1. Introduction

In inertial confinement fusion (ICF), a spherical capsule filled with cryogenic deuterium-tritium (DT) fuel is imploded to reach a high temperature and high pressure to achieve enough thermonuclear reactions [1–3]. ICF is a potential way to realize clean, safe and economic energy source, therefore the research on ICF is very active. In the last few decades, different schemes aiming at ignition have been proposed. In early studies of direct drive (DD) scheme [4,5], laser beams are aimed directly at the capsule. However, researches indicated that during the implosion, the capsule suffers the growth of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) hydrodynamic instabilities [2] which are main obstacles to achieve igintion. The seed of these instabilities are mainly induced by capsule roughness and irradiation non-uniformity. In order to provide sufficiently uniform irradiation, the indirect drive (ID) scheme was proposed in which the laser beam energy is absorbed and converted into x-rays inside a high-Z cavity (hohlraum) to provide a highly uniform x-ray radiation field [1,3]. Recently, a series of hohlraums [6–10] with rugby and spherical shapes have been considered to improve implosion performance. The ID scheme promises high irradiation uniformity at the price of energy-coupling efficiency.

Most large laser facilities in the world are mainly designed for ID thermonuclear fusion, such as the National Ignition Facility (NIF) [11] in USA and the Laser MegaJoule (LMJ) [12] in France, as well as the Shenguang III in China. However, despite of their ID design, it is possible to envisage their use in DD ICF. To employ ID configurations for DD fusion, some methods have been proposed, for instance, the polar direct drive (PDD) technique [13,14] which displaces the laser beams toward the equatorial plane to provide a more uniform irradiation on the capsule. Moreover, the new DD schemes as the shock ignition [15], which would relax the implosion requirement, have induced more favor on DD options. Furthermore, the DD scheme can benefit from the zooming techniques [16,17], which could further improve the laser-capsule coupling efficiency.

For direct drive approach, analytical works of optimizing irradiation systems has been studied by several groups [18–21]. Most of their works are based on geometrical considerations. For an arbitrary number of laser beams, a numerical algorithm to optimize direct drive beam configuration by solving an N-body charged particle simulation is developed [22]. Recently, three-dimensional simulations are also carried out to analyze the symmetry of DD irradiation scheme [23]. To achieve highly uniform irradiation, besides geometrical considerations, another promising method is to adjust the intensity distribution within the laser spots [17,24]. Generally, a super-Gaussian intensity profile $I\left(r\right)=I_0\exp \left[-{\left(r/\Delta \right)}^m\right]$ is assumed and it has two parameters: the half width at 1/e $\Delta$ and the exponent m. Presently, some direct methods have been used for optimizing the intensity distribution, e.g. the scanning method [17], which is simple and easy to implement while is very time-consuming. When there are only a few variables, one can use this method to search optimal solutions. However, there are many parameters that could affect the irradiation uniformity: the PDD displacement quantity d, the capsule radius R, the major semi-axis a and minor semi-axis b (for laser spot with elliptical spatial profile) and so on. For this multi-parameter optimization problem, the particle swarm optimization [25,26] is adopted, which is widely used in computer science while has not been used to any ICF problems, to the best of our knowledge.

The aim of this paper is to address the irradiation uniformity provided during the first few ns of the foot pulse for direct drive scheme. It is assumed that the quality of irradiation could be approximated by the illumination on the capsule during the foot pulse, due to the small density scale length in the plasma corona and by the quasi-stationary of the critical density [24]. In this paper the illumination model is used with a combination of uniform, super-Gaussian elliptical and circular intensity profile is associated to laser beams and about 10 parameters are optimized using multi-parameter optimization approach, leading to high irradiation uniformity.

The present paper is organized as follows. Section 2 describes the illumination model and the multi-parameter optimization approach. A case study based on the Shenguang III laser configuration to valid this optimization method is presented in Section 3. Finally, conclusions are drawn in Section 4.

