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Abstract
Potassium dihydrogen phosphate and its deuteride (KDP/DKDP) are the only tripled frequency crystals used for inertial confinement fusion. The photonic behavior of KDP under laser irradiation is unknown. In this study, the ultraviolet photon transport behavior of KDP with different growth environments has been simulated based on the Monte Carlo method. By comparison, it is shown that the linear absorption of filtered grown crystal is obviously weaker, and the relaxation time is much longer. Moreover, the concentration of defects inside KDP is the critical cause of linear absorption and relaxation time. Finally, the influence of multi-photon absorption on the damage of KDP is discussed.
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1. Introduction
The long-term goal of inertial confinement fusion (ICF) research is to provide clean fusion energy for mankind and solve the energy problem. ICF uses a high-energy laser beam to uniformly irradiate the surface of the target pellet and generate plasma. The plasma sparks outward to produce a backward impulse, compressing the target pellet. When it is at the high temperature and high-density state required for thermonuclear combustion, the reaction is completed within the inertial confinement time, releasing a huge amount of energy. The large laser device is the core of the reaction process. In large laser devices, in order to correctly irradiate the target position and achieve the required wavelength, various optical components need to meet extreme conditions. Thus, the ICF project has very stringent requirements for the selection and production of optical components.
As an excellent nonlinear optical crystal, potassium dihydrogen phosphate and its deuteride (KDP/DKDP) are commonly used to make frequency-tripled elements and optoelectronic switching devices. In particular, KDP/DKDP crystals have the advantages of a high nonlinear coefficient, wide transmission range, and large size [1]. It makes KDP/DKDP crystals become the only 3ω frequency doubling optical element for final optics assembly (FOA) in large laser devices. However, the damage of FOA could limit the performance improvement of large laser devices and the development of ICF. The damage adjustment for KDP/DKDP has been a challenge for researchers to overcome in the present. And the mechanism of laser-matter interaction is not known clearly in this era. Therefore, further figuring out and developing physical models is significant for crystal growth and engineering application.
Accordingly, many researchers have done much work to describe the damage process for KDP/DKDP. At the basic theoretical level, various theoretical models for the interactions between laser and materials have been developed, such as the heat conduction model, the two-temperature model, and the modification of various physical processes with their cross-disciplinary coupling [2–4]. The damage of KDP/DKDP crystals under laser irradiation has been explained in many aspects. Focusing on the defect properties of the crystal, first-principle calculations based on density functional theory (DFT) [5–8] have been used to explain the damage at the microscopic level. It could reveal that impurity elements and intrinsic defects in the crystal can markedly affect the laser damage threshold by reducing the crystal band gap. The molecular dynamics (MD) method coupled with the heat conduction equation [9–11] has successfully explained the dynamic irradiation damage processes at the microscopic level in sapphire, fused silica, and the surface scratch behavior in KDP. At the macroscopic level, the finite element method (FEM) with multiple physical fields [12–15] is more commonly used to study the damage. Based on these theories and experimental foundations, physics models such as impurity-induced thermal damage, defect-induced damage, and multi-photon absorption have been broadly summarized. In experiments, these theoretical models have been successful in some respects. However, the laser is a monochromatic light source consisting of a large number of photons, which is considered as the interaction of photons with matter in the system. In experiments, damage to the crystal caused by laser irradiation is visually observable. Whereas, as the smallest particle, the photon's behavior cannot be observed in detail. And there is no simulation research that has been applied to describe the transport behavior of UV photons in KDP. Therefore, there is an urgent need for the method to explain the absorption and transport behavior of photons to reveal the mechanism of KDP crystals when it is damaged by irradiation.
In this work, the Monte Carlo particle transport method is used to establish the photon model, and two different KDP samples will be simulated to reveal the photonic transport behavior irradiated by laser in different filtered states. Furthermore, the effect of photon damage on the crystal will be demonstrated to investigate the mechanism of laser-induced damage (LID). It is significant to analyze the laser-induced damage threshold (LIDT) of optical elements and enhance the crystal growth technology for engineering applications.
2. Theoretical methods
Monte Carlo (MC) simulation is a powerful and flexible tool for simulating various physical phenomena. It allows transport solutions to be performed effectively and visually, and only physics processes and geometry need to be selected and modeled in the toolkit. Thus, the MC method is more flexible and suitable for analyzing a more comprehensive range of radiation and transport problems. Geant4 (GEometry ANd Tracking) is a Monte Carlo application package developed by the Conseil European pour la Recherche Nucleaire (CERN) based on C++ object-oriented technology [16–19]. Moreover, Geant4 is able to handle low-energy beams in analyzing biological tissues and inorganic matter, making it a suitable tool for studying low-energy particle-matter interactions [20–23]. Geant4 will be used to model photon transport for UV laser irradiation damage to KDP crystals in the model.
