves. (b) Axial resolution.

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From Fig. 6(a), it can be seen that the primary spherical aberration A₀₄₀ρ⁴ results in the axial offset of the zero position of the differential confocal curves, and the axial offset of the zero position Δ₁ is calculated using Eqs. (2) and (15). Figure 6(b) shows that the axial focusing resolution decreases as the primary spherical aberration coefficient A₀₄₀ increases, and the axial focusing resolution is 1.53 × 10⁻³μm when the primary spherical aberration coefficient A₀₄₀ = 0.75λ.

Therefore, the shape accuracy of the capsule out profile is higher because the effect of the primary spherical aberration A₀₄₀ρ⁴ on the focusing accuracy of the inner surface is smaller.

2) Effect of primary astigmatism A₀₂₂ρ²cos²θ at point B on the inner surface

The axial differential confocal response with primary astigmatism A₀₂₂ρ²cos²θ is obtained using Eq. (3), and

\begin{array}{l} I_{D B} (u, u_{M}, 0, ϕ) \\ = \frac{{| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u + u_{M} / 2)} \cdot e^{- 4 π j A_{022} ρ^{2} c o s^{2} θ / λ} ρ d ρ d θ |}^{2} - {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u - u_{M} / 2)} \cdot e^{- 4 π j A_{022} ρ^{2} c o s^{2} θ / λ} ρ d ρ d θ |}^{2}}{{| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u + u_{M} / 2)} \cdot e^{- 4 π j A_{022} ρ^{2} c o s^{2} θ / λ} ρ d ρ d θ |}^{2} + {| \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} e^{j ρ^{2} (u - u_{M} / 2)} \cdot e^{- 4 π j A_{022} ρ^{2} c o s^{2} θ / λ} ρ d ρ d θ |}^{2}} . \end{array}

Using Eqs. (4) and (16), Fig. 7 shows the axial differential confocal intensity curves and axial resolution curve at point B of the capsule’s inner surface for different values of primary astigmatism coefficient A₀₂₂.

Fig. 7 Effect of primary astigmatism. (a) Focusing curves. (b) Axial resolution.

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From Fig. 7(a), it can be seen that the primary astigmatism A₀₂₂ρ²cos²θ results in changes in the zero position and shape of the differential confocal curves, and the axial offset of the zero-crossing position Δ₂ is calculated by Eqs. (2) and (16). Figure 7(b) shows that the axial focusing resolution decreases as the primary astigmatism coefficient A₀₂₂ increases, and the axial focusing resolution is 6.79 × 10⁻³ μm when the primary astigmatism coefficient A₀₂₂ = 0.50λ.

Therefore, the shape accuracy of the capsule out profile is higher as the effect of primary astigmatism A₀₂₂ρ²cos²θ on the focusing accuracy of the inner surface is decreased.

In the actual measurement, the primary goals of the capsule profile measurement are to measure the outer profile of the capsule and to screen the capsule, so the effect of the shape of the outer profile on the focusing accuracy of the inner surface will decrease.

3.3 Capsule eccentricity adjustment error

The capsule center and the center line of the rotation system are not perfectly aligned, so the capsule’s eccentricity adjustment error is not to be omitted from the inner-profile measurement, and its effect on the focusing position of the inner-profile sampling points is as shown in Fig. 8.

Fig. 8 Analysis of effect of capsule eccentricity adjustment error.

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In Fig. 8, O is the gyration center, O′ is the center position of the capsule at the first sampling point of the inner profile, O_i is the center position of the capsule for rotation angle φ_i, W is the zero-reference point, Z_W is the zero point of detection, x₁oy₁ is the coordinate system for the eccentricity adjustment, xoy is the measurement coordinate system, A_i is the theoretical position of the sampling point, A_i′ is the measured position of the sampling point, and ΔA_i is the focusing error that results from the eccentricity.

The focusing error ΔA_i at the ith sampling point of the inner profile can be derived from the geometrical relation shown in Fig. 8, and is given as:

Δ A_{i} = r - \frac{r \cos {θ + φ_{i} + arc \sin [\frac{b}{r} \cos (θ + φ_{i})]}}{\cos (θ + φ_{i})} .

