1. Introduction

Chirped-pulse amplification using pulse-compression gratings remain the prevailing method for generating high-power, ultrashort laser pulses [1]. In this type of laser system, multilayered dielectric gratings (MLDs) are often used to stretch the pulse before amplification and its subsequent recompression while offering the highest laser-induced–damage threshold [2–6]. Owing to their periodic subwavelength groove structure on the surface, MLD gratings produce an enhanced electric field inside the grating pillars. This enhanced field intensity (EFI: the square of the electric-field amplitude normalized by that of the input beam) has been shown to be closely related to the laser-induced–damage threshold (LIDT) of the grating, as well as the location of damage initiation [7–10]. The damage threshold, in turn, limits the maximum output of the laser system.

Significant research effort has been devoted to minimize the field-enhancement effect inside the pillars (within the material optical and mechanical strength tolerance parameter space) by improving the design and fabrication methods [11–15]. However, the improvement of laser damage using optimal grating designs only solves part of the problem. In actuality, and especially for large-aperture laser systems, there is no perfect grating, or grating with an ideally geometric form. Gratings can have irregular features due to manufacturing defects and can be damaged by the laser, causing partial or complete removal of pillar sections and debris that can be redeposited on the surface. Gratings can also be exposed to particle contamination [16] due to handling or the operational environment. To advance understanding of the performance of gratings in laser systems, it is important to study the field enhancement introduced by these inadvertent flaws. This knowledge will help to better understand the damage performance and operational limits of the optics. Specifically, each material constituent of the coating material (silica, hafnia, etc.) has its own damage threshold (which can be measured using monolayers or other methods). The 3D EFI can be used to estimate the maximum input fluence/intensity to avoid any layer (or section of a layer) of the grating (or other optic) to exceed its damage threshold. This knowledge, in turn, may help to design gratings that offer increased damage performance in conditions relevant to the actual operational environment of short-pulse laser systems.

Grating-fabrication–induced defects include wall imperfections that are associated with the etching process. A case example is shown in Fig. 1(a) for a commercially fabricated grating. The scanning electron microscope (SEM) image of the pillars shows the presence of structures on the edges of the walls that appear as ridges. These features are much smaller than the pillar width; an optimally designed grating has rather wide tolerance with respect to its width, therefore, such small features will not significantly affect the grating diffraction efficiency. The electric-field distribution, however, maybe substantially affected by such features. Other types of manufacturing defects include broken pillars or pillars having incorrect spatial dimensions and/or shape.

 

Fig. 1. (a) Slanted scanning electron microscope (SEM) image of a grating to capture the top coating layers and the vertical ridge features on pillar walls that are related to the fabrication process. (b) Damage created on grating pillars viewed from the top under exposure to 0.7-ps pulses incident from the left at 62°.

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Features arising from laser-induced damage in gratings were previously discussed (including the specific type considered in this work, see [9]) and include partially or completely removed pillar sections and the distribution of pillar fragments in the vicinity of the damage site area. When a grating is damaged by picosecond laser pulses, the damage may be limited to the removal of a section of a single pillar or include the removal of sections of few neighboring pillars. Damage initiation with femtosecond pulses typically involves only a partial removal of a pillar, where a notch is created with a representative examples, shown in Fig. 1(b). In either case, this localized change of geometry can cause changes in the field distribution, giving rise to an increase of the EFI in localized regions in both the adjacent pillars or pillar sections and the underlying coating layers. This, in turn, can support the commencement of laser-damage initiation at a lower nominal incidence intensity (compared to that of a “pristine” grating LIDT), thus reducing the damage performance of the optic under subsequent laser irradiation [10]. Furthermore, the damage process can create microscale debris that are deposited either on the pillars or in the grooves. These can also be sources to create local EFIs. Additionally, particle contamination in a typical operational environment is a great concern. This issue was discussed in more detail recently [16] showing that the pulse-compression chamber containing the gratings can have a very high density of contamination particles that are typically smaller than 5 µm. It was hypothesized that these particles can originate from the target chamber (where the laser energy is focused onto targets) as (if) there is no physical barrier to prevent some of the generated ablation byproducts to migrate backward along the beam path.

