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Although the finite-difference time-domain (FDTD) technique has been prevailingly used to calculate the electric field intensity (EFI) enhancement at nodular defects in high-reflection (HR) coatings, the physical insight as to how the nodular features contribute to the intensified EFI is not explicitly revealed yet, which in turn limits the solutions that improve the laser-induced damage threshold (LIDT) of nodules by decreasing the EFI enhancement. Here, a simplified model is proposed to describe the intensified EFI in nodules: 1) the nodule works as a microlens and its focal length can be predicted using a simple formula, 2) the portion of incident light that penetrates through the HR coating can be estimated by knowing the angular dependent transmittance (ADT) of the nodule, 3) strong EFI enhancement is created when the focal point is within the nodule and simultaneously a certain portion of light penetrates to the focal position. In the light of the proposed model, a broadband HR coating was used to reduce the EFI enhancement at the seed by a factor about 10, which leads to a 20 times increment of the LIDT. This work therefore not only deepens the physical understanding of EFI enhancement at nodules but also provides a new way to increase the LIDT of multilayer reflective optics.
© 2015 Optical Society of America
1. Introduction
The laser-induced damage (LID) to optical components has always been a topic of extensive studies since the invention of the laser, and it remains a limiting issue for the development of modern laser systems [1–6]. The optical components are always damaged at the fluences that are much less than the intrinsic LIDTs of the optical materials. The defects, that are unavoidable in real-world optical components, have been identified as the precusors that trigger the LID at low fluences [7–18]. At present, it is generally accepted that the defects induce either strong EFI enhancement and/or additional absorptivity, which results in the enhanced light absorption in vicinity of defects and initiates LID. Among different kinds of defects, the nodules that are embedded in dielectric multilayer coatings are the most extensively studied defects from the aspect of how defects contribute to EFI enhancement and how the intensified EFI affects the LID of optical coatings [19–30]. Nodules grow from seeds (particulates) into an inverted conical shape and they are the main limiting defects for the multilayer reflective optics working in nanosecond or femtosecond regimes [26, 29]. FDTD technique has been successfully used to simulate the intensified EFI in nodules. A direct link between the simulated EFI distributions and the damage morphologies of the nodules has been observed, which convincingly proved that the EFI enhancement played a key role in LID of nodules [28].
It is natural to think that, by reducing the EFI enhancement, it is possible to increase the LIDT of nodules. However, the difficulty lies in how to control the EFI enhancement at nodules. FDTD technique only calculates the intensified EFI, but it does not involve any physical understanding of EFI enhancement at all. The physical insight into the intensified EFI in nodules is limited. Although the microlens model [19–21] and the angular dependent transmission (ADT) model [23] have been proposed to explain the intensified EFI in nodules, these models are over simplified and are not complete enough to resolve the inverse problem like how to change the nodular features to reduce the EFI enhancement. As a result, eliminating nodules rather than reducing EFI enhancement is the predominant way to improve the LIDT of multilayer reflective optics [31–33]. However, the processes that eliminate nodules have their inherent disadvantages or limitations, which make the elimination of nodules still quite challenging [34]. It is therefore imperative to explore the new approach that improves the LIDT of nodules by reducing the EFI enhancement. This new approach is inherently parallel to the nodule elimination processes and can be incorporated into them, which can significantly improve the robustness of nodule control and help to achieve the multilayer reflective optics with higher LIDT.
This work focuses on exploring the underlining physics of EFI enhancement at nodules. First, an example is given to reveal why the current models have difficulty in interpreting the EFI enhancement at nodules. Then, via studying the focusing characteristics and light penetrating behaviors in isolation, a simplified model is established to explain and predict how the nodular features affect the EFI enhancement. In the light of the established model, the possible approaches that can reduce the intensified EFI are proposed. Finally, an example is given to demonstrate that modifying multilayer design is an efficient way to reduce the EFI enhancement at nodules and to improve the LIDT of the near infrared HR coating.
