A coupled model of electromagnetic and heat on nanosecond-laser ablation of impurity-containing aluminum alloy

Abstract

In the emerging field of laser-driven inertial confinement fusion, Joule heating generated via electromagnetic heating of the metal frame is a critical issue. However, there are few reported models explaining thermal damage to the aluminum alloy. The aim of this study was to build a coupled model for electromagnetic radiation and heat conversion of an ultrashort laser pulse on an aluminum alloy based on Ohm's law. Additionally, the application SiO₂ films on aluminum alloy to improve the laser-induced damage threshold (LIDT) were simulated, and the effects of metal impurities in the aluminum alloy were analyzed. A model examining the relation between electromagnetic radiation and heat for a nanosecond laser irradiating an aluminum alloy was developed using a coupled model equation. The results obtained using the finite difference time domain (FDTD) algorithm can provide a theoretical basis for future improvement of the aluminum alloy LIDT.

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1. Introduction

Laser-driven inertial confinement fusion (ICF) can generate large amounts of renewable energy via the chemical reaction of two forms of hydrogen—deuterium (D) and tritium (T)¹—and the reaction is expressed in eqn (1). However, the half-life of tritium (T) is short and it does not occur naturally; furthermore, it is extremely difficult to obtain via reaction. Hence, tritium in fusion reactors needs to be recycled, using doubled neutrons for reaction with lithium followed by recovery of tritium, where tritium functions like a catalyst (reaction shown in eqn (2)).² Because of its potential advantages for quickly generating a large, inexhaustible amount of clean energy.^3–5 A nuclear laser facility can reach 1.8 MJ and 500 TW of 355 nm light in 10⁻⁹ s, which is equivalent to 1000 times the energy consumption of the United States.⁶ The French Megajoule Laser Project (LMJ) supported by the French Atomic Energy Commission (CEA)⁷ and the American National Ignition Facility Project (NIF) are collaborating with Lawrence Berkeley National Laboratory⁸ to realize this goal. For this process, damage to optical elements such as fused silica and potassium dihydrogen phosphate (KDP) is an important topic of research.^9,10 However, laser-induced damage of the aluminum alloy frame for the entire system has not been comprehensively studied, especially via theoretical research.

²₁D + ³₁T → ⁴₂He (3.5 MeV) + ⁰n (14.1 MeV) (1)

⁰n + ⁶₃Li → ³₁T + ⁴₂He + 4.784 MeV (2)

The traditional laser power conversion configuration is usually applied to photovoltaic (PV) cells. Light-to-heat generation and light-to-electricity conversion have recently seen research progress.^11,12 Previous work aimed at improving the efficiency between the input power and the power output by the battery, and some studies attributed the low ceiling to the temperature.^13,14 While laser-induced damage in dielectrics is related to the pulse duration, there are qualitative differences between multi-photon ionization and Joule heating, which include plasma formation and conventional melting.¹⁵ The energy coupling mechanism between an ultrashort pulse laser and optical materials has become an issue of intense research interest, including the exploration of surface damage initiation,¹⁶ the optical breakdown process,¹⁷ electronic excitation–relaxation processes,¹⁸ and particle generation.¹⁹ Studies have been limited to either experimental or single-simulation investigation of the influence of the pulse duration,²⁰ laser fluence, and wavelength²¹ on site damage or the morphology of the optical elements. The impact of the electric-field distribution on field enhancement was simulated²² and a temperature model was established to analyze the interaction between the laser and metallic materials.²³ To investigate impurities present in the crystals, Bhar et al. studied the LIDT and the effect of inclusions in some nonlinear crystals; they found that Na⁺ ion inclusion and the LIDT of the BBO crystals were correlated.²⁴ Some studies investigated the laser-induced damage threshold (LIDT) of the TiO₂/SiO₂ multilayer reflective coatings and determined the ultra-short LIDT of KH₂PO₄ via simulation. They studied the effects of lasers of different wavelengths on the optical properties of dielectric materials experimentally and via simulations.^25,26 This research was aimed at studying the laser damage threshold, the reflectivity curve of a TiO₂ single-layer film (experimentally and fitted), and the GIXRD of SiO₂ and TiO₂ single layers. However, the relationship between the laser fluence and the temperature was not investigated in detail. Hence, the search for a coupled model for electromagnetic radiation and heat conversion during nanosecond-laser ablation of aluminum alloy is still an unresolved issue.