In this work we employ the finite-difference time-domain (FDTD) method to perform 3D modeling of the electric-field distribution in gratings containing inadvertent defect features with a sufficiently high spatial resolution of 1/74 of the laser wavelength while considering a volume of 13.2 λ in the x axis (direction vertical to grating pillars) by 4.3 λ in the y axis (along the direction of the pillars) by 8.6 λ along the z axis (vertical to grating surface). We concentrate on features that are relevant to those observed in laser systems arising from manufacturing defects, contamination particles, and various laser-induced–damage morphologies from both picosecond and femtosecond pulses. The ensuing localized changes in the electric-field distribution is quantified by comparison to that of an ideal, undamaged grating. This FDTD approach offers adequate spatial resolution to encompass the defect features of interest extending to the ≈10-nm spatial scale. The results provide information on the electric-field distribution suggesting the presence of regions of significantly increased intensity in the pillars and the underlying high-index material layer.

2. Modeling approach

The field distribution in the geometrically flawless grating structure has been previously studied using 2D analysis tools by various investigators [2–15] showing that there is a hot spot (a localized region of maximum EFI) in the upper half of the grating pillar. In fact, damage test data show that it is indeed where the initial laser damage is initiated at threshold conditions where the high field causes part of the pillar to be fragmented [7–9]. For a perfect grating, the solution of this problem in 2D is adequate since the s and p polarizations are disentangled as these two polarizations are the eigenmodes of the Maxwell equations. Two-dimensional modeling was also used to analyze electric-field modifications arising from the presence of grating defects and imperfections [17–19]. To properly handle estimation of the electric-field distribution arising from factual structural defects, however, 3D modeling is required that contains all E and H field components, with adequately high spatial resolution over sufficiently large integration volume to accurately capture its distribution of the EFI in the pillars and in the coating layers below the grating pillars.

There are two general approaches to solve the Maxwell’s equations—the spatial finite difference coupled equation and the time-domain approaches. In the former, a 3D grid is established with the E and H fields coupled with adjacent grid points. The coupled equation is solved for steady-state solution of both E and H fields at each point. Commercially available software packages provide such capacities. In this approach, the number of variables for a 3D problem is computationally in proportion to the sixth power of the number of grid pixels in each side of this volume, i.e., both the memory requirement and computation power scales as N⁶, where N is the number of pixels on each side. The second is the time-domain approach, where an initial E and H field distribution propagates according to the Maxwell equation written in finite time-difference form. In this case, the memory requirement scales as N³, and the computation time scales with N⁴. Three-dimensional FDTD has been effectively used to study the field effect caused by defects under coating layers for femtosecond laser pulses [20]. To calculate the effects introduced by the features involved in this study, we need an area of grating that contains several wave periods (wavelengths) in the two dimensions in the grating plane, as well as several wave periods in the vertical direction. Since some defect features under consideration are smaller than the grating pillar width, at least ten or more grid pixels per grating period (we used 74 in this study) are needed to realize adequate spatial resolution. Consequently, the time-domain approach is adopted in this work to perform such calculations and was used to obtain quasi-steady-state solutions. The code was developed in-house to perform FDTD calculations using the IDL (Interactive Data Language) programming language. The aim of this code is to not only enable static modeling in 3D, but also transient behaviors and extend its capabilities to include laser–matter interaction physics in the future. Arguably, this capability is not currently provided by existing commercially available software, at least for the temporal and spatial resolution and volume needed for accurate representation of the related phenomena.

Our 3D FDTD modeling follows the standard Yee’s algorithm [21,22], where two sets of electric E and magnetic H complex field components are defined on a 3D grid with pixel numbers of N_x, N_y, and N_z in each direction. The two sets are offset by a half spatial step and half temporal step, which is the basic feature in Yee’s algorithm. The total number of variable values are maintained and updated in each cycle is (2 × 6 × N_x × N_y × N_z). This numerical analysis technique is widely used in computational electrodynamics and provides good stability when the following stability requirement is met: dx < dt × v, in all materials, where v is the speed of light at a specific location, and dx and dt are the spatial and temporal step sizes.