2. The difficulty in interpreting the EFI enhancement at nodules
The properties of nodules are dependent on the fabrication process of the multilayer coatings. Different processes or even the same process but different parameters can result in quite different nodular features such as seed geometry, seed diameter, lodging depth of seeds, nodular geometry, ADT, etc. Here, via studying the influence of nodular geometry on the intensified EFI, we try to illustrate the difficulty of using previous models to interpret the EFI enhancement at nodules. The geometries of classic nodules can be presented by a simple equation D = sqrt(Cdt), where D is the nodular diameter, C is a constant, d is the diameter of a spherical seed and t is the seed depth. Our previous study has shown that D = sqrt(8dt) and D = sqrt(4dt) nodules exhibited different EFI distributions [28]. Here, these two geometries are further investigated with the focus on the physical insight toward the EFI enhancement at nodules. For practical interest, the nodules investigated here were embedded in the HfO2/SiO2 HR coating working at 1064nm for normal incidence and they were created from 1.9μm SiO2 microspheres on the quartz substrate surface. It is worth to note that fused silica substrate can also be used. Using crystalline or non-crystalline substrate has no influence on the results that are obtained in the following sections. The coating design is [air: L(LH)^13 Substrate], where H and L means a quarter-wave HfO2 and SiO2 layer respectively. A detailed description of the HfO2/SiO2 HR coating, geometrical modeling of nodules and FDTD algorithm used in the present study can be found in our previous paper [28]. For the self-consistency of this work, the spectral curves of the quarter-wave HfO2/SiO2 HR coating, the geometrical models and the EFI enhancement at nodules are briefly reintroduced here. Figure 1(a) gives the ADT curves of the quarter-wave HfO2/SiO2 HR coating. Figures 1(b) and 1(c) show the cross-sectional geometrical models of two kinds of nodules. For the D = sqrt(8dt) nodule, the layers that form above the seed are sections of concentric spheres. Whereas, for the D = sqrt(4dt) nodule, the layers that form above the seed are sections of tangent spheres, and the point of tangency is the intersection point between the spherical seed and the substrate surface. FDTD simulation shows that the two kinds of nodules exhibit quite different EFI distributions, as given in Figs. 1(d) and 1(e). It is worth to note that the EFI peaks at the central axis of nodules are responsible for triggering the LID. Because the P-polarized and S-polarized EFI distributions are very similar for this perspective, the P-polarized EFI distributions are taken as examples for this work. Moreover, it is worth to note that all the coating materials used in our FDTD simulations are assumed to be non-absorbing.
Fig. 1 The D = sqrt(8dt) and D = sqrt(4dt) nodules in the quarter-wave HfO2/SiO2 HR coating. (a) P-polarized and S-polarized angular dependent transmission curves of the quarter-wave HfO2/SiO2 HR coating. (b and c) Geometrical modeling of the D = sqrt(8dt) nodule (b) and the D = sqrt(4dt) nodule (c). (d and e) FDTD-simulated P-polarized EFI distributions in vicinity of the D = sqrt(8dt) nodule (d) and the D = sqrt(4dt) nodule (e). Two nodular geometries show quite different EFI distributions.
Light focusing and light penetrating in nodules are two known effects that contribute to the EFI enhancement. There were already several attempts to treat the nodule as a microlens to describe its focusing characteristics [19–21]. However, these studies dwelt on the qualitative description and failed to describe the nodular focusing characteristics quantitatively. For example, previous microlens models can’t estimate which hotspots in Figs. 1(d) and 1(e) are the focal points of the D = sqrt(8dt) and D = sqrt(4dt) nodules. There are two common arguments against the microlens model. One is that simple geometric optics is probably not applicable for nodules with sizes smaller than about 10 times of the wavelength of light, as is the case here. Another is that the influence of light focusing on the EFI enhancement at nodules is easily shadowed by the complicated interference of multilayer coatings. It is necessary to know how the incident light penetrates through multilayer coatings to make better use of the microlens model.
An ADT model has been proposed to describe light penetrating in nodules [23], which is based on a fact that a nodule is exposed to a broad incident angular range (IAR) as well as to both P-polarization and S-polarization at orthogonal cross sections even though the linearly polarized laser irradiates on the multilayer coating at a fixed incident angle. Figures 1(b) and 1(c) show that the angle of incidence gradually increases from zero to the maximum angle θ when the point of incidence moves from the nodular center to the edge. When the IAR of the nodule is larger than the limited angular reflection bandwidth (ARB) of the HR coating, the incident light can penetrate the HR coating through the nodular edge. The ADT model can partially explain why the D = sqrt(8dt) and D = sqrt(4dt) nodules exhibit different EFI distributions. Table 1 gives a comparison among the IARs of the two nodules and the ARBs of the HfO2/SiO2 HR coating. The IAR of the D = sqrt(8dt) nodule is slightly larger than the P-polarized ARB of the HfO2/SiO2 HR coating. A very high reflectivity occurs over almost the whole IAR of the D = sqrt(8dt) nodule, therefore the EFI peaks occur in the outmost several layers due to the interference and diffraction. Whereas, the IAR of the D = sqrt(4dt) nodule is even larger than the S-polarized ARB of the HfO2/SiO2 HR coating, which results in significantly greater transmission at the outer boundary of the nodule. When the transmitted light arrives at the central axis coherently and in phase, it creates a strong EFI enhancement. However, only the ADT model itself can’t tell whether the light penetrating plays a dominant role in EFI enhancement or not, because the influence of the interference and diffraction within nodules on the intensified EFI can’t be excluded. There was no report on reducing the EFI enhancement by decreasing the light penetration through HR coatings.
