ICF target DT-layer refractive index and thickness from iterative analysis

Tianliang Yan; Dong Liu; Xue Shen; Yuhao Zhou; Zhongming Zang; Tu Shi; Chong Liu

doi:10.1364/OE.26.017781

1. Introduction

In the ICF (inertial confinement fusion) system, the target is the core where all laser beams converge to induce nuclear fusion [1]. To achieve the implosion condition, the solid DT (Deuterium-Tritium)-layer contained in a hollow polymer target must be formed highly uniform and smooth. Parameters of the DT-layer of the target should be strictly controlled, such as uniformity, concentricity, sphericity, surface roughness, etc [2-3]. In that case, the implementation of its quality control is very important to the ignition, which requires a non-destructive, accurate and rapid characterization of DT-layer’s thickness and refractive index as the first step.

As the target is a multi-layer sphere that consists of polymer shell, DT ice and fuel gas, serious deflection will occur as the collimated light passes through. While the size of the target is between the macro and micro, generally ranging from hundreds of microns to a few millimeters, it is difficult to measure the refractive index and thickness of the DT-layer with high precision. Several approaches have been proposed to solve this problem, such as interferometric methods [4–6] and backlit shadowgraph [7–10]. Both of them are non-contact, non-destructive and fast, and can obtain the refractive index or thickness of the DT-layer by analyzing the effect of the light passing through. In interferometric methods, the deformed wave front is employed to obtain the information of the DT-layer. While in backlit shadowgraph, the energy redistribution after refractions and/or reflections is employed to obtain the DT-layer information. In either method, however, the value of refractive index or thickness must be first assumed to obtain the other, and large error will be yielded then.

An exception to this issue occurs in X-ray microradiography, where the thickness measurement of DT-layer can be decoupled from the layer’s refractive index. Unlike in the backlit shadowgraph, the position of the bright ring in the X-ray image of the target is very close to the surface position of the DT-layer, which means that a sensitive measure of the layer thickness can be obtained and is largely unaffected by variations in refractive index. However, to obtain the X-ray image of the target, it takes more time than the interferometric methods or backlit shadowgraph, which makes the measurement vulnerable to environmental vibration and degrades the test accuracy [11,12].

In this paper, an iterative algorithm based on optical path difference (OPD for short) and ray deflection is proposed to obtain the refractive index and thickness of the DT-layer simultaneously and rapidly. In one test, the OPD of different light rays is employed, and the interference fringe obtained contains the information of the refractive index and thickness of the DT-layer. In the other test, the ray deflection caused light intensity redistribution is employed, and the intensity image received on a CCD (Charge-Coupled Device) is analyzed. An obvious bright ring can be observed on the backlit shadowgraph and the bright ring also contains information of the refractive index and thickness of the DT-layer. Considering the two tests, either contains information of the refractive index and thickness of the DT-layer, but neither can solve the two unknowns alone. However, by combining the two tests together, the proposed iterative algorithm can solve the two unknowns precisely and simultaneously.

The iteration can begin from either the refractive index measurement or the thickness measurement with either the interference test or the backlit shadowgraph test. In the iteration beginning from the refractive index measurement with the interference, an initial value of the DT-layer thickness is estimated, and a refractive index value is then solved based on the OPD information obtained from the interference map. The obtained value is substituted as the estimated value of the refractive index to solve the DT-layer thickness based on ray deflection with the backlit shadowgraph test. These steps will be repeated to find the fixed point in the iteration process. To terminate the algorithm, a stopping criterion is set as $| (n_{i} - n_{i - 1}) / n_{i} | < tolerance$ , $| (t_{i} - t_{i - 1}) / t_{i} | < tolerance$ [13]. Where $n_{i}$ and $t_{i}$ are the refractive index and thickness in the $i^{th}$ iteration cycle, while $n_{i - 1}$ and $t_{i - 1}$ are that in the ${(i - 1)}^{th}$ iteration cycle. The tolerance is set as 1/10 of the relative retrieval uncertainties so that the variations of iteration results won’t influence the retrieval results, and the minority retrieval uncertainties of the DT-layer refractive index and thickness is 0.05%, which is detailed in Section 5. Thus, the tolerance is set as 0.005%. When the refractive index and thickness converge within the given tolerance, the iteration cycles break. Simulations show that, the relative retrieval accuracy of the DT-layer refractive index is better than 0.05% after finite converging steps, and that of the thickness is better than 0.1%. The feasibility of the proposed iterative algorithm was verified by experiments. The shell refractive index and thickness can be retrieved with a relative error below $\pm 2 %$ compared with the reference values got with the interferometric method and the X-ray microradiography respectively. The test uncertainties from experiments were also analyzed.