2. Illumination model and multi-parameter optimization approach

2.1 Illumination model

To estimate the direct drive irradiation uniformity, a simple model is established in the following. Neglecting the beam divergence at a capsule scale, one can assume the beam intensity is independent of its propagation along the beam axis, the intensity on the capsule surface resulting from the ith laser beam is then:

A promising method to achieve a very uniform irradiation is to adjust the intensity distribution within the laser spot. In theory, a proper intensity distribution $I_i\left(x,y\right)$will lead to a high irradiation uniformity. Note that $I_i\left(x,y\right)$is a function of two variables that are defined on the plane orthogonal to laser beam axis, we can expand it as $I_i={\displaystyle {\sum }_n{a}_n{f}_n\left(x,y\right)}$ where $\left\{f_n\left(x,y\right),n=1,2,\cdots \right\}$ form a complete set. By changing the coefficients $a_n$, we can get all smooth and continuous functions, which indicates that we can achieve high irradiation uniformity by choosing proper coefficients.

In practice, most laser beams have a super-Gaussian spatial profile. Hence, in the following, a super-Gaussian spatial profile of the incident laser beam characterized by the exponential factor and the half width at $1/e$ is considered, which reads $I\left(r\right)=I_0\exp \left[-{\left(r/\Delta \right)}^m\right]$, where $r=\sqrt{x^2+y^2}$. Such intensity distribution shows that the laser beam intensity is symmetrical around its propagation axis. For laser beams with elliptical profiles, it would be better to introduce the relative length $r=\sqrt{x^2/a^2+y^2/b^2}$where a and b are the major and minor semi-axis, respectively. A combination of uniform intensity, super-Gaussian elliptical and circular intensity profile may lead to better results and is given by:

Besides the coefficients in Eq. (2), the outer radius of the capsule $R$, the major semi-axis $a$and minor semi-axis $b$of the elliptical spatial profile must also be suitably chosen to provide a highly uniform irradiation. Furthermore, considering that the Polar Direct Drive technique (PDD) can reduce the non-uniformity and therefore the displacement d is on the list to be optimized.

The illumination uniformity of a spherical capsule provided by a given irradiation configuration could be estimated in terms of its root mean square (RMS) deviation $\sigma ={\left[{\displaystyle \int {\left(E-\overline{E}\right)}^{2}d\Omega }\right]}^{1/2}/{\left(4\pi \right)}^{1/2}\overline{E}$, where $\overline{E}=\left({\displaystyle \int Ed\Omega }\right)/4\pi$ is the average absorbed energy. Since $\sigma$depends on $A_1$, ${\Delta }_1$, $m_1$, $A_2$, ${\Delta }_2$, $m_2$, a, b, R, and d, it is a function of these parameters, which reads:

Note that beam uncertainties are not included in the RMS deviation $\sigma$ given by Eq. (3). In the reality, the laser beams are affected by unavoidable uncertainties such as laser power imbalance and beam pointing error, which would affect the irradiation uniformity. Usually, power imbalance and pointing error follow Gaussian distributions characterized by statistical deviation ${\sigma }_{PI}$ and ${\sigma }_{PE}$, respectively. To estimate the illumination uniformity when taking into consideration the imperfections, an average $\underset{¯}{\sigma }=\left({\displaystyle {\sum }_i{\sigma }_i}\right)/n$ over a large number $n$ of simulations is adopted.

2.2 Particle Swarm Optimization and its application

As indicated by Eq. (3), the RMS deviation $\sigma$ depends on a series of parameters, therefore, $\sigma$ can be minimized by optimizing all these parameters. For multi-parameter optimization problem, different methods have been developed, i.e., Genetic Algorithms. Here we use the particle swarm optimization (PSO) that was proposed by Kennedy and Eberhart [25] motivated by the social behavior of organisms such as bird flocking or fish schooling. PSO makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. Moreover, the PSO algorithm is easy to understand and implement, has only a few parameters to adjust and converges very fast. In PSO, each potential solution is called “particle” and population of particles is called “swarm”. Each particle in a swarm has its position and velocity, and has a simple memory that stores its own best solution so far. Each particle flies in the search space toward the optimum or a quasi-optimum solution based on its own experience and experiences of other particles.

The objective function is called fitness function in PSO, and its value is called fitness value. In our case, considering that the illumination uniformity $\sigma$ is a function of ten parameters, the optimization problem can be expressed as follows:

The PSO steps to minimize the RMS deviation $\sigma$ can be described as in the following. Note that the fitness value $\sigma$ is calculated by different methods for two cases that accounts for perfectly balanced and unbalanced beams.