The grating type considered in this work is based on a SiO₂/HfO₂ multilayer dielectric coating with an etched top silica to form pillars and mimics the pulse-compression grating design used on OMEGA EP laser system. This grating is designed for operation at 1053 nm [23] and has 1740 line/mm for a 62° angle of incidence with s polarization. The pillar height is 600 nm with a duty cycle of 27%. This grating design has a theoretical efficiency of over 98% and maximum EFI value of about 3 inside the pillars. Modeling was performed using different numbers of high- and low-index material pairs followed by a reflective (metal simulating) layer. The results yielded differences of the order of 1% when more than two pairs were used. To most efficiently utilize the available computational power to achieve the required spatial resolution, the structure was simplified to include the grating pillars (600 nm) and sole (50 nm) and two pairs of high- and low-index material followed by a high reflector, as shown in Fig. 2(a). A 3D coherent pulse with a super-Gaussian of the order of 6 spatially (in each direction) is created in this vacuum space and propagates toward the grating structure at 62°. Allowing time for the field distribution at the center of the grating surface to become steady state, the three electric-field components (E_x, E_y, E_z in the coordinate system as shown in Fig. 2) are recorded at each point of the grating structure. By experimenting with the elapsed propagation time, we found that approximately six optical cycles are sufficient to reach stable field amplitude values (because the field only penetrates the first few layers). Therefore, we consider this approach as adequate for pulses longer than about 20 fs. The array size used in this model is 920 × 300 × 600 pixels and represents approximately 13.2 × 4.3 × 8.6 wavelengths (≈14.0 × 4.5 × 9.1 µm, where each pixel is ≈14 nm or 1/74 of a wavelength), containing exactly 23 periods of grating pillars.

 

Fig. 2. (a) Schematic representation of the grating structure used in the modeling. (b) Three-dimensional depiction of the various defect features located on the pillars and trenches of the grating introduced in a modeling of the ensuing electric-field distribution in 3D; (1) wall ridges (as 14- and 28-nm-thick vertical lines,); (2) fragments or contamination particles on top of pillars and (3) bottom of trenches; and (4) missing pillar sections in one or more pillar rows.

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In order to have a large and flat incident field distribution that covers many grating periods, it is desirable to use the periodic boundary condition in the x direction (orthogonal to the pillars). However, this requires that the dimension in the x direction contains an integer number of grating periods, and also that the incident field at both ends of the x dimension have the same phase. By selecting a width of 23 periods in the x direction for our incident wavelength and incident angle, this phase difference is less than 1% of an optical cycle and produces no noticeable discontinuous effects. In the y direction, the periodic boundary condition implies that the grating contains periodic defects. Consequently, treating this as a single defect is valid as long as the size of the defects is small compared to the field width. For this reason, the area confined with the model volume is maximized so that the EFI values do not noticeably change as we vary the dimension of the defect features. In the z direction, the periodic boundary condition is valid after the time of integration is long enough for the field distribution in the grating to reach quasi-steady state (but before the reflected wave reaches the top boundary). Three-dimensional modeling with such spatial and temporal resolution requires significant computational power. As a result, the array comprising the spatial dimensions and resolution must be carefully selected. We have tested the code in each case using varying parameters to ensure that the EFI values in the parts of the structure of interest, namely the grating pillars and the hafnia layer below, converge to within 2% or better.

We used this 3D FDTD model to study several different representative cases of defect features located on the pillars and trenches of the grating depicted in Fig. 2(b). These flaw features have been observed in gratings and arise from manufacturing defects, laser-induced damage, and contamination due to handling and exposure to a contamination-inducing operational environment. To better relate the increase of the EFI compared to the pristine grating, we express the enhanced electric field in two methods. The first method provides a relative value normalized by the incident intensity (used in the color images), while the second method normalizes by the maximum EFI for the perfect grating and is expressed as a percentage of EFI increase (used in the tables).