Table 1. A comparison among IARs of nodules and ARBs of HfO2/SiO2 HR coating
The difficulty to gain more physical insight into the EFI enhancement at nodules lies in a fact that the contribution of light focusing and light penetrating to the intensified EFI are usually not isolated but rather mutually related with each other. For example, the D = sqrt(8dt) and D = sqrt(4dt) nodules not only have different focusing characteristics but also have different light penetrating behaviors, this prevents the achievement of a clear understanding of both effects. To establish a simplified model to describe the EFI enhancement at nodules, it is necessary to investigate the focusing characteristics and light penetrating behaviors in isolation.
3. A simplified model to describe the EFI enhancement at nodules
Special coating designs are used to study the light focusing and the light penetrating effects in isolation. The focusing characteristics of nodules are first addressed. To carry out a comparative study, a new series of D = sqrt(8dt) and D = sqrt(4dt) nodules were created with two requirements: 1) two kinds of nodules exhibit similar light penetrating behaviors; 2) the dimensions of these new nodules are the same as the dimensions of the nodules in the quarter-wave HfO2/SiO2 HR coating. To meet these requirements, each HfO2 layer in the quarter-wave HfO2/SiO2 HR coating was replaced with a SiO2 layer that has the same physical thickness. As a result, the new series of nodules are in a SiO2 single layer, and they have the same dimensions and the comparable focusing characteristics with the nodules in the quarter-wave HfO2/SiO2 HR coating. Figure 2(a) shows that the ADT of the D = sqrt(8dt) and D = sqrt(4dt) nodules in the SiO2 single layer is quite similar. The difference in focusing characteristics is the main reason that these two nodules exhibit different EFI distributions, as given in Figs. 2(b) and 2(c). Compared to all previous FDTD simulations, a larger simulation domain in the coating-substrate direction is used here to reveal the focusing characteristics of nodules. The maximum EFIs are very similar for these two new nodules, but the focal length of the D = sqrt(8dt) nodule is much longer than that of the D = sqrt(4dt) nodule.
Fig. 2 The D = sqrt(8dt) and D = sqrt(4dt) nodules in SiO2 single-layer coating. (a) P-polarized and S-polarized angular dependent transmission curves of the SiO2 single-layer coating. (b and c) FDTD-simulated P-polarized EFI distributions in vicinity of the D = sqrt(8dt) nodule (b) and the D = sqrt(4dt) nodule (c). Compared to the D = sqrt(8dt) nodule, D = sqrt(4dt) nodule exhibits a shorter focal length.