The paper is constructed as follows: Section 2 describes the principle of the proposed iterative algorithm. In Section 3 and 4, the results of simulation and laboratory experiments are given to demonstrate the feasibility of the proposed algorithm. Section 5 gives some test uncertainty analysis. Some concluding remarks are drawn in Section 6.

2. Principle

The system diagram of the proposed iterative algorithm is shown in Fig. 1, which contains two test configurations: the interference test configuration and the backlit shadowgraph test configuration. In the interference test, the laser is on, and the collimated laser beam is split into two: one goes straight into the reference arm, and the other is reflected into the test arm, where there is a target. The two beams are combined by the beam splitter 2, and the interference fringes are then imaged onto the CCD. The position of the target is corrected so that the back surface of the target is on the conjugate plane of the CCD [14]. Turn off the laser and turn on the LED (Light-Emitting Diode) to switch to the backlit shadowgraph test. The collimated LED beam passes through the aperture and the target, and the longitudinal section of the target is then imaged onto the CCD by the imaging lens, after the position of the target has been adjusted. The reference arm should be blocked when the backlit shadowgraph test is carrying out to prevent the effects of the background light. The apertures are placed in front of the LED and the Beam expander of the laser for the spot size control of the light sources. In experiments, both the spot sizes of the laser and the LED are adjusted to 4 times larger than the target outer diameter. The split ratios of the beam splitters and reflectivities of the mirrors are well selected to ensure optimum visibility of fringe pattern.

Fig. 1 System diagram of the proposed iterative algorithm.

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The interference map and the backlit shadowgraph of the target can be obtained with the interference test and backlit shadowgraph test, respectively. With the obtained interference map and the backlit shadowgraph, the refractive index and thickness of the target DT-layer can be solved simultaneously employing the proposed iterative algorithm. Other parameters of the target (the outer radius of the target $R_{0}$ , the refractive index of the shell $n_{1}$ , and the thickness of the shell $t_{1}$ ) are assumed known. Details on the measurement principle of the two tests are discussed in the following two subsections.

2.1 Refractive index measurement based on OPD (interference)

In this subsection, the principle of refractive index measurement based on the OPD is discussed. As seen in the sub-figure of Fig. 1, a typical interference map from a real target is displayed, consisting by a series of concentric rings. The DT-layer’s thickness and refractive index information is encoded in the ring diameters of the recorded interference map and can be deduced by ray tracing techniques. The detailed trace of one light ray in the test arm of the interference test is shown in Fig. 2. Here, $n_{0}$ , $n_{1}$ , $n_{2}$ , and $n_{3}$ are the refractive indexes of the corresponding media (usually $n_{0} = 1$ , $n_{3} \approx 1$ , $n_{1}$ is about 1.54 and $n_{2}$ is about 1.13 for ICF targets [10]), respectively. m is the imaging system magnification. The radius of the outer shell is $R_{0}$ , $t_{1}$ and $t_{2}$ are the corresponding thicknesses of the layers, respectively, and x is the incident height of one light ray.

Fig. 2 Diagram of refractive index measurement based on OPD.

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Since the CCD is on the conjugate plane of the back surface of the target, the wave front at the CCD is the same as that just leaves the target [13]. The optical path lengths of the fringes (dark rings for example) differ by an amount $k λ$ , where k is the number difference between the two fringes. The optical path of a perpendicularly incident light ray propagating through the symmetric target can be obtained through the ray tracing on the basis of geometry relationships and Snell’s law. The ray is identified by its incident height x, and in Fig. 2 the trace of one light ray with incident height of x ( $0 \leq | x | < R_{0}$ ) is illustrated. Two fringes are applied for the retrieval of the refractive index of the DT-layer, and the corresponding incident heights of the two fringes are assumed to be $x_{1}$ and $x_{2}$ . Then the corresponding OPD between two light rays with incident height of $x_{1}$ and $x_{2}$ in Fig. 2 can be denoted by:

