A High-Resolution Measurement Method for Inner and Outer 3D Surface Profiles of Laser Fusion Targets Using a Laser Differential Confocal-Atomic Force Probe Technique

Weiqian Zhao; Zihao Liu; Lirong Qiu

doi:10.1016/j.eng.2024.05.016

Engineering ›› 2024, Vol. 41 ›› Issue (10) :51 -60. DOI: 10.1016/j.eng.2024.05.016

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A High-Resolution Measurement Method for Inner and Outer 3D Surface Profiles of Laser Fusion Targets Using a Laser Differential Confocal-Atomic Force Probe Technique

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Abstract

The high-resolution and nondestructive co-reference measurement of the inner and outer three-dimensional (3D) surface profiles of laser fusion targets is difficult to achieve. In this study, we propose a laser differential confocal (LDC)-atomic force probe (AFP) method to measure the inner and outer 3D surface profiles of laser fusion targets at a high resolution. This method utilizes the LDC method to detect the deflection of the AFP and exploits the high spatial resolution of the AFP to enhance the spatial resolution of the outer profile measurement. Nondestructive and co-reference measurements of the inner profile of a target were achieved using the tomographic characteristics of the LDC method. Furthermore, by combining multiple repositionings of the target using a horizontal slewing shaft, the inner and outer 3D surface profiles of the target were obtained, along with a power spectrum assessment of the entire surface. The experimental results revealed that the respective axial and lateral resolutions of the outer profile measurement were 0.5 and 1.3 nm, while the respective axial and lateral resolutions of the inner profile measurement were 2.0 nm and approximately 400.0 nm. The repeatabilities of the root-mean-square deviation measurements for the outer and inner profiles of the target were 2.6 and 2.4 nm, respectively. We believe our study provides a promising method for the high-resolution and nondestructive co-reference measurement of the inner and outer 3D profiles of laser fusion targets.

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Keywords

Laser fusion targets / Laser differential confocal-atomic force probe / High-resolution / Nondestructive / Co-reference

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Weiqian Zhao, Zihao Liu, Lirong Qiu. A High-Resolution Measurement Method for Inner and Outer 3D Surface Profiles of Laser Fusion Targets Using a Laser Differential Confocal-Atomic Force Probe Technique. Engineering, 2024, 41 (10) : 51-60 DOI:10.1016/j.eng.2024.05.016

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1. Introduction

Inertial confinement fusion (ICF) is one of the primary techniques for investigating clean energy as well as simulating nuclear explosions and astrophysical evolution [1], [2]. In ICF systems, multiple high-energy laser beams are focused onto a hollow spherical target containing deuterium-tritium nuclear fuel, thereby causing the target to implode and achieve fusion ignition. In ICF devices, the spherical hollow target is a critical core component. A recent study published in the journal Nature showed that slight non-uniformities of ICF targets are amplified during the implosion process, resulting in asymmetrically compressed targets [3]. This significantly affects the Rayleigh-Taylor instability during the implosion process, ultimately resulting in failed ignition experiments [4], [5]. The precision of the shapes of the inner and outer profiles of the fusion target is an important factor affecting the results of ICF experiments. In December 2022, the Lawrence Livermore National Laboratory (LLNL) in the United States successfully achieved fusion ignition for the first time [6], [7], [8], signifying a milestone in the decades-long pursuit of clean energy. One of the key factors in this successful ignition was the use of an almost perfect target. The researchers stated that target measurements allowed the prediction of the effects of target asymmetry on implosion, thereby enabling the selection of the optimal target. Moreover, quantitative characterization of the target surface can improve target manufacturing processes [9]. Therefore, high-resolution measurement and analysis of the inner and outer three-dimensional (3D) surface profiles of a target are crucial.

To measure the outer profile of a target, the LLNL team developed the atomic force microscope (AFM) Spheremapper in 1994 [10], [11] with a measurement resolution of 5-10 nm. By combining this with the rotation axis, data pertaining to the outer profile of the target could be obtained. However, nondestructive measurements of the inner profile of the target were not realized. Digital holographic microscopy (DHM) utilizes a charge-coupled device (CCD) to record holographic images generated by interference between the light reflected from the outer surface of the target and the reference light. When DHM was used to measure the outer profile of a target, approximately 70 holograms of the target’s equatorial profile were obtained using the rotation axis and were stitched together to obtain complete equatorial profile data, indicating an axial resolution of 0.1 nm and a lateral resolution exceeding 400 nm [12], [13]. However, bumps on the brink of the images or approximations during the rotation process can introduce errors, and parasitic fringes caused by the reflection of the inner surface also influence the measurement accuracy. The aforementioned methods can effectively detect the outer profile of a target but cannot measure the target’s inner profile nondestructively.

