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Monolithic large-scale diffraction gratings are desired to improve the performance of high-energy laser systems and scale them to higher energy, but the surface deformation of these diffraction gratings induce spatio-temporal coupling that is detrimental to the focusability and compressibility of the output pulse. A new deformable-grating-based pulse compressor architecture with optimized actuator positions has been designed to correct the spatial and temporal aberrations induced by grating wavefront errors. An integrated optical model has been built to analyze the effect of grating wavefront errors on the spatio-temporal performance of a compressor based on four deformable gratings. A 1.5-meter deformable grating has been optimized using an integrated finite-element-analysis and genetic-optimization model, leading to spatio-temporal performance similar to the baseline design with ideal gratings.
© 2015 Optical Society of America
1. Introduction
Chirped pulse amplification (CPA) has enabled a dramatic increase of the on-target optical intensity obtainable with laser systems [1]. In CPA systems, a low-energy optical pulse is stretched in time before amplification via stimulated emission or nonlinear parametric interactions so that the optical intensity remains below the damage threshold of optical components. Recompression of the amplified pulse into a short, high-energy pulse is most often achieved with a compressor composed of diffraction gratings. Multilayer dielectric (MLD) diffraction gratings are the highly desired pulse compression element to obtain kilojoule, petawatt laser pulses [2–6 ] owing to their high laser-induced damage threshold and high diffraction efficiency. They are typically several times larger than the beam size before the compressor to accommodate the spatial spreading due to angular dispersion and the large use angles. The fluence damage threshold of diffraction gratings is one of the limiting factors for energy scaling of high-energy laser systems, and larger diffraction gratings generally enable the construction of laser systems providing higher on-target energy. Monolithic gratings with size of the order of 1 m in the dispersion direction (perpendicular to the grooves) have been fabricated [7–9 ]. Grating tiling has been implemented on OMEGA EP [4] because gratings with sufficient wavefront quality and diffraction performance were not available with a size of 1.5 m: each composite grating of the four-grating compressor is composed of three smaller gratings (width in the dispersion direction = 0.47 m, height = 0.43 m, thickness = 0.1 m) that are interferometrically aligned to act as a single grating [10,11 ]. A 1.5-m grating having the same length-to-thickness aspect ratio as the grating segments would weigh ~460 kg and would be challenging to manufacture, handle, and mount. Furthermore, the grating wavefront error (WFE) induced by the gravity and coating stress generally increases with increasing grating size. This error degrades the output pulse spatially and temporally and reduces the on-target intensity. The tiled-grating architecture minimizes these issues and has enabled thousands of kilojoule, picosecond laser shots on OMEGA EP, but it requires significant ancillary hardware to maintain the required sub-microradian tile-to-tile alignment. To prevent optical damage from the spatial modulation induced by diffraction from the tile-to-tile gaps, the intensity of the corresponding light is intentionally reduced before the compressor, resulting in a decrease of the achievable energy. A number of other laser facilities have explored grating tiling [12,13 ].
This paper studies the impact of the gratings wavefront error on the spatio-temporal performance of a compressor and treats a parallel approach to the realization of a large-scale grating compressor, i.e., a four-grating compressor where the large-scale monolithic gratings are combined with actuators to optimize their surface figure for optimal compressor performance. A pulse compressor composed of four deformable1.5-m MLD gratings, each having a position-optimized array of actuators on the back of a thin substrate to compensate for the wavefront errors induced by the coating and patterning processes, is simulated. Employing well-developed adaptive–optics techniques minimizes the detrimental space–time coupling in the compressed pulse and maximizes the on-target intensity.
The finite-element analysis of the grating-substrate deformation due to the high-temperature deposition of MLD on 1.5-meter substrates is presented in Section 2. An optical model analyzing the underlying mechanism and impact of grating wavefront on the spatial focusing and temporal compression after the compressor is presented in Section 3. This model shows that the grating wavefront error on each meter-scale grating will lead to significant spatial and temporal broadening at the focus: an in-line deformable mirror at the output of the compressor can correct the spatial aberration of the output beam, but it cannot correct the induced temporal broadening. In Section 4, a monolithic deformable MLD grating with actuators is modeled using finite-element analysis (FEA). The influence function of each actuator is evaluated for various actuator designs and the actuator positions are optimized using a genetic algorithm to enhance the compressor spatio-temporal performance. Sections 5 and 6 present a general discussion of these findings and the conclusions.