OPL (x_{1}, n_{2}, t_{2}) - OPL (x_{2}, n_{2}, t_{2}) = OPD,

where

OPL (x_{1}, n_{2}, t_{2})

and

OPL (x_{2}, n_{2}, t_{2})

are the OPLs (optical path lengths) of light rays with incident heights of

x_{1}

and

x_{2}

respectively, and OPD is the optical path difference between the two optical paths. The geometry distance of the light rays in each propagating progress can be derived, then the OPLs of light rays with different incident heights can be derived by multiplying the geometry distance with the corresponding refractive index in each process. The OPD can be obtained directly from the interference map.

As seen in Fig. 2, $x + Δ x$ is the exiting height of the light ray with incident height of $x$ on the back surface of the target, while the back surface is imaged onto the CCD. Assume that $r$ is the radius of the corresponding circular fringe, and we can get,

x + Δ x = r,

where

Δ x

can be derived from parameters of the target and the incident height

x

of the light ray, then the unknowns to be solved are x,

n_{2}

, and

t_{2}

.

In practical experiments, two arbitrary fringes with radius of $r_{1}$ and $r_{2}$ are applied for the retrieval of the refractive index. Assume that the corresponding incident heights of the two fringes are $x_{1}$ and $x_{2}$ , then the refractive index can be retrieved from,

{\begin{cases} OPL (x_{1}, n_{2}, t_{2}) - OPL (x_{2}, n_{2}, t_{2}) = OPD \\ x_{1} + Δ x_{1} = r_{1} \\ x_{2} + Δ x_{2} = r_{2} \end{cases},

where there are four unknowns in three equations:

x_{1}

and

x_{2}

, the corresponding incident height of the two light rays;

n_{2}

, the refractive index of the DT-layer;

t_{2}

, the thickness of the DT-layer. The OPD, the optical path difference between the two fringes,

r_{1}

and

r_{2}

, which are the radii of the circular fringes, can all be measured from the interference map. Other variables can be got by ray tracing with the parameters of the target: the outer radius of the target

R_{0}

, the thickness of the shell

t_{1}

and the refractive index of the shell

n_{1}

. An initial value must be estimated for the refractive index or the thickness of the DT-layer to solve all the unknowns. Once the average thickness of the DT-layer is known, the refractive index can be solved immediately.

The advantage of this method is the convenience and high accuracy with which the measurement can be made. One can only measure the average distance between two fringes and the center to retrieve the average refractive index, without complicated image processing or other experimental operation.

2.2 Thickness measurement based on ray deflection (backlit shadowgraph)

In this subsection, the principle of thickness measurement based on ray deflection is discussed. In this test, a high-power collimated LED provides backlighting, and the image on the CCD is composed of rays, which undergo multiple refractions and/or reflections at the layer boundaries. Then an obvious bright ring will appear on the backlit shadowgraph [7–10]. In Fig. 3, the trace of one light ray which forming the bright ring in this test is shown. The target structure is the same as that in Subsection 2.1. It should be noted that, in the interference test, the back surface of the target is imaged onto the CCD; while in the backlit shadowgraph, the longitudinal section is imaged onto the CCD. Thus, in either of the two tests, the target position should be adjusted to ensure that the correct plane is imaged onto the CCD. The positioning method will be detailed in Section 4. In both the interference test and the backlit shadowgraph test, the positions of the object surface, the lens, and the CCD are completely the same, then the magnification in the two tests are both m.

Fig. 3 Diagram of thickness measurement based on ray deflection.

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Similar to Subsection 2.1, assume that the distance between the bright ring and the center of the target is $X_{2}$ , then Eq. (4) can be derived from geometry relationship,

X_{2} = X_{1} - Y_{2} \tan φ,

where

X_{1}

,

Y_{2}

, and

φ

can be derived by the parameters of the target and the incident height

X

of the light ray. Detailed derivation process can be seen in works of Nikitenko et al [10].