To measure the inner profile of a target, Stephens et al. [14] and Huang et al. [15] from the LLNL used both the interferometric thickness gauge Wallmapper and the AFM profilometer Spheremapper. The outer profile and shell thickness data were subtracted to obtain the inner profile. However, only low-resolution results could be obtained, due to the limit of the optical spot diameter (100 μm) of Wallmapper. Furthermore, the outer profile and thickness data were not guaranteed to originate from the same trajectory. Scanning electron microscopy can be used to measure the inner profile of a target via the interaction of an electron beam with the surface of the target and can achieve a resolution of 10 nm and a relatively large depth of field. However, the target must be dissected and conductively treated before measurement, which renders the measurement destructive [16], [17]. The aforementioned methods cannot simultaneously achieve both nondestructive and high-resolution measurements of the inner profile of a target.

With the aim of obtaining co-reference measurements of the inner and outer profiles of a target, X-ray phase-contrast imaging is currently widely used to measure the profiles of ICF targets. Artyukov et al. [18] used X-ray phase-contrast imaging to characterize the inner and outer 3D profiles of a glass target, but the maximum pixel resolution achieved was only 0.325 μm owing to the limitation imposed by the CCD pixel size. Wang et al. [19], [20] and Ma et al. [21] used laser differential confocal (LDC) microscopy to perform nondestructive and rapid measurements of the inner and outer 3D profiles of a target. This method improved the axial resolution of the confocal microscopy; however, because of optical diffraction limits, the maximum lateral resolution of LDC microscopy for target profile measurement is only 0.345 μm. The aforementioned methods for the co-reference measurement of the inner and outer profiles of a target require improvements in measurement resolution.

In summary, high-resolution and nondestructive co-reference detection of the inner and outer 3D profiles of laser fusion targets remains a challenge. Hence, we propose a high-resolution method using the LDC method and an atomic force probe (AFP) to measure the inner and outer 3D profiles of laser fusion targets. This method detects the deflection of the AFP using the LDC method to achieve high spatial resolution. It utilizes the high spatial resolution of the AFP to improve the spatial resolution of the measured outer profile of the target, thereby achieving high-resolution detection. The inner profile of a target was detected nondestructively using the tomographic characteristics of the LDC method. By reorienting the target several times using a horizontal slewing shaft, high-resolution and nondestructive co-reference detection of the inner and outer 3D profiles of the target was achieved. Furthermore, a power spectrum analysis was conducted on the entire surface, which facilitated a performance analysis of the target.

2. Inner and outer 3D surface profile measurement principle of a laser fusion target

2.1. LDC linear sensing measurement principle

As shown in Fig. 1, after passing through the expander, polarization beam splitter (PBS), and 1/4 wave plate, a parallel beam emitted from the laser source is focused on the measured point through the measurement objective (Ob). The beam reflected from the measured point is split into two by a beam splitter (BS). Subsequently, the split beams pass through pinholes 1 (PH 1) and 2 (PH 2), located on the front and back sides of the focal point of the positive lens (PL), with the same defocusing offset. Finally, these two beams are received by photoelectric multiplier tubes 1 (PMT 1) and 2 (PMT 2), which are positioned behind the corresponding pinholes. When the axial scanner voice coil motor (VCM) drives Ob for axial scanning, PMT 1 and PMT 2 can obtain the corresponding axial response curves of the system—that is,

{IF (u, - u}_{M})

and

{IB (u,+ u}_{M})

. The normalized differential confocal response curve

{ID (u, u}_{M})

against the sample reflectance can be obtained by subtracting the normalized

{IF (u, - u}_{M})

from

{IB (u,+ u}_{M})

, as follows [19]:

(1)

ID (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}}

where u is the normalized axial coordinate, and

u_{M}

is the normalized defocusing offset of the pinhole from the PL focal point. The property of the LDC response curve indicates that the zero-crossing point of

{ID (u, u}_{M})

corresponds precisely to the focal position of Ob, which is the optical coordinate in the z-direction of the measured point [21]. A linear segment can be observed near the zero-crossing point of

{ID (u, u}_{M})

. Utilizing this linear segment enables a high-spatial-resolution, real-time, and rapid LDC linear sensing measurement [22].