3. Results and discussion

3.1 Pillar ridges

Scalloping features on the pillar walls are modeled assuming thin vertical columns (rectangular and the same height as the pillars) attached on both sides of the pillar walls at a density relevant to that shown in the SEM image of the grating in Fig. 1(a). For the purpose of evaluating the effect of pillar wall ridges on the EFI, we executed calculations assuming that the thickness of the columns is 1 pixel (≈14 nm) or two pixels (≈28 nm). Representative results of the EFI calculated in 3D are shown in Fig. 3 as cross sections along the x–y plane [see Fig. 2(b)] that exhibits maximum EFI in the pillars (at 200 nm from the pillar top). Specifically, Figs. 3(a) and 3(b) represent the EFI inside the pillars assuming 14-nm and 28-nm-thick ridge features, respectively. The insets in Fig. 3 show characteristic sections with 5× magnification to enable better visualization of the locations of induced maximum EFI. The results reveal that the ridge features introduce increased EFI (compared to a pristine grating) in the pillar region between the ridge features, as shown in more detail in the insets, with the EFI value increasing with increasing ridge thicknesses. As a result, the maximum EFI is 13% and 27% higher inside the silica pillars for the two thicknesses (14 nm and 28 nm, respectively) used in this modeling compared to a pristine grating structure. In addition, the maximum EFI inside the topmost hafnia layer is increased progressively with increasing feature size by 5% and 9% compared to a pristine grating structure for the 14-nm and 28-nm feature thicknesses, respectively. The distribution of the EFI in 3D is better shown in Fig. 4 for the case of ridge features having thicknesses of 28 nm, where the attached video represents, frame by frame, the EFI distribution along the z–x plane with each frame shifted along the y axis.

 

Fig. 3. Modeling of the electric-field intensity (EFI) inside the grating structure containing ridge features having thicknesses of (a) 14 nm and (b) 28 nm in the pillar region. Insets show a 5× higher magnification section of a pillar to better visualize the location of the maximum EFI located 200 nm from the top of the pillar. The laser beam is propagating from left to right at a 62° angle of incidence. Axis values are in pixels (1 pixel = 14 nm = 1/74 λ).

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Fig. 4. Modeling of the EFI inside the grating structure containing ridge features having thicknesses of 28 nm in the pillar region at different y-axis positions representing its evolution along the y axis [Visualization 1]. Axis values are in pixels (1 pixel = 14 nm = 1/74 λ).

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3.2 Simulation for missing pillar sections

Laser-induced–damage initiation in gratings (at near-threshold conditions) is manifested with the removal of pillar sections. The general damage morphology depends on the laser pulse duration [9]. In this section we will consider the morphology of both general damage morphologies associated with picosecond and subpicosecond pulses. Missing or geometrically deficient pillars can also be the result of manufacturing or handling errors.

To investigate the electric-field distribution in the presence of modification to the grating structure due to laser-induced damage with picosecond pulses (or fabrication flaws), we investigated the following specific cases; a fully removed (a) single pillar, (b) two adjacent pillar sections, and (c) the case of partially removed pillar(s). The length of the removed pillars used in modeling is relevant to observations discussed in previous work [9] related to grating-damage initiation in the picosecond regime.

Figure 5(a) shows the EFI distribution along the x–y plane at 200 nm below the top of the pillars, simulating a damage site where two small sections of adjacent grating pillars (1.4 µm in length) are completely removed (typical of damage sites generated with picosecond pulses). The EFI is shown along the entire plane, including the area between the pillars while the pillar boundaries are outlined. Figure 5(b) shows the EFI distribution at the top of the underlying hafnia layer. This is the plane where the EFI is the highest in hafnia. Compared to a “perfect” grating, the maximum EFI is 36% higher inside the silica pillars and 29% higher inside the hafnia layer. Figure 6 shows the distribution of the EFI in different y-axis positions and in 3D video form depicting its distribution frame by frame along the z–x plane, where each frame shifted along the y axis.

 

Fig. 5. Modeling of the EFI that contains a modification characteristic of laser-induced damage with picosecond pulses encompassing two detached sections of adjacent pillars, each 1.4 µm in length. The EFI distribution in the x–y plane in the (a) pillar region (pillars boundaries are outlined) and (b) hafnia layer. Axis values are in pixels (1 pixel = 14 nm = 1/74 λ). (a) is at 200 nm below the top of the pillars, and (b) is at the top of the first hafnia layer.

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Fig. 6. Modeling of the EFI inside the grating structure containing two detached sections of adjacent pillars, each 1.4 µm in length, at different y-axis positions representing its evolution along the y axis [Visualization 2]. Axis values are in pixels (1 pixel = 14 nm = 1/74 λ).