The error Δt_i of the optical thickness calculated by the difference between the inner and outer surfaces at the ith sampling point is:

\begin{array}{l} Δ t_{i} = R - r - \frac{R \cos {θ + φ_{i} + arc \sin [\frac{b}{R} \cos (θ + φ_{i})]}}{\cos (θ + φ_{i})} . \\ + \frac{r \cos {θ + φ_{i} + arc \sin [\frac{b}{r} \cos (θ + φ_{i})]}}{\cos (θ + φ_{i})} \end{array}

where R and r are the outer radius and inner radius of the capsule, respectively, φ_i is the rotation angle of the capsule at the ith sampling point of the inner surface profile (φ_i = 2πi/N), N is the number of sampling points of the inner profile, θ is the angle between OO′ and its projection on the y axis, and b is the eccentricity adjustment error.

3.4 Precision spindle error

In the precision spindle error, the radial rotation error has a greater influence on the measurement accuracy of the inner profile of the capsule. Therefore, a high-precision air bearing rotating shafting with a radial gyration error better than 25 nm is used to construct the measurement system, and the radial gyration error of the precision spindle is negligible relative to the measurement error of the capsule’s inner surface profile.

4. Experiments

4.1 Measurement system

Based on the laser differential confocal inner surface profile measurement principle of the ICF capsule shown in Fig. 1, the established laser differential confocal inner surface profile measurement system for the ICF capsule is as shown in Fig. 9(a).

Fig. 9 (a) Laser differential confocal inner-surface profile measurement system for ICF capsule. (b) LDCS.

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The laser differential confocal inner surface profile measurement system for the ICF capsule is mainly composed of a LDCS, precision air adjustment / rotation system, zonal auxiliary rotary system for the capsule, 3D adjustment Table 1 for the LDCS, 3D adjustment Table 2 for the zonal auxiliary rotary system, and a computer that measures and controls the system. From the computer measurement and control system, each component finishes the automatic adjustment of the eccentric capsule, the automatic profile measurement of the equatorial cross section, the automatic adjustment for the capsule zonal auxiliary rotary, and the processing and evaluation of measurement data for the inner profile.

As shown in Fig. 9(b), a semiconductor laser with λ = 405 nm is used in the LDCS. The objective that was used is an objective lens with N.A. = 0.80, and the objective actuator is P725.4CD (PI), which has a travel range of 400 μm and a resolution of 0.5 nm.

4.2 Test experiments for the sensor property

The LDCS objective is driven to rapidly scan the test reflector along the axial direction by the objective actuator P725.4CD. The measured LDCS axial property curve is shown in Fig. 10(a), and the LDCS axial resolution curve for the axial scanning feed step of 1 nm is shown in Fig. 10(b).

Fig. 10 Test of the LDCS properties. (a) Axial focusing curve. (b) Axial resolution.

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As shown in Fig. 10(a), in order to reduce the influence of random noise on the focusing accuracy in the inner profile measurement, the linear range of 0.31 μm of the LDCS axial property curve is fitted with a straight line, and the fitting line used to precisely identify the cross-zero position of the LDCS axial property curve. From Fig. 10(b), it can be seen that the LDCS axial resolution is better than 1 nm, satisfying the requirements of the inner-surface profile measurement for the ICF capsule.

4.3 3D profile measurement experiments for the capsule’s inner surface

An ICF capsule that was screened early was placed on the measurement device shown in Fig. 9(a), which has an outer diameter of D = 1000 μm, an inner diameter of d = 800 μm, and a shell refractive index of n = 1.4. First, the precision air stage was used for the automatic aligning and rotary motion of the capsule, and then the laser differential confocal sensor was used to rapidly focus and measure the equatorial section along the radial direction of the capsule. The number of sampling points on the circumference was set to N = 1024, and the measured profile data within 1024 points on the circumference were processed using Eqs. (7)–(13), and the obtained inner-surface profile data curve of the capsule’s equatorial section is shown in Fig. 11.