2.2. AFP high-resolution measurement principle of the target’s outer profile

The high resolution of the AFP is reflected by the amplification of the atomic interactions between the tip and the measured object using a microcantilever. Measurement of the outer profile of the target is achieved by detecting the deflection of the microcantilever. As shown in Fig. 2, during the contact measurement stage of the AFP, the tip is within the repulsive force range of the outer surface of the measured target. During the measurement process, changes in the surface morphology cause the microcantilever to deflect. The feedback loop maintains a constant microcantilever deflection by propagating the axial scanner to each position [23]. A VCM is used as an axial scanner, which simultaneously carries Ob and the AFP. Before the measurement, a probe-adjustment mechanism is used to move the microcantilever to the vicinity of the focal point of Ob, at which point the microcantilever is located near the zero-crossing point of the LDC linear segment.The VCM is driven to establish contact between the tip and target and to generate a certain amount of deflection, denoted as

b_{0}

, to provide a preload. At this point, the probe reaches the desired state. When the target is rotated, changes in the surface topography causes variations in the repulsive force between the target and tip, thereby changing the deflection

b_{0}

of the microcantilever. The current deflection amount

b_{1}

of the microcantilever is obtained by calculating the normalized differential confocal light intensity returned from the microcantilever. The change in deflection caused by the sample topography (Δ_b) is expressed as follows:

(2)

Δ_{b} = b_{1} - b_{0}

By controlling the VCM to yield

Δ_{b}

, the probe is returned to its initial state. The LDC linear segment has a certain length, and—if the deflection of the microcantilever is small—the VCM does not need to move. Adjustments are only necessary when

| Δ_{b} |

exceeds the threshold

Δ_{m a x}

, while the interaction force between the tip and sample is maintained within a certain range. This ensures that the microcantilever remains within the linear segment. In this case,

Δ_{m a x}

must satisfy the following:

(3)

Δ_{\max} + b_{0} < b_{\max}

where

b_{\max}

represents the maximum allowable deflection of the microcantilever.

b_{\max}

is primarily limited by the physical range of the linear segment, the characteristics of the probe, and the maximum contact force that the sample can withstand. Hence, a continuous adjustment of the VCM is not required to improve the measurement speed. When the target is measured using the aforementioned method, the height data of the measured profile are obtained by fusing the position feedback of the VCM with the normalized optical intensity signal.

2.3. LDC nondestructive measurement principle of the target’s inner profile

As shown in Fig. 3, utilizing the tomographic characteristics of the LDC method [19], the laser beam is focused on the inner surface of the measured target by Ob after passing through the shell. The zero-crossing point of the LDC response curve exhibits the maximum slope and corresponds precisely to the measured point [24]. By utilizing this characteristic, accurate focusing of the inner surface can be achieved. Similarly, the inner profile of the target is detected based on the linear segment of the LDC method [22]. The VCM drives Ob to perform a rapid tracking motion when the target rotates. The height data of the inner profile are obtained by combining the position feedback of the VCM and the normalized LDC intensity. Finally, the inner profile of the target is obtained nondestructively using noncontact measurement.

2.4. Evaluation principle of the power spectrum of the target

Owing to the specific application of the targets, the surface quality is typically characterized using the power spectrum. In this study, the surface topography of the target is measured along a one-dimensional (1D) circular trace, and the power spectrum is defined as the square of the absolute value of the Fourier transform of the height variation along this circular trace [25]. Refs. [25], [26], [27] provide digital representations of the power spectrum of a target surface profile:

(4)

\{\begin{cases} {\hat{P}}_{N} (m) = \frac{1}{N^{2}} {|\hat{Z} (m)|}^{2}, m = 0, N / 2 \\ {\hat{P}}_{N} (m) = \frac{2}{N^{2}} {|\hat{Z} (m)|}^{2}, m = 1, 2, . . ., (N / 2) - 1 \end{cases}

where N represents the number of sampling points, m represents the mode,

{\hat{P}}_{N} (m)

is the digital equivalent of the power spectrum, and

\hat{Z} (m)

denotes the fast Fourier transform of the target profile height data, as expressed in Eq. (5):

(5)

\hat{Z} (m) = \sum_{n = 0}^{N - 1} z (n) e^{- 2 π i m n / N}, - N / 2 \leq m \leq N / 2

where z(n) is the height of the nth sampling point of the measured profile and i represents the imaginary unit.