Step 1: Initialize a population of particles with uniformly distributed random positions and velocities: $x_i~U\left(x_{\min },x_{\max }\right)$, $v_i~U\left(-\left|x_{\max }-x_{\min }\right|,\left|x_{\max }-x_{\min }\right|\right)$.

Step 2: Calculate the fitness value:

A. For perfectly balanced beams:

Evaluate the fitness value for each particle using $\sigma =f\left(x\right)$.

B. Accounting for the beam uncertainties:

The fitness value is calculated as the average value $\underset{¯}{\sigma }=\left({\displaystyle {\sum }_i{\sigma }_i}\right)/n$ over a large number of simulations to incorporate effects of beam uncertainties, i.e., laser power imbalance and beam pointing error.

Step 3: For each particle, compare its current fitness value with its history best. If its current fitness value is lower, then update its history best.

Step 4: Find the current best particle with the lowest fitness value. If the fitness value is lower than that of the global best position, then update the global best.

Step 5: Update the velocity and position of each particle using Eq. (4) and (5).

Step 6: Loop to Step 2 and repeat until the fitness value $\sigma <1\%$or reach a maximum number of iterations.

3. Irradiation uniformity of the Shenguang III laser facility

The Shenguang III laser facility located at Research Center of Laser Fusion, Mianyang, China. It will be completed in 2015 and will be equipped with 48 laser beams distributed over four cones per hemisphere, at polar angles $\theta =28.5°$, $35°$, $49.5°$, and $55°$, as shown in Fig. 1. The number of beams in each cone is 4, 4, 8, and 8, respectively. In each cone, the laser beams are longitudinally equally separated by $90°$(at $\theta =28.5°$ and $35°$) or $45°$(at $\theta =49.5°$ and $55°$) from its closest neighbors. This large laser facility provides two kinds of configurations: one with ~60 kJ total energy in its 1 ns pulse width, and the other with ~180 kJ total energy for a 3 ns pulse width, at 0.351 μm. Beam smoothing technique has been used in the Shenguang III facility and it is expected to produce an uniform circular intensity profile with radius $R_s=250\mu m$. The uncertainties of this laser facility are the beam-to-beam power imbalance ${\sigma }_{PI}=10\%$ and the laser beam pointing error ${\sigma }_{PE}=30\mu m$.

 

Fig. 1 (Left) Shenguang III laser facility and (right) laser beams layout of the Shenguang III around the target (2000 μm from the target center).

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3.1 The initial irradiation uniformity

The Shenguang III laser facility was original designed for indirect drive ICF, therefore its laser beams are clustered in the polar regions, which means that direct illumination of a capsule will cause over-irradiation of the polar areas and under-irradiation of the equatorial zone. Note that the four cones locate at $49.5°$ ($130.5°$) and $55°$ ($125°$) are relatively close to the optimum value for the two-cones configuration direct drive ICF, $54.74°$, as demonstrated by Schmitt [19], therefore the beams at these four cones are suitable for direct illumination. According to the laser spot size, the capsule radius is set to $R=225\mu m$, calculations show that the RMS deviation provided by this configuration is $\sigma =5.62\%$, which is much higher than that required for direct drive ICF.

3.2 Polar direct drive technique

In order to achieve a more uniform irradiation of the capsule, the polar direct drive technique has been considered. Note that there are two cones per hemisphere and they can be displaced by different PDD parameters $d_1$ and $d_2$. The RMS deviation $\sigma$as a function of displacement $d_1$and $d_2$is demonstrated in Fig. 2, which clearly shows that more uniform illumination can be achieved by proper displacements. The black solid circle indicates the original laser configuration whose $\sigma =5.62\%$. Assuming the cones are displaced by the same$d$, PDD has its best performance of $\sigma =2.79\%$ at $d=72\mu m$ (the blue full diamond). The global minimum of $\sigma =2.59\%$is obtained when the outer cone ($49.5°$) is reoriented toward the equatorial plane $d_1=94\mu m$ and the inner cone ($55°$) to the polar region $d_2=44\mu m$, as indicated by the red star. The rectangular area around the black solid circle in Fig. 2 is the region where the laser spot is larger than the capsule and the RMS deviation remains unchanged when each laser spot can cover the whole capsule, as long as the intensity in each laser spot is uniform.