Fig. 11 Measured inner-profile data curve of capsule equatorial section.

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Using Eqs. (2), (15) and (16), the offset between the focus and the converging point on the inner surface, which resulted from the shape and profile of the capsule, was calculated to be 0.22 μm. The capsule has been screened, so its shape and profile have almost the same influence on the focal offset at each sampling point of the inner surface, and this component can be used as a DC component to be removed in the inner-profile measurement of the capsule.

First, the measured data shown in Fig. 11 is expanded in the Fourier series using Eq. (11). The calculated eccentricity error is b = 0.4421 μm, and the measurement error ΔA_i at each sampling point of the inner profile is calculated by substituting b into Eq. (17). Then, the error compensation on the measured data shown in Fig. 11 is done point-by-point, resulting in precise inner-profile data, as shown in Fig. 12(a). Finally, the measurement data shown in Fig. 12(a) is evaluated by the MZC, and the result is shown in Fig. 12(b), where the wave number of the standard Gauss filter used is 1-50 upr.

Fig. 12 (a) Inner- profile data after compensation. (b) Roundness evaluation result.

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To further verify the stability of the measurement system, the inner-surface profile of the equatorial section was measured 10 times, resulting in the roundness measurement results and profiles shown in Fig. 13, where the average roundness is 0.6475 μm with a standard deviation of 0.0153 μm.

Fig. 13 Repeated measurements of the inner-surface profile. (a) Roundness evaluation results. (b) Shape of inner profile.

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To achieve sphericity in the target inner profile, the zonal auxiliary rotation system shown in Fig. 9 rotates the capsule by 10° along the zonal direction of the capsule according to the path shown in Fig. 5. This makes the capsule inner profile of the second tested section lie on the equatorial position, and the inner profile is then measured. In turn, the capsule turns to the next measured cross-section profile in the equatorial position, and its inner surface profile is measured until the inner-profile data of the 18 circumferential sections are obtained, which are required for the sphericity evaluation of the capsule’s inner surface. Then, the MZC method was used to assess the sphericity of the capsule’s inner profile, and Fig. 14 shows the 3D reconstructed profile of the capsule’s inner surface when the Gauss wavenumber used for the filter is 1-50 upr.

Fig. 14 3D reconstructed profile of the capsule’s inner surface.

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From Fig. 14, it can be seen that the assessed sphericity of the capsule inner profile is 1.248 μm and the eccentricity is 0.462 μm, where the eccentricity in direction x is −0.073 μm, the eccentricity in direction y is 0.136 μm and the eccentricity in direction z is −0.435 μm. In Fig. 14, the color represents the relative height of the inner-surface profile, and the color bar from top to bottom represents the gradual decrease in the relative height.

5. Conclusions

In this paper, a new laser differential confocal inner-profile measurement method is proposed for ICF capsules. The proposed method has a high axial resolution and insensitivity to the surface reflectivity, and a laser differential confocal inner-profile measurement system was developed based on this method. Experiment results indicate that for the first time, the developed measurement system achieves high accuracy, non-contact, and nondestructive 3D measurement for the capsule’s inner surface profile, and the repeated measurement accuracy can reach 15 nm in the inner surface profile measurement. This method provides an effective approach for achieving the high-precision nondestructive measurement of 3D inner surface profiles in laser fusion targets.

Funding

National Natural Science Foundation of China (No. 51422501 and 61475020); National Key Research and Development Program of China (No. 2016YFF0201005).

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Laser differential confocal inner-surface profile measurement method for an ICF capsule

Abstract

1. Introduction

2. Laser differential confocal inner surface profile measurement principle of ICF capsule

2.1 Laser differential confocal measurement principle

2.2 Radial focusing property of laser differential confocal ICF capsule measurement

2.3 Sectioning measurement of radial position for inner and outer surfaces of ICF capsule

2.4 Laser differential confocal 3D profile measurement of inner surface of ICF capsule