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The example results shown in Figs. 5 and 6 show the locations of increased EFI that can be the source of damage growth initiation under subsequent exposure to a laser pulse. A series of simulations was carried out to elucidate the change of the maximum EFI as a function of (a) the length of the missing pillar, (b) the percentage of reduction of the height for partial pillar removal, and (c) the number of missing pillars. The results are summarized in Tables 1–3.

Table 1. Change in the maximum EFI compared to a pristine grating for the case of complete single pillar removal as a function of the missing pillar length.

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Table 2. Change in the maximum EFI compared to pristine grating for the case of 1.4-µm-wide section of a single pillar partially removed as a function of the pillar height removed.

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Table 3. Change in the maximum EFI compared to pristine grating for the case of complete removal of a 1.4-µm-wide section adjacent pillars as a function of the number of missing pillars.

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Table 1 represents the case of complete removal of a single pillar as a function of the length of the removed pillar. Table 2 represents the case of a 1.4-µm-wide section of a single pillar partially removed as a function of the percentage of pillar height removed (the original height is 600 nm). The results show that the EFI values do not increase monotonically with the volume of pillar removed. One possible interpretation is that the locally increased EFI is the result of interference between the field pattern of the undamaged grating and the scattered field by the defect. Both the amplitude of the scattered field component and its phase relative to the undisturbed field matters. Therefore, it appears that there is a worst-case height for the missing pillar. Table 3 represents the case of a 1.4-µm-wide section pillar complete removal as a function of the number of removed adjacent pillars.

The results presented in Tables 1–3 show that missing pillars or sections of pillars can introduce a significant increase in the maximum EFI in adjacent pillars, which will proportionally reduce the ability of that section of the grating to withstand the laser intensity (reduction of the local LIDT) [10]. This could be associated with initiation of expansion (growth) of the damage site at the locations of the increased EFI. For example, for the s polarization used in this work, growth initiation will be manifested with an increase of the dimensions of the damage site in both directions along the y axis, while the increase along the x axis will be more directional and on the opposite side from the direction of the incoming laser beam. This behavior has been previously experimentally observed in laser-damage growth [19].

The results also indicate that as the dimensions of the damaged (removed) pillar sections increases (number and length of pillar removed), the EFI inside the hafnia layer significantly increases. This can give rise to damage initiation inside the hafnia layer, which will constitute the propagation of the damage site inside the MLD layers via a cascade process.

The damage morphology on the grating structure under exposure to subpicosecond laser pulses encompasses the removal of pillar sections toward the back wall of the pillars in localized areas as shown in Fig. 1(b) [9]. To simulate this damage morphology, we assume partial removal of a pillar section in notched (chipped-off) form. This morphology (in the pillar region) is evidenced in Fig. 7(a), where the EFI distribution in the x–y plane (at 200 nm beneath the top of the pillars) is shown. The results suggest that hot spots are formed next to the removed section of the pillar accompanied by a smaller increase to the undamaged pillar to the right. A better visualization of this field enhancement is provided in Fig. 7(b) that shows the EFI distribution in the y–z plane along the x axis that is just inside the back wall of the damaged pillar. These results suggest that the initial damage site under subsequent exposure to subpicosecond laser pulses can start growing in size accompanied by possible damage initiation (depending on laser intensity) in the neighboring pillar.

 

Fig. 7. The EFI distribution in a grating structure that contains a detached, notched section of a pillar (which is characteristic of laser-induced damage with subpicosecond pulses) along the (a) x–y plane in the pillar region at a height shown by the arrow in (b); and in the (b) z–x plane, about at the opening of the notch, as shown by the arrow in (a). (a) is the plane that passes through the bottom of the notch, and (b) is the plane passing through the side of the pillar containing the notch opening. Axis values are in pixels (1 pixel = 14 nm = 1/74 λ).