The trace of one light ray can be obtained from the geometry relationship, while the light rays which compose the bright ring remain unknown. The intensity of each point on the image is determined by different light rays undergoing different paths. The bright ring will be defined by the following equation [7,10],

\frac{d X_{2}}{d X} = 0,

The relationship _{$| d X_{2} / d X |$} produces a zero point, which means that when the incident height is $X$ , the most rays converge to the corresponding height $X_{2}$ on the image, while the intensity difference of the rays can be ignored [10], thus the bright ring which has the largest intensity in the shadowgraph is formed. Then the thickness of the DT-layer can be retrieved from,

{\begin{cases} X_{2} = X_{1} - Y_{2} \tan φ \\ \frac{d X_{2}}{d X} = 0 \end{cases},

In Eq. (6), the distance between the bright ring and the center of the target $X_{2}$ , can be measured from the backlit shadowgraph. The other variables can be got by ray tracing, with the light incident height $X$ and the parameters of the target. There are three unknowns in the previous two equations: $X$ , the corresponding incident height of the light ray; $n_{2}$ , the refractive index of the DT-layer; $t_{2}$ , the thickness of the DT-layer. An initial value must be estimated for the DT-layer refractive index or thickness to solve all the three unknowns. Thus, similar to Subsection 2.1, once the refractive index of DT-layer is known, the thickness can be solved immediately.

In addition, all of the above equations (Eqs. (1)–(6)) can be applied to the single-layer targets. To satisfy the situation of testing single-layer targets, just set all of the DT-layer refractive index $n_{2}$ and thickness $t_{2}$ in the above equations as 0, then the unknowns to be solved are $x_{1}$ , $x_{2}$ , $X$ , $n_{1}$ , and $t_{1}$ .

2.3 The iterative algorithm based on OPD and ray deflection

According to the previous two subsections, for the interference test, there are four unknowns to solve with three equations; while in the backlit shadowgraph test, there are three unknowns to solve with two equations. The analytical solutions of the five unknowns ( $x_{1}$ , $x_{2}$ , $X$ , $n_{2}$ , and $t_{2}$ ) are hard to be got simultaneously with the five equations, as the equations are all composed of complex and nonlinear formulas. However, numerical solutions can be found by some means. In the interference method, once the thickness is provided, the refractive index can be solved; in the backlit shadowgraph method, once the refractive index is provided, the thickness can be solved. If the nominal thickness value can be used as the initial value for the interference method, then the solved refractive index value can be used as the initial value for the backlit shadowgraph. The refractive index and thickness are solved back and forth as the iteration performs.

The iterative algorithm for retrieving the refractive index and the thickness of the ICF target DT-layer is proposed. The specific process is shown in Fig. 4 and described below.

Fig. 4 Flow chart of the proposed iterative algorithm.

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(1) At the beginning, basic parameters of the target ( $R_{0}$ , $n_{1}$ , and $t_{1}$ ) are known. Other parameters such as $r_{1}$ , $r_{2}$ , OPD and $X_{2}$ need to be measured through the interference map and backlit shadowgraph respectively in order to retrieve the remaining parameters.
(2) An initial value is estimated for the DT-layer thickness, usually the nominal value or values around it.
(3) Then a refractive index value $n_{2}$ is solved based on OPD (interference). The obtained value is substituted to solve the DT-layer thickness $t_{2}$ based on ray deflection (backlit shadowgraph).
(4) Check if the iteration stopping criterion is met. To terminate the algorithm, a stopping criterion is set as $| (n_{i} - n_{i - 1}) / n_{i} | < tolerance$ , $| (t_{i} - t_{i - 1}) / t_{i} | < tolerance$ [13]. The tolerance is set as 1/10 of the relative retrieval uncertainties so that the variations of iteration won’t influence the final retrieval results, and the minority retrieval uncertainties of the DT-layer refractive index and thickness is 0.05%, which is detailed in Section 5. Thus, the tolerance is set as 0.005%. When the refractive index and thickness converge within the given tolerance, the iteration cycles break. If the stopping criterion isn’t met, step (3) will be repeated. When the iteration cycles break, the current $n_{2}$ and $t_{2}$ are the final iteration values.