Moreover, energy conversion resulting in laser-induced damage of a thin film has not been widely studied, and other aspects of thin films used for PV cells have generally received more attention. For example, the double-pulse temporal method was applied to a thin foil film, the results showed that the energy conversion efficiency was 15% higher than with a single pulse,²⁷ and Cu₂ZnSnS₄ thin films are widely applied in solar cells. These studies focused more on pulsed laser film deposition,²⁸ energy conversion efficiency,²⁹ and the fabrication of thin films.³⁰ Silica is used in optical elements because of its antireflection properties in high-power laser systems, which can improve the LIDT of the optical components.^31,32 Studies have mostly been confined to the laser parameters, including the pulse duration,³³ laser fluence,³⁴ and wavelength,³⁵ as well as to the film processing parameters including the thickness and organic groups³⁶ that affect the growth of the silica thin film, such as its morphology and growth rate. However, studies have not modeled the relation between electromagnetic radiation and heat in the silicon dioxide thin films. Therefore, a coupled model that is related to the aluminum alloy and the silicon dioxide thin film used as a surface coating needs to be investigated; this should extend beyond the relation between electromagnetic radiation and heat and also include the silicon dioxide film and its substrate.

In this work, a coupled model was built for electromagnetic radiation and heat conversion in an aluminum alloy with a nanosecond-laser. We first investigated the effect of copper impurities at a certain depth in the aluminum alloy under UV laser irradiation via FDTD simulations. Then, the light-to-electricity conversion was studied. The results of the light intensity enhancement factor (LIEF) of the aluminum alloy before and after coating with the SiO₂ thin films were determined. Finally, the effect of metal impurities (copper and aluminum) in the aluminum alloy on the temperature rise was simulated and the conclusions are outlined.

2. Coupled geometric modeling of electromagnetic and heat

The coupled model of electromagnetic radiation and heat was based on Ohm's law. Electromagnetic radiation to heat conversion in the structure can be assigned to the response to ohmic loss and Joule heating. The power dissipated is equivalent to:

P = J × E, (3)

where J is the current density and E is the electric field. When the equation was applied to the coupled model of electromagnetic and heat conversion, the applied heat energy transfer rate (W m⁻³) is Q = P. According to research about the metal,^37,38 the heat transport equation can be expressed as follows:

Q = εω₀Im|E(r)|², (4)

where ω₀ is the angular frequency and ε is the permittivity of the metal. This equation shows the relationship between the incident electric field and the imaginary part of the permittivity. Generally, metal particles like Cu and Al are usually located at a certain depth in the alloy or film. However, the difference between plasma formation, ablation, and conventional heating of materials depends on the duration of laser action.¹⁵ In this case, two processes were responsible for the laser-induced damage of the material. The damage was first due to ablation by high-power laser-induced multiphoton ionization, and then heat conduction and thermal explosion based on absorption by the impurities occurs. The surrounding metal particles are in contact with the air after the first step, and then the ultrashort pulse laser acts as a heat source and leads to a rise in temperature. The power dissipated can be equivalent to P = σE², where σ is the electrical conductivity of the metal. According to Maxwell's equations, the relationship between the magnetic field H and the current density J can be expressed based on Ampere's law:

∮H × dl = ∫J × ds. (5)

This indicates that the line integral of the magnetic field along any loop is equal to the current intensity in any curved surface with the loop as the boundary. The linear relationship between the magnetic field H and the current density J shows that the heat energy transfer rate Q has a quadratic relationship with the rotational magnetic field H₀. Eqn (3) indicates that the heat energy transfer rate of the metal particles is related to the magnetic field, which provides us with an approach to decrease the magnetic field around the metal particles and reduce the ohmic loss.

The calculation flow chart for the coupled model of electromagnetic radiation and heat on nanosecond-laser ablation is presented in Fig. 1. The steps to simulate the electromagnetic radiation to heat conversion using the coupled model is as follows:

Fig. 1 Flow chart for the electromagnetic and heat coupling analysis.

• Step 1: after irradiation with a high-fluence nanosecond laser, the electromagnetic field was first calculated until the field reached a steady-state inside the object.

• Step 2: the power, which dissipated because of ohmic loss, was next calculated based on eqn (3), which was then applied to the heat transport equations as the heat energy transfer rate.

• Step 3: once the desired simulation time was reached, the simulation was stopped.