By applying Parseval’s theorem, we can further derive the relationship between the root-mean-square (RMS) deviation

R_{q}

and the power spectrum, as shown in Eq. (6):

(6)

\sum_{m = 0}^{N / 2} {\hat{P}}_{N} (m) = R_{q}^{2}

The RMS deviation within the mode range from

m_{1}

m_{2}

can be obtained from the corresponding power spectrum, as follows:

(7)

R_{q_{m_{1} \to m_{2}}} = \sqrt{\sum_{m = m_{1}}^{m_{2}} {\hat{P}}_{N} (m)}

Generally, for the measured profile of a target, the surface 1D power spectrum and RMS deviation can be calculated using Eqs. (4), (7), respectively. The power spectrum of the entire 3D surface of the target is represented as the average power spectrum of multiple circular traces.

3. System and experiments

3.1. Measurement system

The measurement system shown in Fig. 4 was constructed based on the measurement principle shown in Fig. 1. The system mainly comprised an LDC-AFP sensor, a rotary shaft system, and a control system. The LDC-AFP sensor used a semiconductor laser (wavelength λ = 405 nm) as the light source. It employed an Ob with a numerical aperture (NA) of 0.50, focal length of 3.60 mm, and magnification factor of 50.00. The probe used was a ContAl-G (manufactured by Budjet Sensors, Bulgaria). In addition, a VCM developed by our group was used as the axial scanner. The VCM featured a traversal range of 5.0 mm and a maximum motion resolution of 0.5 nm. It was equipped with a Heidenhain grating ruler (Germany) with a picometer-level position resolution to provide closed-loop position feedback. The VCM was driven by an ACS controller (Israel), and its highest frequency response for a 500 nm traversal distance was 100 Hz. The VCM simultaneously carried Ob and the AFP and provided a rapid tracking motion during measurement. The rotation system included vertical air-slewing shaft A and horizontal slewing shaft B. An eccentricity adjustment mechanism was used such that the eccentricity between the target and shaft A could be adjusted rapidly and automatically before measurement. 3D adjustment table A was used to adjust the position of the LDC-AFP sensor, whereas 3D adjustment table B was used to align the suction nozzle on shaft B with the target. The system used a high-speed multifunction data-acquisition card (NI USB6353, USA), with a 16-bit analog-to-digital converter to synchronously acquire the LDC optical signal and VCM position feedback. The acquired data were processed and analyzed using a computer to obtain the inner and outer 3D profile data of the target.

3.2. Analysis

3.2.1. Spatial resolution of sensor

The theoretical spatial resolution of the sensor in LDC mode can be calculated using Eq. (1) and the differential confocal response curve. When the laser wavelength λ is 405 nm and the NA of the Ob lens is 0.50, the theoretical axial resolution of LDC mode can be expressed as [19]

(8)

Δ u = \frac{λ}{3.32 \times {NA}^{2} \times SNR} = \frac{405}{3.32 \times 0 . 50^{2} \times 300} nm \approx 1.6 nm

where SNR is the signal-to-noise ratio.

The theoretical lateral resolution of LDC mode can be expressed as

(9)

Δ v = \frac{0.436 λ}{NA} = \frac{0.436 \times 405}{0.50} nm \approx 353.2 nm

where v is the normalized lateral coordinate.

In the AFP mode, the theoretical axial resolution is based on Eq. (8). Because of the metallic layer on the back of the microcantilever, the reflectivity can be enhanced to improve the SNR, thereby increasing the axial resolution of the AFP mode. The position of the AFP tip relative to the LDC optical axis affects the axial resolution of the AFP, as shown in Fig. 5. Under the action of force F, the microcantilever undergoes elastic deflection; in this case, the deflection at the end of the microcantilever is expressed as [28]

(10)

d (L) = \frac{L^{3}}{3 E I} F

where d is the deflection of the microcantilever, L is the length of the microcantilever, E is the Young’s modulus of the microcantilever, and I is the moment of inertia of the microcantilever. If the LDC optical focal point at the back of the microcantilever deviates from the tip in the y-direction, that is, when L₁ < L (L₁ is the distance in the y-direction from the optical axis to the fixed end of the microcantilever), the measured cantilever deflection (d₁) is less than d. At this point, the axial resolution of the AFP decreases. Therefore, before measurement, the tip must be adjusted to the optical axis as much as possible.