The proposed iterative algorithm also can be applied to the single-layer target test. The only parameter need to be obtained in advance is the outer radius $R_{0}$ of the target, which can be measured by the microscopes. Similar to the double-layer target test, set an initial value of the thickness of the shell, then the iteration process begins as that in the double-layer target test. Thus, to obtain all of the information ( $R_{0}$ , $n_{1}$ , $t_{1}$ , $n_{2}$ , and $t_{2}$ ), the targets can be tested when they were single-layer using the proposed iterative algorithm. After the inner DT-layer formed, the remaining unknowns $n_{2}$ and $t_{2}$ can be solved using the algorithm and the system proposed in this paper.

3. Simulation

Simulations were carried out to verify the effectiveness of the proposed iterative algorithm. In the simulation, four typical real ICF targets [10] were employed and their parameters were listed in Table 1. The parameters including outer radius, shell thickness, DT-layer thickness, shell refractive index and DT-layer refractive index vary widely.

Table 1. Parameters of four targets

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Before the proposed iterative algorithm is simulated, the methods based on the OPD and ray deflection were verified respectively in previous. One set of the interference map and backlit shadowgraph of target 1 were simulated with Zemax (Zemax Development, Bellevue, WA) and ASAP (Breault Research, Tucson, AZ) respectively, as shown in Fig. 5.

Fig. 5 Simulations of (a) interference map, and (b) backlit shadowgraph.

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The OPD, $r_{1}$ , and $r_{2}$ from the interference map is 7.7647λ, 100 $μ m$ , and 200 $μ m$ , respectively. Applying the obtained OPD, $r_{1}$ , and $r_{2}$ into the proposed refractive index measurement based on OPD, the retrieved refractive index is 1.15963 (unit), compared to the theoretical value 1.16 (unit), the relative error is 0.03%. The obtained X₂ is 333.2100 $μ m$ , and the retrieved thickness based on it is 65.1768 $μ m$ , the relative error is 0.27% compared to the theoretical value 65 $μ m$ . The errors of the two simulations come from the pixel resolution of the interference map and the backlit shadowgraph. Both of the retrieved values of the refractive index and the thickness of the DT-layer are extremely similar to the theoretical value, which verifies the correctness and feasibility of the two measurement algorithms separately.

After the parameters ( $r_{1}$ , $r_{2}$ , OPD and $X_{2}$ ) have been got from the interference map and the backlit shadowgraph, the proposed iterative algorithm based on OPD and ray deflection is implemented. First, an initial value of the thickness of the DT-layer is assumed (the initial value will not influence the retrieved results). Second, a refractive index value is solved based on OPD (interference), and the obtained value is substituted as the estimated value of the refractive index to solve the thickness based on ray deflection (backlit shadowgraph). Third, the derived refractive index and thickness are used for verifying that whether the stopping criterion is met, and the stopping criterion to terminate the algorithm is set as $| (n_{i} - n_{i - 1}) / n_{i} | < tolerance$ , $| (t_{i} - t_{i - 1}) / t_{i} | < tolerance$ [13]. Where $n_{i}$ and $t_{i}$ are the refractive index and thickness in the $i^{th}$ iteration cycle, while $n_{i - 1}$ and $t_{i - 1}$ are that in the ${(i - 1)}^{th}$ iteration cycle. The tolerance is set as 1/10 of the relative retrieval uncertainties so that the variations of iteration results won’t influence the retrieval results, and the minority retrieval uncertainties of the DT-layer is 0.05%, which is detailed in Section 5. Thus, the tolerance is set as 0.005%. If the refractive index and thickness converge within the given tolerance, the iteration cycles break. If the stopping criterion isn’t met, the iteration steps will be repeated. When the iteration cycles break, the current $n_{2}$ and $t_{2}$ are the final iteration values.

The iteration processes of the DT-layer refractive indexes and thicknesses of 4 double-layer targets are shown in Fig. 6, It is easy to find that the iteration processes for the double-layer targets are oscillating convergences.

Fig. 6 The iteration process for the 4 double-layer targets.

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The iteration processes in Fig. 6 show that the values of the refractive index and thickness of the DT-layer converged rapidly as the iteration cycles increases, and the values of refractive index varied with the values of thickness. After the iteration cycles break, the relative retrieval accuracy of the DT-layer refractive index was better than 0.05%, and that of the thickness was better than 0.1%, for all of the 4 targets. The retrieval errors come from the errors of $r_{1}$ , $r_{2}$ , OPD, and $X_{2}$ got from the backlit shadowgraph and interference tests. Therefore, the DT-layer refractive indexes and thicknesses were retrieved accurately with the proposed iterative algorithm.