2. Grating wavefront description and prediction of surface deformation using finite element analysis
The grating wavefront error, i.e. wavefront of the diffracted wave, consists of a mirror term and a holographic term. The former refers to the surface deformation induced by substrate gravity, mounting and coating stress; the latter is caused by spatial patterning imperfections such as spatial variations of the groove spacing, depth, or parallelism across the entire grating surface. The WFE of a grating usually increases with increasing aperture and with decreasing substrate thickness. MLD gratings for pulse compression are coated at high-temperature (e.g., 200 °C) but are operated at room temperature (e.g., 20°C) in vacuum, therefore thermal load induces large amount of surface deformation. A FEA model of a MLD grating, measuring 1.5 m × 0.43 m was built using the commercial software ANSYS® [14] to predict the surface deformation due to the coating process as a function of the grating thickness. The grating model includes a BK7 substrate and alternating layers of low- and high-refractive index material, silica (SiO2) and hafnia (HfO2), respectively. The model is capable of predicting the surface deformation caused by gravity, actuator movement, thermal gradients, and mounting-induced surface deformation, but only thermal gradients and actuator movements (when applicable) are taken into account in this article. The surface deformation was predicted over the temperature range from 200 ̊C to 20 ̊C, representing the range experienced by the substrate during the coating run and subsequent cool-down to room temperature.
The model shows that the surface deformation is inversely proportional to the square of the substrate thickness [Fig. 1(a) ]. The surface deformation increases linearly with the temperature change [Fig. 1(b)]. The thermal-load–induced surface deformation exhibits a parabolic shape [Fig. 1(c)]. The predicted deformation is significant, e.g., reaching a value of 6.4 µm for a 20-mm substrate and a deposition temperature of 200 ̊C. Reduction of this surface deformation has been experimentally achieved using fused-silica substrates because of their lower thermal expansion coefficient, but this requires a low-temperature ion-assisted deposition coating process [9,15 ]. In this paper, a wavefront of 1.5 waves at 1053 nm across the 1.5-meter grating is considered for all simulations that aim at demonstrating the spatio-temporal performance degradation and compensation by deformable gratings.
Fig. 1 (a) Surface deformation as a function of the substrate thickness for a 1.5 × 0.43 m substrate experiencing a temperature change from 200 ̊C to 20 ̊C. (b) Surface deformation at 20 ̊C as a function of the temperature during coating deposition for a 20-mm-thick substrate (c) Surface map of the thermally induced deformation for the 1.5m × 0.43m substrate at 20 ̊C after coating deposition at 200 ̊C.
3. Grating compressor modeling to analyze the spatiotemporal degradation caused by grating wavefront errors
3.1. Description of the optical model and design baseline
Chirped pulse amplification systems vary greatly in terms of compressor parameters, in particular size, stretch factor, and gratings groove density. This article uses the OMEGA EP laser system compressor as a test case for optical modeling of the spatiotemporal effects associated with gratings wavefront errors and compensation using deformable gratings. This technology is well adapted to such system because the beam in a high-energy laser system must be large so that the fluence remains below the damage threshold of optical components. Highly dispersive large-groove-density gratings are required in these systems because a large dispersion is required to ensure that the chirped pulse, with relatively low spectral bandwidth, remains below the intensity damage threshold and does not accumulate significant nonlinear phase.