The theoretical lateral resolution of the AFP can be represented by the contact radius of the tip. Based on the Hertzian theoretical model, the contact radius of the tip (a) is defined as [29]

(11)

a = {(F R / K)}^{1 / 3}

where F can be obtained from the microcantilever’s elastic coefficient of 0.2 N·m⁻¹ and a preload of 200 nm; and R is the radius of curvature of the tip, which is 10 nm in this study. The elastic constant K is expressed as follows:

(12)

K = 4 E_{p} / [3 (1 - v_{p}^{2})]

where E_p is the Young’s modulus of the tip and ν_p is the Poisson’s ratio of the tip. For the tip used in this study, which was fabricated using single-crystal silicon, E_p = 130 GPa and ν_p = 0.21. Therefore, Eq. (11) yields a theoretical value of 1.3 nm for the lateral resolution of the AFP.

3.2.2. Co-reference measurement of the outer and inner profiles

In the system shown in Fig. 4, the measurement of the outer and inner profiles of the target in the same section only requires switching between the AFP and LDC modes of the sensor. In other words, the target does not need to be shifted, which ensures a unified measurement reference. As shown in Fig. 5, a misalignment between the AFP tip axis and the LDC optical axis can result in deviations in the sections of the measured outer and inner profiles. Before measurement, the tip of the AFP was adjusted to the optical axis as much as possible using a microscope. At this point, the deviation of the tip from the optical axis did not exceed 15 μm. For a target with a diameter of 853.312 μm, the measured outer and inner profiles can be considered to be in the same section.

3.3. Experiments

3.3.1. Feasibility of the LDC-AFP method

To verify the feasibility of the LDC-AFP method, a standard elliptical cylinder calibrated by the National Institute of Metrology (NIM) was measured. The calibration roundness provided by the NIM was 1.7 μm. Figs. 6(a) and (b) respectively show the results of ten repeated profile measurements under the AFP mode and its roundness repeatability with an average value of 1.7084 μm and a standard deviation of 7.9 nm. Figs. 6(c) and (d) respectively show the results of repeated measurements under the LDC mode and its roundness repeatability with an average value of 1.7133 μm and a standard deviation of 7.2 nm. The roundness of the standard elliptical cylinder obtained using the LDC-AFP method was consistent with the calibration result, and the repeatability was favorable. This verifies the feasibility of this method.

3.3.2. Sensor spatial resolution test

To measure the axial resolution of the AFP mode, the VCM drove the AFP to approach and establish contact with a planar reflector. Once the probe reached a predetermined position, the VCM performed a rapid step motion while simultaneously obtaining optical intensity signals from the two PMTs. Similarly, to measure the axial resolution of the LDC mode, the microcantilever was first shifted to a more distant location. Subsequently, Ob was driven by the VCM to focus on the planar reflector. The VCM propagated via rapid step movements, and the differential confocal intensity signal was simultaneously obtained. As shown in Fig. 7(a), the axial response signals of the AFP and LDC modes under a 0.5 nm step size differed significantly. The signal of the AFP mode displayed a clear step, indicating that its axial resolution can reach 0.5 nm, whereas the LDC mode signal could not distinguish each step. Fig. 7(b) presents the axial response of the LDC mode signal with a 2.0 nm step size, indicating that the axial resolution of the LDC mode can reach 2.0 nm.

To verify the lateral resolution of the sensor, a standard step sample (STEP-OX-0.5) calibrated by NIM was measured. The step height was 500 nm, with a period of 5 μm. The rising width of the step calibrated using a commercial AFM by members at the NIM was 1.067 μm, which was considered inherent. The measurement from the AFP mode, as shown in Fig. 7(c), yielded a measured rising width of approximately 1.079 μm, whereas the measurement from the LDC mode, as shown in Fig. 7(d), showed an rising width of approximately 1.447 μm. We concluded that the measurement result obtained in the AFP mode was closer to the calibrated value. In addition, the lateral resolution of the LDC mode was approximately 400 nm, which was limited by the optical diffraction limit. The AFP method described herein significantly enhanced the lateral resolution of the sensor by utilizing the high spatial resolution of the probe.