Here, we take a step away from the conventional opinion that simple geometric optics is not applicable for describing the focusing characteristics of nodules and revisit the microlens model [35]. The spherical feature of the nodule in the SiO2 single layer actually makes it a SiO2 microlens, its focal length is determined by
where f is the focal length, n is the refractive index of the medium and the r is the radius of curvature of the microlens. Equation (1) reflects that bigger radius of the curvature leads to greater focal length. Comparing the Figs. 1(b), 1(c), 2(b) and 2(c), it can be seen that the trend reflected by Eq. (1) is consistent with the FDTD simulation results. For example, the D = sqrt(4dt) nodule has smaller radius of curvature and it actually has a shorter focal length. However, the focal lengths calculated by Eq. (1) and the FDTD algorithm show considerable deviation. Table 2 shows that the focal lengths obtained from the FDTD simulations are about 20% smaller than those calculated using Eq. (1). Since the correctness of our FDTD simulation results have been proven through experimental studies [28], Eq. (1) needs to be slightly modified to beto correctly estimate the focal lengths of nodules. k(r, n) is a correction factor that is given in Table 2. Then Eq. (2) can be used to predict the focal lengths of the nodules.Table 2. A summary of the focal lengths calculated using the microlens formula and the FDTD algorithm
Equation (1) also reflects that bigger refractive index leads to shorter focal length. To check whether this trend is pronounced or not, the EFI enhancement at the D = sqrt(4dt) nodules in two more single layers were simulated using the FDTD algorithm. One coating material is HfO2 whose index is 1.962 and another coating material is an artificial one whose index is 3.2. The above thick SiO2 single layer was replaced respectively with these two single layers that have the same physical thickness. The ADT curves in Figs. 3(a) and 3(b) show that the transmittance of both single layers over the IAR of the D = sqrt(4dt) is still very high, and a significant portion of light can penetrate through the nodules. Figures 3(c) and 3(d) present the FDTD simulation results. Equation (1) once again reflects the right trend that higher refractive index of the coating material leads to shorter focal length. It is also found that the difference between the focal lengths calculated by Eq. (1) and the FDTD algorithm is larger when the refractive index of the coating material is lower. The dependence of the correction factor k(r, n) on n and r can be interpreted by taking the wave property of the light into account. As the diameter of the nodule becomes smaller or the refractive index of the coating material becomes lower, the wave property of the light plays a more important role in determining the focusing characteristics of the nodules. So the correction factor needs to become smaller to account for the discrepancy between the simple geometrical optics and the electromagnetic simulations, as given in Table 2. As to the nodule in multilayer coatings, its focal length can be correctly estimated by knowing the equivalent refractive index of multilayer coatings.
Fig. 3 The D = sqrt(4dt) nodules in two single-layer coatings. (a and b) P-polarized and S-polarized angular dependent transmission curves of the HfO2 single-layer coating (a) and the artificial material single-layer coating (b). (c and d) FDTD-simulated P-polarized EFI distributions in vicinity of the nodules in the HfO2 single-layer coating (c) and the artificial material single-layer coating (d). The focal length of the nodule decreases with the increasing refractive index of the medium and the EFI enhancement gets stronger in the medium having higher refractive index.
Figures 2(c), 3(c) and 3(d) also reflect that higher refractive index of coating material results in stronger EFI enhancement. The ADT model fails to explain this phenomenon, because the nodule with greater reflectivity over the IAR shows stronger EFI enhancement. The diffraction effect of the nodule must be taken into consideration to qualitatively explain the observed dependence of the EFI enhancement on the refractive index of coating materials. If the nodule is treated as a diffracting circular aperture having a focal length, it produces an Airy function at its focal plan. The diameter of the central Airy disc is reversely proportional to the refractive index of the medium, so bigger refractive index of the coating material leads to stronger EFI enhancement.
Using the above single layers, the focusing characteristics of nodules were studied in isolation, which lays a good foundation for exploring the influence of light penetration on the intensified EFI in isolation. To highlight the contribution of light penetrating behaviors to the EFI enhancement at nodules, an extreme condition using omnidirectional HR coating is considered. SiO2 and the artificial material whose index is 3.2 were used to create the omnidirectional HR coating. The coating design is [air: L(LH)^16 Substrate]. The key point here is also to keep the total thickness of the omnidirectional HR coating almost equal to that of the above coatings. Therefore the D = sqrt(4dt) nodule in the omnidirectional HR coating has the comparable focusing characteristics as the above D = sqrt(4dt) nodules, and then the contribution of light penetration to the EFI enhancement at nodules can be studied in isolation. Figure 4 gives the ADT curves of the omnidirectional HR coating and the corresponding EFI distribution at the nodule. Compared to the above multilayer and single layer coatings, no portion of light can penetrate through the omnidirectional HR coating to the focal position of the D = sqrt (4dt) nodule. The maximum EFI is only about 3, which is significantly reduced compared to the maximum EFIs in the above D = sqrt(4dt) nodules. This example using the omnidirectional HR coating further proves the effectiveness of the ADT model in a reliable way. For nodules having comparable focusing characteristics, the portion of the incident light that can penetrate through the coatings to the focal position plays a dominant role in determining the EFI enhancement at nodules.