In order to simulate steadily, virtual simulation scenarios need to be defined with geometric media, a primary photon source, and physics processes specifically in Geant4. And the kernel will track all interactions of primary and secondary particles throughout the virtual construct, including all material properties setting and selected physical processes calculation. In this simulation, the track of photons is visible. Figure 1 has shown the whole simulation model after finally modeling.
2.1 Geometric model
Firstly, simulation geometry needs to be registered. The simulation scenario consists of a cubic world body with a smaller cube in the center. The central cube is filled with KDP crystal material, and its geometric center is located at the center of the simulated environment, coinciding with the coordinate origin of the system. Detailed data are shown in Table 1.
Table 1. Geometric parameter of the simulation model
2.2 Source and physics model
Secondly, the primary photon source needs to be set up as a UV laser source. Surface damage is in 1-on-1 damage experiments with an 11.5 J/cm2 6.8 ns UV pulsed laser [24]. In this study, energy densities are applied as the laser source, which is similar to pulse parameters with actual damage experiments. In addition, the number of emitted photons will be ensured to be 106, and the single photon energy will be 3.5 eV. The energy density of the spot is Gaussian distributed with radius. It is assumed that the radial radius of the laser source will not change with the incident depth due to the tiny quantity compared with the crystal size, and the spot area is fixed at 4.87 nm2. Since the laser source momentum direction is perpendicular to the crystal surface and the photon polarization direction is set perpendicularly to the optical axis, the refractive index of O light will be used in this study for laser irradiation.
Two samples of KDP crystals with same cut type are used for simulation and analysis. All the parameters are shown in Table 2. The actual simulation results will differ from these values to some extent. As shown in Table 2, sample #1 was unfiltered grown, while sample #2 was continuously filtered. The absorption length is the average distance that a photon travels before the medium absorbs it. Clearly, the absorption length of sample#1 is lower than that of sample #2.
Table 2. Simulation parameters of two KDP crystals of same type [25]
Finally, physics processes will be set up in the program. The optical electromagnetic physics process in Geant4 is applied in this work, which consists of Rayleigh scattering of photons, surface boundary reactions (reflection, refraction, and transmission), and linear absorption of photons. Photons will be absorbed and deposited as energy after a series of scattering, reflection, and refraction on the surface and internal crystal, while the remaining photons will leave the crystal after transmission and reflection. Although Rayleigh scattering is a type of elastic scattering and the physical process does not ultimately make energy deposition, it will affect the distribution of photons. Thus, the scattering process needs to be discussed. Photon absorption is a critical physical process, and linear and nonlinear crystal absorption is a fundamental cause of damage to the KDP crystal elements, so it must be appropriately treated in the tracking of optical photons. Research in recent years has shown that the nonlinear properties of KDP crystals originate mainly from the PO4 group, with the H-O bond contributing most of the linear absorption at 1064 nm laser irradiation; at 532 nm and complex multi-photon absorption behavior of KDP crystals has been observed at 355 nm [26,27]. Since Geant4 does not include any nonlinear physical processes, this study will mainly focus on the linear absorption behavior of KDP crystals under high-flux UV laser irradiation and corroborate the complex multi-photon absorption process of KDP crystals from the side.
3. Results & discussions
To get photon transport behavior, and obtain photon energy deposit and time evolution data in KDP crystals with different filtration states, two samples were researched in this study. To guarantee the stability and quality of the data, energy deposit units with 100-bin precision values were employed. Firstly, the simulation was verified using the Gaussian light sources, which was performed for 1-on-1 pulse width and spot radial direction as shown in Fig. 2. It is shown that the laser source generated from the simulation result could meet expectations, and Gaussian random sampling of the particle number has been set on the radius. The density of the points in Fig. 2 is the photon number density. The half-height width of Gaussian sampling is 0.6, the spot diameter is 2.5 nm after analysis. The Z-axis radius random sampling rate is slightly lower than the Y-axis, which is in line with the area setting of the light source. Therefore, the Gaussian light source set by the program can achieve the expected purpose.