Fig. 2 shows the process used to obtain the coupled model for the electromagnetic field and heat during nanosecond laser ablation. It can be seen from Fig. 2(a) and (b) that the thickness of the aluminum alloy for its own micro arc oxide layer and after coating with a silicon dioxide thin film remains consistent (h = 10 μm). Fig. 2(c)–(e) show the laser-induced damage process for the thin films doped with metal particles. The plasma formation and ablation first happened as shown in Fig. 2(a), and then the metal particles were exposed to air; for an irradiation time > 100 ps, both the particles and substrate were subject to conventional melting and boiling, as seen from Fig. 2(b). Because the temperature rise of the metal particles was faster, the temperature changes of the metal particles were primarily simulated after being exposed to air (see Fig. 2(c)).

Fig. 2 Process diagram for the thin-film electromagnetic field and heat coupling damage under laser irradiation. (a) Metal particles underneath the micro arc oxide layer (Al₂O₃ film). (b) Metal particles under the micro arc oxide layer (Al₂O₃ film) and silicon dioxide films (SiO₂ film). Both (a) and (b) with dimensions: p = 20 μm, h = 10 μm. (c) Irradiation begins. (d) Multiphoton ionization damage of the thin films. (e) Metal particles exposed to air and Joule heating.

3. Discussion and numerical simulation

3.1 Spectral characteristics of Al₂O₃ before and after coating with a SiO₂ thin film

The electric-field intensity in Al₂O₃ before and after coating with a silicon thin film was simulated using the FDTD method. To ensure accurate results for the simulation, the simulation domain was gridded with spacings at 12 samples per wavelength according to the Courant stability condition. Furthermore, to ensure uniformity of film thickness for each layer, the x- and y-directions adopted periodic boundary conditions while the z-direction adopted a perfectly matched layer-absorbing boundary condition.

The spectral characteristics took the form of a plane wave as the initial condition and the incident direction was in the z-direction at a normal incidence of 1.064 μm. According to the Fresnel reflection theory, a formula for calculating the transmission coefficient of the TE mode material at normal incidence can be expressed as follows.

where r is the transmission coefficient and n₁ and n₂ are the refractive index of air and the material, respectively. The refractive indices at 1.064 μm for Al₂O₃ and SiO₂ are 1.754 and 1.453, respectively. The E_t of the transmission wave for Al₂O₃ and SiO₂ was 0.726 V m⁻¹ and 0.815 V m⁻¹, respectively, while E₀ was the normalized electric field. Based on eqn (5), the internal light intensity of Al₂O₃ and SiO₂ under nanosecond-laser irradiation was calculated, which were 1.2277 × 10⁻³ W m⁻² and 1.2809 × 10⁻³ W m⁻², respectively.where I is the light intensity value, S is the Poynting vector, ε is the dielectric constant of the medium, and μ is the magnetic permeability of the medium. Fig. 3 demonstrates the FDTD numerical simulation results for the internal electric field intensity in Al₂O₃ before and after being coated by the thin film, and the amplitudes of the electric field intensity were 0.721 V m⁻¹ and 0.812 V m⁻¹, respectively, and their internal light intensities were 1.2108 × 10⁻³ W m⁻² and 1.2714 × 10⁻³ W m⁻², respectively. The relative error was within 2%, which means that the fit of the simulation results was in good agreement with the Fresnel theory. Hence, this method could be used to simulate the light intensity (I) well.

Fig. 3 FDTD simulation results for the TE mode E distribution at a normal incidence for non-metal doped (a) Al₂O₃ and (b) SiO₂ thin films.

Fig. 4 shows the light intensity distributions for the Al₂O₃ thin films before and after coating with the SiO₂ thin films with mental copper cube particles of edge lengths 0.3, 0.6, 0.9, 1.4, and 1.9 μm. The differences in the light intensity distributions with the SiO₂ thin film coatings were not remarkable. The light intensity enhancement factor (LIEF) is widely applied to characterize the intensity enhancement of an aluminum alloy before and after coating by thin films with metal particles:

where I_max is the maximum light intensity of the metal particle doped film and I₀ is the light intensity of non-metal particle doped film. The LIEFs of the Al₂O₃ thin films and SiO₂ thin films are listed in Table 1, as calculated using Fig. 4. The LIEFs of the SiO₂ thin films were ∼50% and 25% those of the Al₂O₃ thin films when the edge length of the copper cubes were 0.3 and 0.6 μm, respectively. When the edge lengths of the cubes were 0.9 and 1.9 μm, the LIEFs of the SiO₂ thin films were ∼85.9% and 78.4% of those of Al₂O₃ thin films. It is worth noting that the LIEFs of the 1.4 μm-edge length cube were almost the same. However, in general, after being coated by the SiO₂ thin films, the LIEFs were sharply reduced.

Fig. 4 FDTD simulation results for the TE mode light intensity distribution at y = 0, x–z plane: Al₂O₃ before (a1–a5) and after (b1–b5) being coated by SiO₂ thin films that are both doped with copper cubes of different edge lengths.