Fig. 4 The D = sqrt(4dt) nodule in the omnidirectional HR coating. (a) P-polarized and S-polarized angular dependent transmission curves of the omnidirectional HR coating. (b) FDTD-simulated P-polarized EFI distribution in vicinity of a nodule. The scale bar is set the same with the one in Fig. 1(e) to show that the omnidirectional HR coating significantly reduces the EFI enhancement at the nodule.
Now a simplified model describing the EFI enhancement at nodules can be given: strong EFI enhancement is created when some portion of the incident light penetrates to the focal position of nodules. The focal length of a nodule can be predicted using a modified formula that originates from the microlens model. And the portion of the incident light that penetrates to the focal position can be roughly determined by comparing the IAR of the nodule and the ARB of the coating.
4. The applicability and limitations of the proposed model
The proposed model can succinctly explain and predict the EFI enhancement at different kinds of nodules. Here two examples are discussed to show the applicability of the proposed model. The first example is to estimate the EFI enhancement at different D = sqrt(Cdt) nodules. As to the same-sized seed, smaller constant C presents smaller nodule. For the focusing characteristics, smaller nodule exhibits a smaller radius of curvature and then a smaller focal length. For the light penetrating behaviors, smaller nodule exhibits a greater IAR. Taking the limited ARB of quarter-wave HR coating into consideration, the portion of the incident light that can penetrate through the smaller nodule is likely larger. As a result, the nodules having smaller aspect ratio tend to create stronger EFI enhancement at shallower positions. This can perfectly explain our previous FDTD simulations that the D = sqrt(2.5dt) nodules induced stronger and shallower EFI peaks than the D = sqrt(4dt) nodules [28]. Another example is to explain the dependence of EFI enhancement on the lodging depth of the seed. For the same-sized seed, shallower lodging depth of the seed generates a smaller nodule. This not only results in a smaller focal length but also leads to a greater IAR of the resulting nodule. So the nodules initiating from shallow seeds tend to induce stronger EFI enhancement at shallower positions, which can explain the previous report that D = sqrt(8dt) nodules originating from shallower seeds exhibited stronger and shallower EFI peaks especially for the normal incident case [24]. In addition, the proposed model also tells that previous FDTD simulation domain in coating-growth direction was not large enough to expose the EFI enhancement at the focal position of the D = sqrt(8dt) nodules even though some portion of light could penetrate to there.
The proposed model also has its limitations. Some features of the intensified EFI, for example, the hotspots in the outmost several layers of the nodules in Figs. 1(d) and 1(e), can’t be interpreted using the proposed model. The interference and diffraction among the first several layers develop these EFI peaks, but a simple model describing these complicated effects is still absent. In addition, the previous report of the abnormal dependence of the EFI enhancement on the seed diameter also can’t be explained using the proposed model [24]. It has been speculated that the resonant effect played a dominant role in creating the abnormally strong EFI enhancement [36, 37]. However, the resonance effect at nodules is still not well understood yet, more work is needed to clarify it.
Above all, the proposed model is thought to reflect the primary mechanisms of the EFI enhancement at nodules, which not only lays a good foundation for further exploration of other mechanisms but also provides the guidance to control the nodular features to reduce the EFI enhancement at nodules.
5. One approach to reduce EFI enhancement at nodules
According to the proposed model, the EFI enhancement at nodules can be controlled via changing the focusing characteristics or the light penetrating behaviors of the nodules. Compared to changing the focusing characteristics through the modification of the nodular geometry, altering the light penetrating behaviors can be easily achieved by modifying the design of multilayer coatings. So in this work, we demonstrate an approach to suppress the EFI enhancement at nodules via decreasing the portion of the incident light that can penetrate through the multilayer coating to the focal position.
A broadband HfO2/SiO2 HR coating that works at 1064nm for normal incidence is designed with its ARB bigger than the IAR of the D = sqrt(4dt) nodule. Its total thickness is about 9.1μm. Figure 5(a) shows that P-polarized and S-polarized ARBs of the broadband HfO2/SiO2 HR coating are [-60, 60] and [-90, 90] degree respectively. An electron beam evaporation process was used to deposit the broadband HfO2/SiO2 HR coating over the 1.9μm SiO2 and hafnium-coated SiO2 seeds on quartz substrates. The details of the preparation of the artificial nodules were given in our previous work [30]. Focus ion beam technology was used to reveal the cross-section patterns of the prepared artificial nodules. Figure 5(b) confirms that this nodule also exhibits the D = sqrt(4dt) geometry. It is worth to note that, for the same-sized seed, increasing the coating thickness leads to a decreased IAR of the resulting nodule. The IAR of this new nodule is about [-50, 50] degree, which is smaller than the ARBs of the broadband HfO2/SiO2 HR coating. Figure 5(c) gives the P-polarized EFI distribution at the nodule. The broadband HR coating prevents the light from penetrating to the focal position and successfully suppresses EFI enhancement. The maximum EFI in this nodule is only 5, which is more than three times less than the maximum EFI at the D = sqrt(4dt) nodule in the quarter-wave HfO2/SiO2 HR coating.