3.1 Scattering behavior of photons
Although the process of elastic scattering is not the primary focus of the problem for laser-induced damage, it can affect the direction of the photon's momentum and must be observed. In this investigation, the amount of sample #1's Rayleigh scattering under laser illumination was measured. Figure 3(a) displays the scattering statistics within the crystal from -4.9 mm to 5 mm in the direction of laser emission. All Rayleigh scattering energy within the crystal in the X-direction is averaged for data processing since the crystal is in a three-dimensional condition. The scattering energy decreases with the laser incidence depth, and the decay rate gradually decreases. At the sample length of 7 mm, there is a minimum of energy deposition. As shown in Fig. 3(b) and 3(c), the scattering process on the first crystal surface perpendicular to the laser transmission direction is counted from two directions, where the peak in the Z-axis direction is slightly lower than that in the Y-axis, which is the result of random sampling error of the spot. At the laser spot position, the scattered energy reaches its peak and shows a radial decrease from the spot to the surrounding area. It is shown that the scattering energy and laser radial and propagation distance are strongly related. In the laser spot position, there are the maximums for photon number and the probability for photon collision. Along with laser incidence depth variation, a series of photon diffusion, transmission, and absorption processes will take place, resulting in a reduced probability of photon scattering. Thus, the Monte Carlo calculation also reflects the probability density of the scattering process occurring at each position of the crystal, which illustrates the transmission behavior of the crystal irradiated by the laser, and the scattering probability decreases gradually when the photons are transmitted from the front surface of the crystal to the back surface of the crystal.
3.2 Absorption of photons
The photon energy density deposition in sample #1 after UV laser irradiation was analyzed. The energy density deposition in a 1 cm3 volume of crystal #1 is depicted in the three coordinate directions in Fig. 4(a), (b), and (c). Since the energy deposition solely results from the photon absorption process, this two-dimensional graphic shows the distribution of the absorbed photons passing through the sample. The findings demonstrate the change in photon absorption energy density in the XY and XZ planes, as illustrated in Fig. 4(a) and (b). The position near the crystal surface, which is close to the order of 10−4 J/cm2, has the highest longitudinal photon energy density. In the volume of 1 cm3, the photons are not evenly distributed. In both the laser radial and propagation directions, energy density has a gradually decreasing tendency to approximative zero. As shown in Fig. 4(c), the energy density deposition at the crystal surface reaches its highest values at the laser irradiation's focal point, which are on the order of 10−4 J/cm2. The energy density deposition shows a radial pattern in the radial direction. Same as other surfaces, and the photon energy density deposition on the crystal surface also fails to fill the whole space. The energy density statistics do not represent the distribution of photons. In the blank position in the figure, there will also be a small number of photons deposited.
Fig. 4. Two-dimensional graphs of photon energy density deposition of #1 sample irradiated by 11.5 J/cm2 7 ns UV pulse laser.
In Monte Carlo simulation, only the energy deposition due to the linear absorption of photons is shown in the figure. This deposition can explain the cratered appearance of the surface damage of sample #1 irradiated by laser light. The photon distribution is in agreement with the experimental results which have been done in the stress wave in KDP crystal [28,29]. Since a large amount of photon energy is deposited at the front surface spot, melting may occur and thermal stress waves will be generated, resulting in crystal damage. Thus, it can be regarded that the massive deposition of photons is the main cause of changes in the temperature and stress fields. In large laser devices, the speed and degree of damage that occurred on the back surface of KDP crystals are often more significant than that on the front surface. Furthermore, a large number of photon depositions is one of the critical reasons for the damage on the front surface of the crystal and the cause of the damage on the rear surface of the crystal is the transmission of stress waves. It fits with the experimental data under high-flux laser irradiation [30].
It was focused on the particular photon deposition phenomena in one-dimensional energy deposition. The correlation between the laser single pulse time and action time is shown in Fig. 5(a). According to the data, the half-height width of the reacting time, the single laser pulse's peak position, and the reaction time's peak position are 7.92 ns, 14.25 ns, and 15.51 ns, respectively. The single laser pulse's pulse width is 7.06 ns. After a series of physics processes, photons are finally absorbed by the sample. The time scale of various physics processes occurring in sample #1 is extended by approximately 0.9 ns compared to the laser source pulse width. This is because the photons have a relaxation time when they interact with the crystal lattice, reaching the equilibrium of the laser-matter interaction. What’s more, it is also illustrated the time for photon linear absorption.