3.3.3. Repeated measurements of the inner and outer 1D profiles of a target

First, the outer profile of a target with a 853.312 μm diameter and 15.227 μm shell thickness (measured using the LDC mode) was measured at high resolution using the AFP mode with ten repeated measurements (16 384 samples). Subsequently, the probe was shifted to a more distant location, and the inner profile of the same cross-section was measured for ten noncontact repeated measurements using the LDC optical probe under the same conditions (8192 samples). Figs. 8(a)-(c) present the height data, 1D power spectrum curve, and RMS deviations of the ten repeated measurements of the outer profile, and Figs. 8(d)-(f) show the corresponding data for the inner profile.The first harmonic component of the power spectrum was caused by the eccentricity of the target mounting and was excluded from the analysis. Because the power spectrum is related to the square of the RMS deviation, the repeatability of the profile RMS deviation—excluding the first harmonic component—was calculated to intuitively demonstrate the stability of the method. The standard deviations of the RMS deviation for the ten measurements of the outer and inner profile were 2.6 and 2.4 nm, respectively, thus confirming the favorable repeatability of this method.

3.3.4. Inner and outer 3D profile measurement of a target

The inner and outer 3D profiles of the target were measured using a meridian-parallel trace scheme, as shown in Fig. 1, based on an equidistant meridian measurement scheme. After the major circular trace in each meridian direction was measured, the target was translated up and down in the y-direction at uniform intervals to measure a set of parallel lines. After the meridian measurements were complete, the target was rotated by 90° to add a set of “belt” trace measurements, thus achieving the measurement of the inner and outer 3D profiles of the target. During the measurement, the eccentricity of the target was first adjusted to not exceed 0.3 μm. For the same circumference trace, the outer profile of the target was first densely sampled using the AFP. Subsequently, the probe was moved further away to detect the inner profile nondestructively using the LDC optical probe. The high-precision 3D adjustment table A was used to adjust the sensor in the y-direction, where an offset of 45 μm was ensured between each meridian line and the maximum circular meridian line. After completing one set of meridian and parallel line measurements, the target was repositioned using the precise horizontal slewing shaft B shown in Fig. 4, with a rotation angle of 10°. Finally, 19 sets of 57 trace lines were measured for both the inner and outer 3D profile measurements of the target. Figs. 9(a) and (b) show the results of the 3D profile measurements of the outer and inner surfaces, respectively. Figs. 9(c) and (d) show the 9th and 18th outer meridian line measurement results, respectively, and Figs. 9(e) and (f) show the 9th and 18th inner meridian line measurement results, respectively. The least-squares sphericities of the measured outer and inner surfaces of the target were 0.9596 and 1.0643 μm, respectively.

3.3.5. Power spectrum evaluation of entire inner and outer surfaces of a target

Using Eq. (4), the mode-power spectrum curves of each meridian and parallel line were calculated based on the surface profile data of the target presented in Section 3.3.4. The average mode-power spectrum curve of the target surface can be obtained by averaging the power spectrum values of each meridian and parallel line. Figs. 9(g) and (h) show the average power spectrum curves of the outer and inner surfaces, respectively. The power spectrum curves of the measured target exhibited high disturbance amplitudes in the low-mode range. The low-mode part (modes 2-10) of the power spectrum is primarily influenced by the sphericity error. The calculated RMS deviation values in the range of modes 2-10 were 103.9 nm (outer surface) and 145.1 nm (inner surface), validating the measurement results with significant sphericity error. Both the outer and inner surface power spectra exhibited significant fluctuations around mode 150, indicating that the surface quality of the target near this mode is relatively poor. This also implies a strong correlation between the surface morphologies of the outer and inner surfaces. The RMS deviation within modes 101-8192 for the outer surface was 8.9 nm, whereas that for the inner surface within modes 101-4096 was 21.3 nm.

4. Uncertainty evaluation

4.1. Uncertainty of the axial measurement

Based on the experiments described in Section 3.3.2, the proposed method exhibits inherent uncertainty in axial measurements. The uncertainties in the measurements of the outer and inner surfaces of the target can be represented using the axial resolutions of the AFP and LDC modes, respectively. The axial measurement uncertainty of the AFP mode (Δu_a) was 0.5 nm, and the axial focusing uncertainty of the LDC mode (Δu_b) was 2.0 nm.