A compressor optical model [Fig. 2(a) ] was developed using the commercial software FRED® [16] combined with MATLAB® [17] analysis routines to analyze the spatiotemporal properties of a grating pulse compressor. The model was based on the geometry of the OMEGA EP grating compressor [11] but it consists of four identical monolithic diffraction gratings (G1, G2, G3, and G4) with dimensions 1.41 m × 0.43 m × 0.02 m, instead of the four tiled grating assemblies implemented in the actual laser system. The incident angle on the first grating is 72.5° and the slant distance between the gratings in the pairs (G1,G2) and (G3,G4) is equal to 3.23 m. The groove density of all four gratings is 1740 lines / mm. The input beam has a square profile without the spatial modulation introduced in the laser system to attenuate light incident on the gaps of the gratings in the actual laser system [18]. Although imperfect beam fluence profile and wavefront modulation generally degrade the focusability, a theoretical 20th-order supergaussian input beam profile with a full-width at half-maximum (FWHM) of 37 cm and a flat wavefront have been used for the purpose of establishing the design baseline. The spectral density corresponds to a measured OMEGA EP front-end optical spectrum, i.e., approximately a 20th-order supergaussian spectrum at a central wavelength of 1052.8 nm with a FWHM of 7.5 nm. The spectral phase is chosen to be equal to the opposite of the spectral phase introduced by the ideal grating compressor described above. This choice of spatial and spectral phases allows to establish the spatio-temporal baseline for an ideal compressor and quantify the impact of non-ideal gratings. The complex electric field at the entrance pupil of the final focusing element, an f/2 off axis parabola, was calculated by tracing the optical path length (OPL) at each spatial location (X, Y) of the input beam for each spectral component λ [Eq. (1) ].
The compressed pulse at the focal plane of the focusing optic was obtained by Fourier transforming the electric field in the spatial and temporal domains after removal of the OPL obtained by propagation in an ideal compressor. This assumes that there is no wavefront or spectral-phase distortion in the laser system other than the one induced by the aberrated compressor. The spatio-temporal performance of the compressor is evaluated with the temporal pulse duration FWHM and spatial R80 of the focal spot. R80 is the radius of a circle centered on the focal-spot centroid inside which 80% of the total focused beam energy is enclosed. Figure 2(b) shows the performance baseline for an ideal input beam and compressor, resulting in a diffraction-limited spot (R80 = 2.4 µm) and a Fourier-transform-limited pulse duration (FWHM = 0.41 ps). We focus in this article on the performance degradation due to gratings wavefront errors, but the model can also take into account the fluence modulation and wavefront of the input beam, and the compressor alignment errors (e.g., non-parallel gratings).Fig. 2 (a) Optical model of the compressor and focusing optics consisting of four gratings G1, G2, G3 and G4 and an f/2 off-axis parabola (OAP). (b) Spatio-temporal distribution at the focal plane across and along the grating groove directions (top and bottom, respectively).
3.2. Impact of the grating-wavefront error on the spatiotemporal performance
To demonstrate the effects of grating wavefront errors on the spatio-temporal performance of a grating compressor, a conical wavefront with a peak-to-valley of 1.5 waves at 1053 nm was added onto the surface of each of the four compressor gratings [Fig. 3(a) ]. The conical shape was chosen for illustration purposes due to the distinctly different grating wavefront slope along the dispersion direction. Figure 3(b) shows the resulting significant spatio-temporal focal-spot degradation: the focused pulse splits in both time and space, resulting in R80 = 17.9 μm and FWHM = 0.97 ps. The spatial phases of the central and two edge wavelengths at the entrance pupil of the off-axis parabola were analyzed to investigate the cause of the spatio-temporal split. Figure 3(c) illustrates that the conical spatial phases of the central and edge wavelengths spatially couple with each other; it also clearly shows that two opposite wavefront slopes across the grating groove direction contributed to the split of the focal spot in the spatial domain. The temporal split is explained by the spatially varying group delay defined as the slope of the spectral phase φ(ω). Figure 3(d) shows that the left side of the beam arrived earlier in time than the right side of the beam, leading to the temporal splitting.
Fig. 3 (a) Conical wavefront (peak-to-valley = 1.5 λ) added onto each of the four grating surfaces. (b) Spatio-temporal distribution at the focal plane across and along the grating groove directions (top and bottom, respectively). (c) Spectral phase versus spatial position across the grating groove direction for the central and two edge wavelengths. (d) Group delay across the grating groove direction, showing an approximately constant early arrival for x<0 and approximately constant late arrival for x>0.