4. Experiment

With the proposed iteration algorithm, experiments were carried out to test two single-layer targets (Only standard single-layer targets can be accessed at the present stage subject to fabrication conditions). When test single-layer target, set all of the DT-layer refractive index $n_{2}$ and thickness $t_{2}$ in the equations [Eqs. (1)–(6)] as 0. The outer radii of the two targets were 476.05 $μ m$ and 469.66 $μ m$ , measured by the Zeiss Axio Imager A2m Microscope (Carl Zeiss, Germany).

The experiment system is shown in Fig. 7, which is set up according to the system diagram in Fig. 1. A CCD with pixel size of 4.65 $μ m$ is used to receive the interference map and the backlit shadowgraph. A Nikon imaging lens with a magnitude of −3 is used for imaging. Beside the imaging lens sits the target. The split ratios of the beam splitters and reflectivities of the mirrors are well selected to ensure optimum visibility of fringe pattern. The reference arm should be blocked with a baffle when the backlit shadowgraph test is carrying out to prevent the effects of the background light. The laser and the LED are illumination sources with wavelengths of 632.8nm and 660nm respectively, which can be switched to select the interference test or the backlit shadowgraph test.

Fig. 7 (a) The setup of the interference test; (b) The setup of the backlit shadowgraph test.

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The received interference map and backlit shadowgraph are shown in Fig. 8. As the backlit shadowgraph exhibits a good image contrast, it can be set as a reference when positioning the target. When the bright ring of the backlit shadowgraph is the clearest, the longitudinal section of the target is on the conjugate plane of the CCD. And the sharpness of the bright ring can be judged by gradient functions (Laplacian gradient functions or Brenner gradient functions). After the backlit shadowgraph was shot, move the target toward the CCD with the distance of the radius of the target, then turn off the LED and turn on the laser to switch to the interference test.

Fig. 8 (a) The interference map; (b) The backlit shadowgraph.

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The repeatability of positioning method was verified experimentally. The target was moved back and forth crossing the camera focus plane to make the bright ring on the backlit shadowgraph changing from blurry to clear, and then to blurry, and this step was repeated 5 times. The backlit shadowgraph of each position was saved and the corresponding sharpness of the bright ring was analyzed. The fitted relationship between the sharpness and the positions is shown in Fig. 9. It can be seen that the fitted curves and the largest sharpness positions in the five tests are almost the same. The positioning accuracy is dependent on the mechanical precision, which is typical 5 $μ m$ in this experiment.

Fig. 9 The fitted relationship between sharpness and positions in 5 tests.

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The positions of the fringes in the interference map and the bright ring in the backlit shadowgraph were got from the figures as those in Fig. 8. It can be seen that at 7 o’clock position in Fig. 8 (a) there is a distortion in the fringe, indicating the existence of a small local defect. As the target is assumed uniform (if not uniform, Ref [15]. may be referred), the local defect should be ruled out to exclude its effect on the retrieval of the average refractive index and thickness of the target shell. As seen in Fig. 10, the positions of the interference fringes and the bright ring are marked (in the interference map, the first fringe is marked in blue, and the second fringe is marked in orange; the bright ring in the backlit shadowgraph is marked in blue). The average distance of the first and second dark fringes from the center are $r_{1}$ and $r_{2}$ , and that between the bright ring and the center is $X_{2}$ .

Fig. 10 The obtained (a) first and second fringes positions, and (b) bright ring positions.

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With the proposed iterative algorithm, the refractive indexes and the thicknesses of the ICF targets shells can be retrieved, as seen in Table 2. For both targets, the experiments were repeated three times.

Table 2. Iteration results of two targets

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The retrieved results are compared with the results from the interferometric method and the X-ray microradiography. As the refractive indexes of the shells measured by the interferometric method are 1.5432 and 1.5760 respectively, and the thicknesses of the two targets shells measured by the X-ray microradiography (detailed description of the device and method can be seen in [12]) are 19.67 $μ m$ and 17.88 $μ m$ respectively, the relative errors of the iteration results can be got. It can be seen from Table 2 that both the shell refractive indexes and thicknesses can be retrieved with a relative error below $\pm 2 %$ compared with the results measured by the interferometric method and the X-ray microradiography respectively, which proves the feasibility of the proposed iteration method.