4.2. Uncertainty introduced by the focusing position of the LDC optical probe on the microcantilever

Based on Section 3.2.1, and as shown in Fig. 5, if the fixed focus of the LDC optical probe on the back of the microcantilever deviates from the tip in the y-direction, that is, L₁ < L, then the measured deflection, d₁ < d, introduces a measurement error. To compensate for this error, we obtained the LDC response curve before the profile measurement. First, we adjusted the cantilever to the defocused state and then drove the VCM to make the AFP probe contact the target and generate deflection. In this way, an LDC response curve was obtained. We found that a linear function could be used to satisfactorily fit ID and the measured deflection. Finally, this error was compensated for by calibrating the proportional coefficient between the LDC-normalized light-intensity signal and the measured deflection.

4.3. Uncertainty introduced by the radial rotation error of shaft A

The precision of the vertical air-slewing shaft A in the measurement system shown in Fig. 4 is one of the key factors ensuring measurement accuracy during the target’s 3D profile measurements, as its rotation error directly influences the measurement. To determine the effect of this error on the measurements, a standard glass hemisphere (Taylor Hobson, UK) was used as a calibration tool. The standard glass hemisphere was placed on shaft A, and, after eccentric adjustment, the rotation of the measurement ring was measured using a TALYMIN4 inductive sensor (Taylor Hobson, UK). The least-squares roundness of the measurement ring was obtained, which can be considered to be the radial rotation error of shaft A. To verify the rotational stability of shaft A, the radial rotational error was measured ten times; the results are listed in Table 1.

The radial rotation error of shaft A was a stable systematic error. After the circular trace of the target was measured, the rotation error was separated using data processing,thereby eliminating the measurement uncertainty introduced by this error.

4.4. Uncertainty introduced by the environment and system noise

Measurement errors can be introduced during the measurement process owing to various factors such as stray environmental light, detector noise, airflow disturbances, and workspace vibrations. The errors introduced by these factors can be considered random errors. Fig. 10 shows the noise in the air sampling when the LDC optical probe was focused on the microcantilever, indicating an overall noise level of approximately 1 nm.

4.5. Composite uncertainty

The aforementioned error sources are independent of each other, and the composite uncertainty associated with the measurement of the target’s outer surface (Δ_outer) can be expressed as

(13)

Δ_{outer} = \sqrt{Δ u_{a}^{2} + Δ_{s}^{2}} \approx 1.1 nm

where Δ_s is the uncertainty introduced by the environment and system noise.

The composite uncertainty of the measurement associated with the target’s inner surface (Δ_inner) can be expressed as

(14)

Δ_{inner} = \sqrt{Δ u_{b}^{2} + Δ_{s}^{2}} \approx 2.2 nm

5. Conclusions

To conclude, an LDC-AFP measurement method was proposed to achieve high-resolution and nondestructive co-reference detection of the inner and outer 3D profiles of laser fusion targets. This method enhances the spatial resolution of the outer surface measurement by exploiting the high spatial resolution of the AFP. The deflection of the AFP is monitored via an LDC method. Nondestructive detection of the inner surface in the same section is achieved using the tomographic characteristics of the LDC method.

The experimental results revealed that the axial resolution of the AFP mode for the outer surface profile measurement was 0.5 nm, whereas the lateral resolution was 1.3 nm. The axial resolution of the LDC mode for measuring the inner surface profile was 2.0 nm, and the lateral resolution was approximately 400.0 nm. The repeatabilities of 1D profile RMS deviation measurements for the outer and inner surfaces of the target were 2.6 and 2.4 nm, respectively. The 3D profile data of the outer and inner surfaces of the target were obtained, and a power spectrum evaluation was performed on the outer and inner surfaces. The low-mode (2-10) RMS deviations of the outer and inner surface were 103.9 and 145.1 nm, respectively, which is consistent with the sphericity measurement. The high-mode (> 101) RMS deviations of the outer and inner surfaces were 8.9 and 21.3 nm, respectively. Our study provides a feasible method for high-resolution and nondestructive co-reference measurements of the inner and outer 3D profiles of laser fusion targets.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (52327806 and U22A6006).

Compliance with ethics guidelines

Weiqian Zhao, Zihao Liu, and Lirong Qiu declare that they have no conflict of interest or financial conflicts to disclose.

References

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