To further predict the effects of a more realistic grating-wavefront error on the spatio-temporal performance of the grating compressor, a theoretically predicted parabolic-shaped wavefront with a peak-to-valley of 1.5 waves at 1053 nm [Fig. 4(a) ] was added onto the surface of each of the four compressor gratings. Figures 4(b) and 4(c) show significant spatio-temporal distortion, resulting in R80 = 16.2 μm and FWHM = 0.94 ps. Because of the continuous nature of the parabolic wavefront error compared to the conical wavefront error, the spatio-temporal broadening was continuous in both space and time.
Fig. 4 (a) Parabolic wavefront (peak-to-valley = 1.5 λ) added onto each of the four grating surfaces. (b) Spatio-temporal distribution at the focal plane across and along the grating groove directions (top and bottom, respectively).
Parabolic wavefronts with increasing peak-to-valley magnitude were added onto the surface of each of the four gratings to quantify the impact on the spatio-temporal performance of the compressor. Figures 5(a) and 5(b) show that the spatio-temporal distribution at the focus broadens with increasing wavefront errors in both the temporal and spatial domains. Owing to the increasing opposite local wavefront slopes on the left and the right side of the beam and the increasing group delay across grating groove, the focal spot eventually splits into two spots in both the spatial and temporal directions. For a given temporal and spatial broadening tolerance, such sensitivity study determines the largest grating wavefront error that a compressor system can tolerate.
Fig. 5 (a) Sensitivity of pulse broadening to the wavefront error on each grating. (b) Sensitivity of R80 to wavefront error on each grating.
3.3. Investigation of the spatiotemporal correction ability of an inline deformable grating
To investigate whether the spatio-temporal degradation caused by individual grating wavefront error can be corrected by a deformable mirror located at the output of a grating compressor, the opposite of the combined wavefront errors from the four gratings, each having the 1.5-wave wavefront error shown in Fig. 3(a), is numerically added to the beam. Figure 6(a) demonstrates that a near diffraction-limited focal spot is achieved (R80 = 2.6 μm) with the inline deformable mirror correction, but such correction is unable to recompress the pulse (FWHM = 1.04 ps). This is due to the fact that the wavefront error on each grating surface degrades the optical pulse in both space and time, with a coupling between these two domains [as seen in Fig. 3(c)]. An in-line deformable mirror at the compressor output can at best correct the spatial degradation averaged over all frequencies but cannot correct the temporal pulse broadening. The spatially varying group delay before [Fig. 3(d)] and after [Fig. 6(b)] the deformable mirror are identical, therefore showing significant degradation of the obtainable pulse duration after compression. As expected, a similar conclusion was reached for the parabolic wavefront profile.
Fig. 6 (a). Spatio-temporal distribution at the focal plane across and along the grating groove directions (top and bottom, respectively). (b) Group delay as a function of position after the deformable mirror.
4. Compensation of grating wavefront error using deformable gratings
4.1 FEA modeling of deformable gratings
In this section, we consider deformable gratings with actuators on the back surface and study the spatio-temporal performance of such a compressor. The surface bending ability is determined by the thickness of the grating substrate, the actuator actuation depth, and the geometrical layout of the actuators. A deformable MLD-grating model was built using the FEA software ANSYS®. The magnitude of the surface-figure correction that can be generated by an array of actuators is in general inversely proportional to the substrate thickness: a thinner substrate is desired to achieve larger actuator actuation depth and reduce the grating weight, but it is subject to more surface deformation [as shown in Fig. 1(a)] caused by the manufacturing and operation process as well as the actuator print-through noise. An optimal substrate thickness can be determined by considering the tradeoffs among surface deformation, substrate manufacturability, and the correction ability of a particular actuator design. A thickness of 20 mm was chosen for the subsequent wavefront error prediction and correction.
The parabolic wavefront shown in Fig. 4 (peak-to-valley of 1.5 waves at 1053 nm) was used. The initial nine-actuator layout design [Fig. 8(a)] was created based on the magnitude and spatial frequency of the wavefront to be corrected. The number of actuators was arbitrarily chosen as a compromise between wavefront-correction ability and complexity. The spatial wavefront is sampled every 1 mm, i.e. on N = 1500 × 410 = 61,500 points. The FEA model was used to derive the influence function of each actuator, i.e. the N × M matrix of coefficients dj,k (j = 1 to N, k = 1 to M). Equation (2) expresses the linear relation between the displacements of the M actuators (D1 through DM) and the wavefront values at the sampled points (Z1 through ZN).