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3.3 Particle contamination

The electric-field distribution arising from the presence of contamination particles on nominally pristine grating structures is investigated as a function of their size, shape, and optical properties. Specifically, spherical, cubic, and pyramidal particles are examined as general types of model contamination particles (encapsulating the phenomena that might be present for the general case of nominally random shapes) having sizes (diameter or length) of 1 µm, 500 nm, and 250 nm. Particles of these sizes were shown in recent work to be in very high concentration in the pulse-compression vacuum chamber of the OMEGA EP Laser System with most of the particles believed to be transported from the target vacuum chamber [16]. The 1-µm and 500-nm particles are assumed to be positioned on top of the pillars, while the smaller particles (250 nm) are between the pillars (on top of the grating trenches). Two types of materials are assumed: glass (to represent the general case of transparent particles) and metal (which represent about half of the particles reported in [16]). Figure 8 shows the EFI distribution in the x–z plane for the case of 1-µm-diam (a) glass and (b) metal spherical particles located on top of the pillars in a nominally pristine grating. Similar calculations were carried out for the shapes and sizes mentioned above. The results are summarized in Table 4, where the maximum increase of the EFI (compared to the pristine grating case) inside the pillars and the underlying hafnia layer is reported. In general, the maximum EFI for transparent particles is on the right side of the particle (along the laser beam propagation direction) and on the opposite side for reflective particles (due to induced reflection of the impinging light). Table 4 summarizes the change of the maximum EFI compared to pristine grating for the case of particles sizes of 500 nm and 1 µm (located on top of the pillars) and 250 nm (assumed to be located between the pillars) for different particle materials (glass and metal) and different particle shapes (spherical, cubic, and pyramidal).

 

Fig. 8. Modeling of the EFI distribution in the x–z plane caused by a 1-µm-diam spherical particle made of (a) glass and (b) metal. The particle outline is shown with dashed line. Axis values are in pixels (1 pixel = 14 nm = 1/74 λ).

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Table 4. Change of the maximum EFI compared to pristine grating for the case of particles located on top of the pillars (for sizes of 500 nm and 1 µm) or between the pillars (250 nm) for different particle materials (glass and metal) and different particle shapes (spherical, cubic, and pyramidal).

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The simulation results suggest that particles smaller than a quarter-wavelength cause a few percent of EFI increases. Such increase is likely to be equal or smaller than the EFI increase due to fabrication imperfections. We thus hypothesize that contamination particles of such size on gratings will not reduce its overall damage performance. For larger particles, metal particles generate smaller EFI increase than glass and exhibit smaller dependence on the shape or size (for example, in the first row, an increase of 4 × in size produces a 23 × EFI increase for glass, but only a 2 × increase for metal). This suggests different dependence between size and EFI increase for metal and transparent particles. The relative increase in EFI in the hafnia layer can be comparable or higher than the relative increase of EFI in the pillars. This means grating-damage initiation due to debris can be caused by hot spots in both the pillars and the underlying (topmost) hafnia layer.

Although this work is devoted on understanding the electric-field distribution, which governs the damage-initiation mechanism, the final result of a damage event is dictated by the laser-material interactions and the subsequent response of the material and/or contamination particles to localized superheating. For example, it is noted above that metal particles generate a smaller EFI. Experimental work on model contamination particles on mirrors, however, has demonstrated that metal particles cause a much more-significant secondary contamination of the surface than transparent particles [24]. This previous work also demonstrated that the LIDT for glass particles is much lower than for metal particles, in agreement with the present work. Therefore, we expect that the knowledge obtained from [24] regarding the secondary effects of the interactions of the laser pulse with contamination particles is directly applicable to the case of gratings.

4. Conclusion

Three-dimensional FDTD modeling was used to calculate the electric-field distribution in gratings inside the dielectric material layers (pillars and MLD layers) arising from a variety of inadvertent flaws including manufacturing flaws, particle contamination, and laser-induced damage. We also evaluated the effect of contaminating by particles having size of the order of one wavelength or smaller. From the various defect features discussed in this work, the contamination particles are the most potent because they introduce a significantly higher increase in EFI. The results suggest that this modeling approach can help predict and/or interpret grating-damage behaviors by providing estimates of the (reduced) damage threshold of realistic and imperfect gratings, as well as gratings that already have been damaged. In the latter case, this could be used to predict the location of damage growth initiation and help design gratings that are less susceptible to laser-induced damage and damage growth. This method can also be used to study transient or non-steady-state field distribution for femtosecond scale pulses.