Fig. 5 The D = sqrt(4dt) nodule in the broadband HfO2/SiO2 HR coating. (a) P-polarized and S-polarized angular dependent transmission curves of the broadband HfO2/SiO2 HR coating. (b) Cross-section pattern of a D = sqrt(4dt) nodule. (c) FDTD-simulated P-polarized EFI distribution in vicinity of a nodule. The scale bar was set the same with the one in Figs. 1(e) and 4(b) to show that the broadband HfO2/SiO2 HR also significantly reduces the EFI enhancement at the nodule. (d) The damage morphologies of a nodule that was created from a SiO2 microsphere. The damage occurred at the central location where the EFI is the strongest. (e) The damage morphologies of a nodule that was created from a hafnium-coated SiO2 microsphere. The damage initiated at the seed where the absorption is the highest.
To confirm that reducing EFI enhancement can effectively improve the laser damage resistance of multilayer reflective optics, the LIDT of the nodules in the broadband HR coating was measured and compared to that of the previously prepared nodules in the quarter-wave HfO2/SiO2 HR coating [30]. Table 3 shows that the broadband HR coating significantly increased the LIDT of nodules. The damage morphologies of the ejected nodules were also characterized to explain the observed LIDT increment. The cross-section patterns of the ejected nodules in the broadband HfO2/SiO2 HR coating are given in Figs. 5(d) and 5(e). The damage morphologies of the nodules in the quarter-wave HfO2/SiO2 HR coating can be found in our previous work for reference [30]. For the nodules initiating from the SiO2 seeds, the LID initiates at the location where the EFI is the strongest. As to the broadband HfO2/SiO2 HR coating case, the upper part of the multilayer coating is damaged. Whereas, for the quarter-wave HfO2/SiO2 HR coating case, the SiO2 seed is melted [30]. Compared to the quarter-wave HfO2/SiO2 HR coating, the broadband HfO2/SiO2 HR coating reduces the EFI enhancement by a factor of 3 and leads to about 2 times increment of the LIDT. Moreover, for the nodules initiating from the absorbing hafnium-coated SiO2 seeds, the whole nodules are ejected from the bottom for both HR coatings because the high absorptivity of the seeds plays a dominant role in triggering the LID. Compared to the quarter-wave HfO2/SiO2 HR coating, the broadband HfO2/SiO2 HR coating reduces the EFI enhancement at seeds by a factor more than 10, and the LIDT increases by 20 times. Above all, modifying multilayer design can effectively reduce the EFI enhancement at nodules, increase ejection fluences of nodules and improve the laser damage resistance of multilayer reflective optics.
Table 3. A summary of LIDTs of two kinds of HfO2/SiO2 HR coating (1064 nm, 10 ns)
6. Conclusion
Light focusing and light penetrating have been proved to be the main mechanisms for creating the intensified EFI in nodules. Strong EFI enhancement is developed when a certain portion of the incident light penetrates to the focal position of the nodules. A simplified model was established to describe the EFI enhancement at nodules, where the focusing characteristics of the nodule can be predicted by a simple formula and the light penetrating behaviors can be estimated by knowing the ADT of the nodule. The proposed model not only can explain the dependence of EFI enhancement on nodular features but also can provide guidance on how to change the nodular features to reduce the EFI enhancement. According to the proposed model, a broadband HR coating was prepared to prevent the incident light from penetrating to the focal position of the nodule, which reduced EFI enhancement at nodules and then significantly increased the ejection fluences of nodules and the LIDT of multilayer reflective optics. Moreover, the proposed model also lays a good foundation for exploring the other mechanisms that contribute to the EFI enhancement at nodules.
Acknowledgments
National Natural Science Foundation of China (61235011, 51475335), the National Major Research Equipment Development project (ZDYZ2013-2), the National Key Scientific Instrument and Equipment Development Project (2014YQ090709).
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