 

Fig. 2 RMS deviation $\sigma$ as a function of PDD displacement $d_1$and$d_2$.

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3.3 Multi-parameter optimization approach

In this section, multi-parameter optimization approach described above is applied to further reduce the non-uniformity. Two cases that account for perfectly balanced and unbalanced beams are both analyzed in the following.

3.3.1 Perfectly balanced beams

In this case, the particle swarm optimization is performed with power imbalance and pointing errors neglected. Evolution of the RMS deviation $\sigma$ is given in Fig. 3, which clearly shows that $\sigma$reduces as the PSO proceeds. The optimal parameters are obtained and summarized in Table 1. Highly uniform irradiation with $\sigma =0.23\%$is achieved by using such intensity distributions and the intensity illumination of the capsule surface is shown in Fig. 4.

 

Fig. 3 Evolution of the RMS deviation $\sigma$ as a function of the PSO iterations.

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Table 1. Laser parameters that optimize the capsule irradiation with perfectly balanced beams.

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Fig. 4 Detailed laser intensity distribution on the capsule surface using the adjusted intensity profile in Table 1.

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To see the evolution of energy deposition non-uniformity in detail, spectral analysis is carried out in the following. The total absorbed energy density on the capsule surface $E\left(\theta ,\phi \right)$can be expanded as $E\left(\theta ,\phi \right)={\displaystyle {\sum }_{l=0}^{\infty }{\displaystyle {\sum }_{m=-l}^l{a}_l^m{Y}_l^m\left(\theta ,\phi \right)}}$ where $Y_l^m\left(\theta ,\phi \right)$are the spherical harmonics and $a_l^m$ its coefficients. Using the orthogonal property of spherical harmonics, the RMS deviation $\sigma$can be rewritten as ${\sigma }^2={\displaystyle {\sum }_{l=1}^{\infty }{\displaystyle {\sum }_{m=-l}^l{\left|C_l^m\right|}^2}}$ where $C_l^m=a_l^m/\sqrt{4\pi }$. All $C_l^0$with odd l cancel due to the symmetry around the equatorial plane and the evolution of $C_l^0$with even l are describes in Fig. 5. As clearly shown, the main contributions to $\sigma$ are due to terms $C_2^0$ (polar flattening), $C_4^0$ (overpressure at $\pm {45}^{\circ }$ latitudes) and $C_6^0$. As the PSO proceeds, $C_2^0,C_4^0,C_6^0$ are reduced from ${10}^{-2}$ to ${10}^{-4}$ and $C_8^0,C_{10}^0$ from ${10}^{-3}$ to ${10}^{-6}$, suggesting that this multi-parameter optimization approach is effective.

 

Fig. 5 Evolution of spherical harmonics coefficients as a function of the PSO iterations.

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3.3.2 Including laser power imbalance and pointing errors

The analysis of the RMS deviation $\sigma$is then performed taking the beam uncertainties into consideration and $\sigma$is estimated by averaging over 1000 simulations. The uncertainties assumed in our calculations are the beam-to-beam power imbalance ${\sigma }_{PI}=10\%$ and the laser beam pointing error ${\sigma }_{PE}=30\mu m$. The average non-uniformity $\underset{¯}{\sigma }$provided by the initial configuration (with the uniform intensity profile and the laser beams remain at their original positions) and the optimized one using the parameters summarized in Table 1 is 6.4% and 4.6%, respectively, which is much higher than that without any beam uncertainties.

To achieve the minimal illumination non-uniformity with beam errors, multi-parameter optimization is carried out and the parameters are obtained and given in Table 2. With such an intensity distribution, the average non-uniformity $\underset{¯}{\sigma }$ is reduced to about 3.5% .

Table 2. Laser parameters that optimize the capsule irradiation when taking into account beam errors.