In the emission direction near the surface, as in Fig. 5(b), the energy deposition distribution is at -4.9 mm ∼ 5 mm, and it reaches a maximum of 1.73 × 10−4 J/cm2 at the -4.9 mm location and declines with the depth of the laser incidence. The highest energy density deposition at the crystal surface is 2.1 × 10−4 J/cm2, and the minimum value of surface attenuation reduces to a value near 0 at 4 mm vs -4 mm, as illustrated in Fig. 5(c) and (d). The slight difference between the maximum values of the Z-axis and Y-axis is due to random sampling error of the spot. It is found that the absorption pattern of photons is closely consistent with the scattering pattern of photons in the crystal, which implies the appearance probability of photons at various positions. And the locations with high scattering probability are accompanied by a high probability of photon absorption. In another way, the linear absorption of photons in KDP crystals irradiated by UV laser is regulated by elastic scattering assistance. In this sample simulation, 22,863 photons were absorbed, accounting for 2.2% of the total photon count, and the total energy density of absorption was 0.26 J/cm2. The number of reflected process photons was 99,478, accounting for 9.9% of the total photon count. And the highest number of photons occurred in the transmission process, accounting for 87.9%.
Besides, the photon energy density deposition of sample #2 was studied, which was by the filtration of the growth solution and different from sample #1. As shown in Fig. 6 (a) and (b), the photon energy density deposition inside sample #2 shows a scattered pattern, with only a few photons absorbed at an average magnitude of 10−7 J/cm2. As shown in Fig. 6(c), the scattered distribution is denser near the crystal surface, and energy density was an average magnitude of 10−7 J/cm2.
Fig. 6. Two-dimensional plot of photon energy density deposition of sample #2 irradiated by 11.5 J/cm2 7 ns UV pulse laser.
Since this sample's total energy density deposition shows a scattered pattern, the analysis process uses a Gaussian fit as shown in Fig. 7(a). The fitted Gaussian curve has a half-height width of 11.4 ns and a peak location of 17.24 ns. With a relaxation time that is about 4-5 times longer than the correlation for sample #1, the response time of sample #2 is nearly 4 ns slower than the light source. Repeated filters were used for the sample #2 growing solution. The crystal's impurity particles were greatly decreased, and the duration of the absorption was increased. The simulation shows how the impurities substantially impact the linear absorption relaxation time of photons in KDP crystals when exposed to UV laser irradiation. This is because defects and impurity particles will significantly decrease the energy bandwidth and offer an abundance of free electrons, increasing the probability of crossing into the conduction band. According to the Drude model [31], the free electron number density is inversely proportional to the relaxation time. When the impurity concentration is lower and the number density of free electrons decreases, the relaxation time scale will be longer.
The greatest energy density deposited on the crystal surface is calculated to be 4 × 10−7 J/cm2, as illustrated in Fig. 7(b), (c), and (d). In Fig. 7, the scatter points of the X-axis, Y-axis, and Z-axis distributions of sample #2 show similar variation curves as those of sample #1, but the energy density deposition of sample 2 is much lower. In this sample simulation, a total of 137 photons were absorbed with an absorption energy density of 1.58 × 10−3 J/cm2, accounting for 0.01% of the total number of photons. The number of reflected process photons was 91,917, accounting for 9.1% of the total number of photons. The largest number of photons occurred in the transmission process, accounting for 90.8% of the total number of photons.
Additionally, the energy density deposition of the two KDP samples was compared, which was shown in Fig. 8. In the X-, Y- and Z-axis views, the linear photon absorption of sample #1 is 2.3%, while sample #2 is 0.01% based on the same reflectance of samples. The photon absorption energy density of the repeatedly filtered KDP tangential crystals is much lower than that of the unfiltered KDP crystals. Therefore, the filtering technique can significantly reduce the linear absorption of photons in the UV band and increase the transmittance of photons. Because linear absorption is one of the critical causes of laser irradiation damage, this simulation also illustrates that impurity elements and defects in the crystal will seriously affect the laser damage threshold in the UV band.