5. Test uncertainty analysis of the double-layer targets

In this section, we discuss the ICF DT-layer refractive index and thickness test uncertainties by considering the uncertainties contributed from the proposed iterative algorithm.

According to Eqs. (3) and (6), it is easy to deduce that the measurement uncertainties of $r_{1}$ , $r_{2}$ , and $X_{2}$ will influence the uncertainty of the DT-layer refractive index $n_{2}$ and thickness $t_{2}$ . As the equations are all composed of complex and nonlinear formulas, analytical relationship between $r_{1}$ , $r_{2}$ , $X_{2}$ and $n_{2}$ , $t_{2}$ are hard to derive. However, numerical methods can be applied.

Relative uncertainties of $r_{1}$ , $r_{2}$ , and $X_{2}$ were introduced quantitively, ranging from −0.4% to 0.4% with a step of 0.1%. The corresponding uncertainties of $n_{2}$ and $t_{2}$ were shown in Fig. 11.

Fig. 11 (a) Relative test uncertainties of $n_{2}$ from $r_{1}$ , $r_{2}$ , and $X_{2}$ ; (b) Relative test uncertainties of $t_{2}$ from $r_{1}$ , $r_{2}$ , and $X_{2}$ .

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As illustrated in Fig. 11, the relative uncertainties of $n_{2}$ and $t_{2}$ are approximately linear with the relative measurement uncertainties of $r_{1}$ , $r_{2}$ , and $X_{2}$ , which means that the iteration is linearly convergent in this range. After performing linear fit in MATLAB (MathWorks, Inc., Natick, MA), the specific relationship can be obtained,

\begin{matrix} {\begin{matrix} δ n_{2} = 0.1205 δ r_{1} - 5.2526 e - 04 \\ δ n_{2} = - 0.6099 δ r_{2} - 5.2522 e - 04 \\ δ n_{2} = 0.6413 δ X_{2} - 5.1912 e - 04 \end{matrix} \\ {\begin{matrix} δ t_{2} = 0.3831 δ r_{1} + 0.0011 \\ δ t_{2} = - 1.9388 δ r_{2} + 0.0010 \\ δ t_{2} = - 2.9411 δ X_{2} + 0.0010 \end{matrix}, \end{matrix}

The sensitivity coefficient $c_{i}$ of the input quantity $x_{i}$ is given by $c_{i} = \partial f / \partial x_{i}$ , where $f$ is the functional relationship between the input quantity and the measurement result [16]. Thus, the sensitivity coefficient of $δ r_{1}$ , $δ r_{2}$ , and $δ X_{2}$ to $δ n_{2}$ are 0.1205, −0.6099 and 0.6413, and that of $δ r_{1}$ , $δ r_{2}$ , and $δ X_{2}$ to $δ t_{2}$ are 0.3831, −1.9388 and −2.9411.

In the practical experiments, the CCD pixel size is 4.65 $μ m$ , the radius of the target is 383 $μ m$ , and the image magnification is 3. Thus, the relative uncertainty come from $r_{1}$ , $r_{2}$ , and $X_{2}$ are all $4.65 / (383 \times 3) = 0.40 %$ , since the obtaining methods are the same. According to the sensitivity coefficient, the relative uncertainties of $n_{2}$ coming from $r_{1}$ , $r_{2}$ , and $X_{2}$ are 0.05%, 0.24%, and 0.26%, and that of $t_{2}$ from $r_{1}$ , $r_{2}$ , and $X_{2}$ are 0.15%, 0.78% and 1.2%, respectively. It can be seen that the relative uncertainties of both $n_{2}$ and $t_{2}$ from $X_{2}$ are the largest while that from $r_{1}$ are the smallest. The minority of the relative uncertainties of both $n_{2}$ and $t_{2}$ are used for the set of the tolerance in the iteration process seen in Subsection 2.3.