The optimal set of actuator displacements for a specific wavefront correction is obtained by solving Eq. (2). The residual wavefront (difference between the wavefront to be corrected and the surface figure obtained using the calculated actuator displacements) serves as the performance estimate to evaluate the surface-wavefront correction ability of a specific layout of the nine actuators.
4.2 Genetic optimization of actuator positions to achieve a minimized residual wavefront
The FEA ANSYS® model was integrated with a MATLAB® genetic optimization model [19] to optimize the deformable grating design in terms of actuator positions, i.e., minimize the residual wavefront. Figure 7 illustrates the optimization process. For a given wavefront to be corrected, an initial design was first constructed with a preliminary actuator layout. The genetic optimization routine randomly creates a user-defined number of populations of actuator configurations. The integrated ANSYS® deformable-grating design model simulates the influence function of each actuator and the surface figure that can be created by a particular actuator layout design for each population. A weighted average of the root-mean-squared (RMS) and peak-to-valley (PV) of the residual wavefront error (10 RMS + PV) was used as the fitness function. At each generation, the population of actuator configurations was modified by passing on elite actuator configurations achieving relatively low residual wavefront error and mutating or combining the other configurations after each generation. The actuator configurations were forced to be spatially symmetric because of the symmetry of the target wavefront. The optimization input parameters include the fitness function, the initial actuator positions, the boundary conditions, the forced symmetric constraint, the population size, the tolerance value and the cross-over ratio [19]. The optimization process ends when an actuator design meets the tolerance criteria (RMS < = 0.02 λ and 10 RMS + PV < = 0.3 λ).
Figure 8(a) shows the initial and optimized positions of the nine-actuator design (blue and red markers, respectively). The black markers correspond to the actuators positions across all members and generations. These intermediate positions extend over most of the grating area, giving confidence that the optimization spanned a large set of populations. Figure 8(b) shows that the fitness value averaged over all the members of the population decreases at each generation.
Fig. 8 (a) All the sampled actuator positions with red, blue and black markers representing the initial, optimized, and intermediate positions, respectively. (b) Fitness value averaged over the population as a function of the generation during the genetic optimization.
Figures 9(a) and 9(b) show the distinctively different initial and optimized actuator positions. Figures 9(c) and 9(d) show the residual grating wavefront errors for the initial and optimized designs. The initial design corresponds to a residual wavefront with RMS and PV values equal to 0.046 λ and 0.300 λ, respectively. The residual wavefront is significantly reduced for the optimized design, with RMS and PV values of equal to 0.016 λ and 0.090 λ, respectively. The optimum specifications for boundary conditions, crossover ratio and forced symmetry constraints were critical to achieve the optimized design. Figures 9(e) through 9(h) illustrate the spatio-temporal performance of a compressor composed of four identical deformable gratings with either the initial or optimized actuator configuration. The initial and optimized designs both achieved a Fourier-transform-limited pulse width, but the spatial focusing is improved by a factor two after optimization, resulting in no significant spatio-temporal performance difference compared to the baseline ideal compressor.
Fig. 9 (a) Initial nine-actuator design. (b) Optimized nine-actuator design. (c) Residual wavefront error of initial design (RMS = 0.046 λ, PV = 0.30 λ). (d) Residual wavefront error of optimized design (RMS = 0.015 λ, PV = 0.09 λ). (e), (f), (g) and (h) Spatio-temporal distributions after focusing for a pulse compressor composed of four deformable gratings with the initial actuator configuration (FWHM = 0.41 ps, R80 = 5.1 µm) and the optimized actuator configuration (FWHM = 0.41 ps, R80 = 2.5 µm).
The optimization process has been run with two distinctively different initial designs, leading to similar performance. The actuator design populations for which the RMS wavefront deviates by less than 5% from that of the optimized design cover an area extending ~110 mm in the horizontal direction and ~50mm in the vertical direction. This indicates that the optimized design has relatively high flexibility and stability and shows that optimal compensation does not depend very precisely on the location of the actuators.