Table 1 LIEFs of Al₂O₃ before and after coating with the SiO₂ thin films

Cube (μm)	Al2O3		SiO2
	E (V m^−1)	LIEFs	E (V m^−1)	LIEFs
0.3	1.9	6.9	1.5	3.4
0.6	3.0	17.3	1.6, 6.6	3.9
0.9	2.1	8.5	2.2	7.3
1.4	2.0	7.7	2.3	7.8
1.9	2.4	11.1	2.4	8.7

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Differences in the LIEFs between the Al₂O₃ thin films and SiO₂ thin films could be attributed to the different types chemical bonds in the two materials. Photo-ionization and impact ionization led to an increase in the free electron density, which destroyed the metal frame because of the interaction between the strong laser field and the dielectric materials. SiO₂ is an atomic crystal, here, a Si atom forms four covalent bonds with four O atoms, while one oxygen atom forms two covalent bonds with two Si atoms. The covalent bond is a type of chemical bond, here, two or more atoms use their outer electrons to reach a state of electronic saturation under ideal circumstances. This makes for a relatively stable chemical structure. The valence bond is the interaction between atoms via the sharing of electron pairs. The ionic bond refers to a chemical bond between anions and cations formed via electrostatic interaction, which occurs after two or more atoms or chemical groups lose or gain electrons and become ions. An electrostatic interaction is present between oppositely charged ions. When two oppositely charged ions are close, they are attracted to each other; there is mutual electrostatic repulsion between electrons and electrons and the nuclei and nuclei, when the electrostatic attraction and repulsion reach an equilibrium, ionic bonds are formed. The melting and boiling points of the SiO₂ atomic crystal are higher than those of Al₂O₃, which is an ionic crystal. This can explain the larger LIEF observed for the Al₂O₃ thin films than for the SiO₂ thin films.

3.2 Electromagnetic radiation and heat model for impurities-containing Al₂O₃

Fig. 5 shows the electromagnetic and heat model of Al₂O₃. For the structure in Fig. 5, the copper cube dimensions were 90 nm, and the Al₂O₃ substrate 20 μm × 20 μm × 10 μm was held at a laser fluence of Φ = 8 J cm⁻² with a laser wavelength of 1064 nm. We first ran the simulation and visualized the absorbed power density profile (shown in Fig. 5(a)), and then the absorption data were loaded in the heat source. Once the simulation was carried out, the results were loaded in the HEAT solver region and the temperature was visualized in Fig. 5(b) and (c). The result of the coupled electromagnetic and heat model showed that the highest temperature for copper was 1520 K, which was above the melting point (1357.77 K) and led to damage of the material (see Fig. 5(c)).

Fig. 5 (a) The absorbed power density profile. (b) Temperature profile at the copper and the top surface of the substrate. (c) Temperature profile underneath the superstructure at the center region, taken in the x–z plane (local magnification images on the lower right hand).

To investigate the effect of impurities on the temperature rise under different laser fluences (Φ), copper and aluminum cubes were placed in the Al₂O₃ substrate, with cube edge lengths between 50 and 90 nm in 10 nm intervals. The temperature rise curves are shown in Fig. 6(a) and (b), where the temperature rise (T) vs. laser fluence (Φ) is plotted. We can see that the temperature increased linearly with the laser fluence, as follows:

T = kΦ + b, (9)

where k is the proportionality growth coefficient and b is a constant, and Table 2 indicates the specific values for the copper and aluminum cubes under laser irradiation. R² is the coefficients of determination, and the data obtained for the two materials are shown to correspond well (R² ≥ 0.99).

Fig. 6 Temperature rises along with increasing laser fluence and edge length of (a) copper and (b) aluminum.

Table 2 Comparison of simulation data for copper and aluminum impurities

Cube (nm)	Copper			Aluminum
	k	b (K)	R^2	k	b (K)	R^2
50	29.61	300.21	0.99	48	300	1
60	39.59	300.08	0.99	71.35	300	0.99
70	73.74	300.00	0.99	120	300	1
80	92.52	299.58	0.99	156.26	299.58	1
90	152.49	300.08	1	232.09	303.7	0.99

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Table 2 shows that the temperature growth coefficient (k) increased with increasing cube size. The temperature growth coefficient for the copper cubes was smaller than for aluminum cubes, with the former being approximately two-thirds of the latter. Furthermore, the growth rate of the growth coefficient (k) increased a lot with increasing cube edge length. For a copper cube with a side length of 50 nm, the k was 29.61, and the k went up to 152.49 for a length of 90 nm, which is ∼5 times larger. A similar trend was also observed for the aluminum cubes. Both of their constants (b) were ∼300 K, and the results fit extremely well.