6. Summary

To achieve the implosion condition, the implementation of quality control of the target in ICF is very important to the ignition. In this paper, an iterative algorithm based on OPD and ray deflection is proposed to obtain the refractive index and thickness of the DT-layer simultaneously. Simulations show that the retrieval accuracy of the DT-layer relative refractive index is better than 0.05% after finite iteration, and that of the thickness is better than 0.1%. Experiments show that both the shell refractive indexes and thicknesses can be retrieved with a relative error below $\pm 2 %$ compared with the results measured by the interferometric method and the X-ray microradiography respectively, which proves the feasibility of the proposed iterative algorithm and corresponding system. The thickness and refractive index of the shell have been measured at the room temperature at the present stage to verify the proposed algorithm. The measurement in the same conditions as the DT-layer will be conducted in our future work. A comprehensive analysis of the measurement uncertainties of contributed from other factors (such as the uncertainty of the refractive index and thickness of the ablator, the positioning uncertainty, the test uncertainty of the outer radius of the target, etc.) will be seen in our next work.

Funding

National Natural Science Foundation of China (NSFC) (61475141); the Public Welfare Project of Zhejiang Province (2016C33004); the Fundamental Research Funds for the Central Universities (2017QNA5001); State Key Laboratory of Modern Optical Instrumentation Innovation Program (MOI2017MS01).

Acknowledgment

The authors would like to express their great appreciation to the anonymous reviewers for the valuable advice to promote the quality of this paper.

References and links

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		Thickness of shell ( $μ m$ )	Thickness of DT-layer ( $μ m$ )	Refractive index of shell (unit)	Refractive index of DT-layer (unit)
1	383	25	65	1.56	1.16
2	450	3	100	1.59	1.13
3	1000	50	150	1.50	1.14
4	2000	100	300	1.50	1.14

Target	Experiment	$r_{1}$ ( $μ m$ )	$r_{2}$ ( $μ m$ )	$X_{2}$ ( $μ m$ )	$n_{1}$ (unit)	$t_{1}$ ( $μ m$ )	$δ n_{1}$	$δ t_{1}$
1	1	88.8123	163.9380	0.9485*R₀	1.5571	19.3099	0.90%	−1.83%
	2	74.8448	156.5143	0.9485*R₀	1.5728	19.2891	1.92%	−1.94%
	3	87.6337	163.6656	0.9485*R₀	1.5516	19.3174	0.54%	−1.80%
2	1	101.9210	172.1540	0.9507*R₀	1.5612	18.1991	−0.94%	1.78%
	2	95.1866	168.6023	0.9507*R₀	1.5611	18.1992	−0.95%	1.79%
	3	103.6861	173.4910	0.9507*R₀	1.5540	18.2080	−1.40%	1.83%

		Thickness of shell ( $μ m$ )	Thickness of DT-layer ( $μ m$ )	Refractive index of shell (unit)	Refractive index of DT-layer (unit)
1	383	25	65	1.56	1.16
2	450	3	100	1.59	1.13
3	1000	50	150	1.50	1.14
4	2000	100	300	1.50	1.14

Target	Experiment	$r_{1}$ ( $μ m$ )	$r_{2}$ ( $μ m$ )	$X_{2}$ ( $μ m$ )	$n_{1}$ (unit)	$t_{1}$ ( $μ m$ )	$δ n_{1}$	$δ t_{1}$
1	1	88.8123	163.9380	0.9485*R₀	1.5571	19.3099	0.90%	−1.83%
	2	74.8448	156.5143	0.9485*R₀	1.5728	19.2891	1.92%	−1.94%
	3	87.6337	163.6656	0.9485*R₀	1.5516	19.3174	0.54%	−1.80%
2	1	101.9210	172.1540	0.9507*R₀	1.5612	18.1991	−0.94%	1.78%
	2	95.1866	168.6023	0.9507*R₀	1.5611	18.1992	−0.95%	1.79%
	3	103.6861	173.4910	0.9507*R₀	1.5540	18.2080	−1.40%	1.83%

ICF target DT-layer refractive index and thickness from iterative analysis

Abstract

1. Introduction

2. Principle

2.1 Refractive index measurement based on OPD (interference)

2.2 Thickness measurement based on ray deflection (backlit shadowgraph)

2.3 The iterative algorithm based on OPD and ray deflection

3. Simulation

4. Experiment

5. Test uncertainty analysis of the double-layer targets

6. Summary

Funding

Acknowledgment

References and links

Cited By

Figures (11)

Tables (2)

Equations (7)

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