5. Discussion
This article uses the OMEGA EP compressor geometry to illustrate the effect of the grating wavefront on the spatiotemporal degradation at the focus. For this particular geometry, the effect in relation to spectral bandwidth and beam size was further studied by analyzing the spatially dependent spectral phase and group delay of the parabolic wavefront (peak-to-valley of 1.5 wave) added on each of the four grating surface as shown in Fig. 4(a). Figure 10(a) shows the resulting parabolic spatial phases for different wavelengths in the optical spectrum. These spatial phases mostly differ by a linear term. Figure 10(b) shows the resulting group delay at three points in the beam, with a slight variation of the group delay over the optical spectrum at each point in the beam, and more importantly, a large variation in the average group delay as a function of the position in the beam. The former effect leads to a small temporal broadening that can in principle be partially compensated by a change of the compressor parameters. The latter effect leads to spatio-temporal coupling and the majority of the pulse-duration increase observed at the focus. A beam-size increase for a given spectral width leads to a larger group delay variation across the beam, hence a proportionally larger increase of the pulse duration at focus. It also leads to a larger spatial phase across the beam. A bandwidth increase for a given beam width increases the pulse duration primarily because of the group-delay variation at each point in the beam. The range of group delay due to spatio-temporal coupling remains approximately the same. These effects therefore lead to a relatively larger decrease in performance because the larger bandwidth should lead to a shorter pulse in ideal conditions. It should be noted that either a significant beam size increase or bandwidth increase will require larger diffraction gratings. The resulting different wavefront magnitude and geometry will require additional simulations.
Fig. 10 (a) Spatial phase across the beam at three different wavelengths. (b) Group delay as a function of the optical frequency at three different locations in the beam. These results correspond to a 1.5-wave parabolic wavefront on each compressor grating, and are complementary of the results presented in Fig. 4.
Small changes in the compressor design, including changes in the incident angle and inter-grating distance, will not significantly modify the results presented here because the obtained geometry will lead to similar optical-path variations between gratings. Extenstion of these results to significantly different compressor geometries, for example the compressor in a broadband Ti:sapphire laser system with lower groove density gratings and induced dispersion, will require to use the modeling strategy described in this article and apply the model to the new set of parameters. Generally, it is expected that the alignment tolerance (angle and inter-grating distance) becomes tighter as the gratings groove density increases and the gratings operate at a higher incidence angle [20]. The wavefront error on each grating essentially leads to spatially varying angles of incidence across the beam on each grating surface and introduces a spatially dependent spectral phase that leads to the spatially varying group delay. It is therefore expected that the absolute magnitude of these effects decreases for a compressor with gratings having a lower groove density. However, the relative magnitude of these effects, e.g. the on-target duration caused by the spatio-temporal coupling, might be larger because of the lower value of the Fourier-transform-limited duration for a pulse with broader bandwidth.
6. Conclusion
Monolithic meter-size gratings are desired to increase the on-target energy generated by large-scale CPA laser systems. The tiling of smaller gratings generally requires shadowing of the inter-grating gaps which reduces the output energy. Large gratings can be manufactured, but their wavefront error is detrimental to the spatio-temporal properties of the output pulse, resulting in lower on-target intensity because of poorer focusability and compressibility. A spatio-temporal optical model has been developed to investigate the mechanism of grating wavefront errors and their impact on the spatio-temporal performance of the focused pulses. A new deformable-grating compressor has been proposed and studied to correct the gratings wavefront errors, enabling simultaneous compensation of the spatial and temporal aberrations. A diffraction-limited and Fourier-transform-limited focused pulse was achieved with a grating compressor composed of four 1.5-meter deformable gratings, each having an identical nine-actuator design optimized by a genetic algorithm. This approach is promising to scale up the energy of large-scale CPA systems that are currently limited by the size of diffraction gratings.
Acknowledgments
The first author started this work at the Laboratory for Laser Energetics (LLE), University of Rochester and finished it at the Rochester Institute of Technology. The secondand third authors were undergraduate interns at LLE when the relevant work was conducted.This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-NA0001944, the University of Rochester,and the New York State Energy Research and Development Authority. The support of DOEdoes not constitute an endorsement by DOE of the views expressed in this article.This work was also supported by the first author's new-faculty startup fund provided by the Rochester Institute of Technology.
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