During the growth process of KDP crystals, it is necessary to use a high-purity raw material of potassium dihydrogen phosphate. However, there are still many impurities. For example, the growth solution would contain some microorganisms, some precipitates, and other impurity ions (Fe3+, Cr3+, and sulfate). On the one hand, in the 1980s, some studies have shown that microorganisms and impurities can affect the bulk damage threshold of KDP crystal [32,33]. On the other hand, the material is dissolved in the growth solution to start the growth, which contains trivalent impurity ions such as sulfate, some Fe3+ and Cr3+ [34]. The addition of trivalent metal ions forms partial hydrates in the solution and tiny clusters appear in the grown crystals. Moreover, the metal ions will cause a significant decrease in the energy band at the location of the particle calculated by the first principle [35–37], increasing the probability of electron tunneling to the conduction band. Especially, it has been shown that Fe3+ and Cr3+ at 220 nm ∼ 370 nm significantly will cause the absorption of laser energy in the crystal by UV absorption spectrum [38], which is consistent with the results of the simulation. In addition, the sulfate structure is similar to phosphate, so it can connect the KDP lattice by H-bonding and electron attraction [5]. It is easy for sulfur impurities to replace phosphorus ions and consequently enter the KDP lattice during crystal growth. These impurities could change the local structure to form dislocations in the crystal, while others could form cluster and bulk defects. However, the absorption characterization of the above defect was determined by absorption length in the calculation parameter. Samples, with a relative higher concentration of defects have a lower absorption length. Whereas, crystals grown by continuous filtration have a lower concentration of defects and a longer absorption length. Therefore, the linear absorption through filtration is significantly lower than that of non-filtration by two orders of magnitude, which is the result of those defects. To sum up, if there is a defect inside the crystal, a large amount of photon absorption will occur and lead to local damage, which causes stress waves and visible mechanical damage to the crystal. The continuous filtration growth technique can improve the purity of the grown crystal, by reducing microorganisms, precipitates, and impurity particles in the growth solution. The simulation result is fit with the experimental studies [39,40] that crystals grown by continuous filtration have significantly higher transmittance, lower absorption, and higher LIDT. So that in filtered KDP crystals, linear absorption no longer makes a significant contribution in the analysis of LIDT.
In the final, the damage mechanism of filtered KDP crystal needs to be discussed briefly. UV photons could only interact with the outermost electrons of an atom due to the low energy. Considering the crystal structure of KDP from atom physics and condensed matter physics, K+ ions are linked to H2PO4- groups by ionic bonding and have the lowest first ionization energy of 4.32 eV. But it does not play a vital role in damage [8]. The first ionization energy data for the constituent elements are given in Table 3.
Table 3. The first ionization energy of the atoms of the constituent elements
At the point of atomic view, the outermost orbital of the phosphorus atom needs to absorb more than 10.3 eV of photons to ionize an electron. Phosphorus is more prone to three-photon absorption than other elements of the group under UV light with energy at 3.5 eV. In other way, it has a higher probability of multi-photon ionization compared to other elements. And phosphorus is a very central element in terms of the crystal structure. Therefore, the analysis for nonlinear absorption should be focused on phosphate element. If the crystal is under high-flux pulsed laser irradiation, multi-photon ionization will be more easily triggered. And if the crystal still contains a few of impurities and defects, it will reduce the band gap width, making it more prone to irreversible damage. The laser-induced damage process for KDP can be regarded as the result of coupling several damage theories mentioned above. Thus, the main damage pathway of the crystal is multi-photon absorption for near-perfect KDP crystals at 355 nm. The Monte Carlo particle transport simulations corroborate the first-principles calculations for the energy band changes of defective and doped KDP crystals under laser irradiation and provide strong evidence of the energy density distribution inside the crystal and on the surface in a visual way.
4. Conclusions
To visualize the photon transport behavior and demonstrate the damage cause in the photon aspect for KDP irradiated by UV laser, this study explored this problem based on the Monte Carlo method by using Geant4 software. By simulating the sample, it is found that the aggregation of a large number of UV photons on the front surface is a considerable cause of surface damage in KDP crystal. The photon distribution is highly in agreement with the stress field in the existing experiment research. And the distribution of photons shows a radial pattern, which is highly consistent with the cratered appearance shown in the actual damage experiments. By comparing the filtered crystals, it is found that the linear absorption relaxation time of the crystals is significantly related to the impurity concentration, and the relaxation time of filtered crystals is significantly longer. The impurities contribute most of the linear absorption, and the continuous filtration technology can make the linear absorption of photons significantly reduced, which is in accordance with first-principle calculations and experimental studies. The multi-photon absorption process is the crucial cause of damage to the crystal by laser irradiation. Therefore, the quality of filtration for crystal growth can directly affect the threshold of laser damage. It is meaningful to provide guidance for engineering work to improve the application of the optical element.
Funding
University of Science and Technology Beijing.
Acknowledgements
This work was supported by the Department of Physics, School of Mathematics and Physics, University of Science and Technology Beijing.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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