Fig. 6 intuitively compares the temperature (T) growth calculated using Lumerical's DGTD solver. The linear fit shown on the plot indicates that the 60 nm-length aluminum-cube almost reached its melting point (933.16 K) at a laser fluence (Φ) of 8 J cm⁻², which had a temperature of 871 K. However, the copper cube's temperature was only 617 K under the same conditions. When the side length of the copper cube was 90 nm, the temperature abruptly increased to 1520 K under 8 J cm⁻² laser irradiation, which is approximately two times that at 4 J cm⁻² (880 K). The reasons for the temperature response to laser fluence and edge-length between copper and aluminum could be attributed to their different melting points. The lower laser fluence can lead to an aluminum temperature of up to 890 K, and this temperature is far from the melting point of copper (1357.77 K). Additionally, aluminum is a type of active metal. Under laser irradiation, a series of collisions occur between energetic particles and the lattice atoms of the metal. Aluminum can easily lose its electrons and be damaged by irradiation.

To validate the hypothesis that the rate of temperature rise is related to the edge length of the cubic impurities, we analyzed the average temperature rise rates (TRR). The TRR was calculated using the simulation. The growth curves for the relationship between the TRR and copper/aluminum cubes can be seen in Fig. 7(a) and (b). These graphs plot the TRR vs. cubes length (50–100 nm, 10 nm intervals). It can be seen from the graph that the TRR of the copper and aluminum cubes was similar, and that they could both take the form of a quadratic equation. However, as the laser fluence reached 8 J cm⁻², the temperature of the 100 nm aluminum cube was 2520 K, while that of copper was 1740 K, hence, the TRR of the aluminum cube was larger than that of the copper cube. The reasons were previously explained and are mainly determined by the characteristics of the material itself.

Fig. 7 Temperature rises rates with edge length of (a) copper and (b) aluminum cubes.

4. Conclusion

In this paper, we investigated a coupled model for electromagnetic radiation and heat conversion in a metal frame for laser-driven inertial confinement fusion. The process of reacting deuterium (D) and tritium (T) could generate a large amount of clean and recyclable energy, and the success of the laser ignition process relies on a clean optical system. The proposed coupled model can guide the choice of laser fluence, provide a method to improve the LIDT of materials in the system, and reduce impurities in the laser system. This work could provide an effective theoretical reference to control the nuclear reaction. On the basis of the simulation, the specific conclusions can be summarized as follows.

(1) The laser-induced damage threshold of the aluminum alloy could be improved by coating with a SiO₂ thin film. The light intensity enhancement factor of Al₂O₃ thin films could be significantly reduced after surface treatment.

(2) The existence of metal impurities in the aluminum alloy could result in melting of the metal frame and may cause impurity contamination of the laser system. Furthermore, the contaminated-environment will reduce the efficiency of laser transmission and would eventually lead to a failure of ignition; hence, further research on the metal impurities in the aluminum alloy frame is important.

(3) The temperature growth of the aluminum alloy is related to the edge length of the metal cube. At the nanoscale (<100 nm), the temperature growth for a cube of a certain length was found to be linear with increasing laser fluence. With increasing cube length, the growth rate of the temperature was larger.

In this work, a coupled model was built to verify the laser induced damage threshold of an aluminum alloy frame before and after being coated with SiO₂, and the results demonstrate the effect of nanoscale metal impurities on the temperature rise of the frame. It was shown that while metal impurities in the aluminum alloy frame were inevitable, surface treatment could provide effective protection and gives direction for the future design of such systems.

Nomenclatures

^21D	Deuterium
H	Magnetic field
^21T	Tritium
r	Transmission coefficient
^21He	Helium
n1	Refractive index of air
^0n	Neutron
n2	Refractive index of materials
^63Li	Lithium
Et	Electric field under transmission wave of materials
P	Power dissipated
E0	Normalized electric field
J	Current density
I	Light intensity value
E	Electric field
S	Poynting vector
Q	Heat energy transfer rate (W m^−3)
μ	Magnetic permeability
ω0	Angular frequency
T	Temperature
ε	Permittivity
Φ	Laser fluence (J cm^−2)

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (No. 2017YFA0701200), the State Key Program of National Natural Science Foundation of China (No. 51535003), the State Key Laboratory of Precision Measurement Technology and Instruments (No. PILAB1701).

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