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The average uniformity as a function of the beam uncertainties ${\sigma }_{PI}$ and ${\sigma }_{PE}$ is described in Fig. 6. As clearly shown, the intensity on the capsule becomes more uniform as ${\sigma }_{PI}$ and ${\sigma }_{PE}$ decrease. Figure 6 also indicates that the non-uniformity $\sigma$ is more sensitive to laser power imbalance ${\sigma }_{PI}$ and $\sigma$ will decrease to about $2\%$ when ${\sigma }_{PI}$ is reduced to 5% while ${\sigma }_{PE}=30\mu m$. It is worth to note that if every direction of irradiation is associated with a bundle of $N_B$ beams, the direction-to-direction power imbalance will be reduced to ${\sigma }_{PI}/\sqrt{N_B}$ due to the statistical weight. For example, in the LMJ facility, 160 laser beams are grouped in 40 quads and $N_B=4$, therefore power imbalance ${\sigma }_{PI}$ is reduced to $10\%/\sqrt{4}=5\%$.

 

Fig. 6 Average RMS deviation $\sigma$ as a function of laser power imbalance and pointing error.

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The optimized laser parameters summarized in Table 1 and 2 are obtained assuming that absorption fraction of incident energy follows a cosine law. Now we use another absorption efficiency [22] to estimate the illumination uniformity. The absorption efficiency of a laser ray with an incident angle $\psi$ is ${\eta }_a\left(\psi \right)=1-{\left(1-{\eta }_{\perp }\right)}^{{\cos }^3\left(\psi \right)}$ where ${\eta }_{\perp }$ is the absorption efficiency of a normally incident laser and it depends on the electron-ion collision frequency at the critical point and the plasma scale length. Taking the beam uncertainties into consideration (${\sigma }_{PI}=10\%$ and ${\sigma }_{PE}=30\mu m$), calculations show that the average non-uniformity provided by the initial configuration is $\underset{¯}{\sigma }=7.6\%$, a little higher than $\underset{¯}{\sigma }=6.4\%$ that obtained when a “cosine law” is assumed. With the laser parameters in Table 2, the non-uniformity is reduced to about $\underset{¯}{\sigma }=4.4\%$ ($\underset{¯}{\sigma }=3.5\%$ for “cosine law”). These results show that though their values are a little different when different absorption efficiencies are adopted, the tendency is the same, suggesting that our method is effective to optimize the illumination uniformity of the capsule

It is also possible to extend our results to more general situations by including all the essential physical effects with full 3D hydrodynamic simulation. For a given irradiation configuration, the relation between non-uniformity $\sigma$ and laser intensity profile can be obtained by 3D hydrodynamic simulation. With this relation, we can use the PSO to optimize the illumination uniformity of the capsule. Corresponding steps are similar to that described above, and the only difference is that in Step 2, the fitness value $\sigma$ is given by full 3D hydrodynamic simulation.

4. Conclusion

The illumination uniformity on a spherical capsule during the initial imprinting phase directly driven by laser beams has been analyzed. A novel approach to minimize the illumination non-uniformity $\sigma$ based on multi-parameter optimization (particle swarm optimization) is proposed, which enables us to obtain a set of optimal parameters over a wide range. A way to realize high-uniform irradiation is to adjust the intensity distribution of the laser spot. With this aims, a combination of uniform, elliptical and circular super-Gaussian intensity profiles has been considered. Thus, there are about 10 parameters to be optimized, which has been done using the particle swarm optimization method. Finally, this method is applied to improve the direct drive irradiation uniformity provided by the Shenguang III that is designed to work as an indirect drive ICF laser facility. Compared with Polar Direct Drive technique, which can reduce the RMS deviation $\sigma$ from 5.62% to 2.59%, the multi-parameter optimization further reduces $\sigma$ to 0.23%, for perfectly balanced beams. Spectral analysis is also carried out to see the evolution of non-uniformity in detail and results show that $C_2^0$, $C_4^0$, $C_6^0$ are reduced to about ${10}^{-4}$ and $C_8^0$, $C_{10}^0$ to about ${10}^{-6}$. Furthermore, the analysis of the average RMS deviation $\underset{¯}{\sigma }$ is performed taking power imbalance (10%) and pointing error (30μm) into consideration, which shows that $\underset{¯}{\sigma }$ is reduced from 6.4% to 3.5%. It is found that $\underset{¯}{\sigma }$ is more sensitive to laser power imbalance ${\sigma }_{PI}$ and $\underset{¯}{\sigma }$ will decrease from 3.5% to about 2% if ${\sigma }_{PI}$ reduces from 10% to 5%. It is also possible to extend our results to more general situations by full 3D hydrodynamic simulation and multi-parameter optimization is still effective.