## Abstract

A tiled-grating assembly with three large-scale gratings is developed with real-time interferometric tiling control for the OMEGA EP Laser Facility. An automatic tiling method is achieved and used to tile a three-tile grating assembly with the overall wavefront reconstructed. Tiling-parameters sensitivity and focal-spot degradation from all combined tiling errors are analyzed for a pulse compressor composed of four such assemblies.

© 2007 Optical Society of America

## 1. Introduction

The size requirements of diffraction gratings for the compressor of large-scale chirped-pulse-amplification systems cannot be met with the current state-of-the-art diffraction gratings [1], which makes grating tiling an alternative solution to achieve multipetawatt laser systems [2]. Kessler *et al*. have demonstrated the coherent addition of two small-scale gold-coated gratings using a far-field-based approach [3]. Cotel *et al*. also demonstrated that the measured far-field intensity distribution of tiled small-scale gratings agrees well with the wavefront measured by an interferometer [4]. However, due to the wavefront error of large gratings and the general difficulty of achieving diffraction-limited, far-field performance with a large-size beam, the aberrations in the far-field imaging system normally lead to an inconsistency between the tiled wavefront and the measured far-field performance, i.e., the optimum focal spot does not correspond to optimum tiling. For a highly aberrated large-size beam, which is subject to disturbance induced by turbulence and vibration, the far-field image may have multiple split spots even if there is no tiling error, or the far field might not show the expected symmetrical split when there is a piston misalignment value of *π*. We have observed both phenomena while conducting large-aperture grating tiling. Therefore we think the measured far field does not necessarily represent the quality of tiling, and it should not be the final criterion for tiling. Instead, the near-field wavefront of all tiles after alignment presents tiling quality without ambiguity, and it should be used as the final criterion for tiling quality.

In this paper, a near-field tiling method is explored, and the practical issues related to large-aperture-grating tiling are addressed. The fully reconstructed wavefront of the full-size tiles after alignment is used as the final tiling criterion to achieve automatic tiling for the first time. This large-aperture grating tiling method was developed, automated, and was successfully used for tiling a tiled-grating assembly composed of three full-size gratings (0.47 m × 0.43 m, 0.5 m in diagonal) with interferometric precision, therefore realizing a 1.41-m grating. Section 2 describes the configurations of a pulse compression system having four such tiled-grating assemblies for the OMEGA EP Laser Facility (presently under construction [5]) at the Laboratory for Laser Energetics (LLE), University of Rochester. In Sec. 3, tiling-parameters sensitivity and focal-spot degradation from all combined tiling error for various input-beam wavefronts are thoroughly analyzed for such a compressor. The factors determining final tiling accuracy are discussed. The metric for analyzing the performance of a tiled-grating compressor for the experimentally verified tiling accuracy is, for the first time, presented and used to predict the focal-spot performance of the OMEGA EP compressor. Section 4 presents the near-field tiling technique, by which the wavefront of each individual tile was reconstructed, the differential phase error and the wavefront of all three tiles after alignment were minimized. The automatic interferometric tiling process is developed and the results of tiling three large-aperture gratings using this process are also reported.

## 2. Pulse compressor and tiled-grating assemblies for a petawatt laser system

Figure 1(a) shows the pulse-compression scheme for OMEGA EP. The compressor consists of four sets of tiled-grating assemblies (TGA’s) of three 0.47-m×0.43-m gratings. The beam size of the OMEGA EP laser is 0.37 m×0.37 m. The incident angle on TGA1 is 72.5°, which offers a large pulse-compression ratio and relaxes the damage-threshold requirements for the gratings. Figure 1(b) shows the rear view of one TGA, which holds three full-size OMEGA EP grating tiles. Each tile is mounted on a triangular support frame. All three tile support frames are mounted on a mechanical platform, with the center support frame fixed to the mechanical platform to provide structural stability. The mechanical platform is positioned on a rotary stage, which allows the entire TGA to rotate between -165° to +165° (*θ _{y}* rotation). Tip (

*θ*) and in-plane rotation (IPR/

_{x}*θ*) movements are provided by two motorized linear actuators mounted on the back of the mechanical platform.

_{z}Tile-to-tile alignment is realized by maintaining the central tile static and by moving the two outboard tiles relative to the central tile. For each outboard tile, there are six degrees of freedom relative to the central tile: tilt *θ _{y}*, tip

*θ*, IPR

_{x}*θ*, lateral shift

_{z}*d*, longitudinal shift, also referred to as piston

_{x}*d*, and relative groove-spacing change Δ

_{z}*. The six parameters form three independent pairs: piston and lateral shift, tip and IPR, and tilt and groove spacing change.*

_{d}The two parameters within each pair compensate each other [6]; therefore, each of the outboard tile support frames incorporates only three electrostrictive actuators to provide tile-to-tile alignment by modifying tip, tilt, and piston. Each actuator is paired with a capacitive sensor to form a closed control loop to hold its position. The resolution for holding the position of an actuator is ±4 nm. For coarse alignment, the TGA is positioned at normal and Littrow angles iteratively to remove the tip and IPR of the central tile by adjusting the two linear actuators. The grating grooves of outboard tiles can be aligned to that of the central tile by manually adjusting the tip and IPR through three screws underneath the tile support beam. The initial aligned position is determined by interferometric analysis. The changes from the aligned positions in terms of piston and lateral shift, tip, and IPR are monitored by two pairs of capacitive sensors across the tile-gap mounted on the top and bottom surface of the tile substrates. The aligned position is then maintained in real time by compensating the temporal drift of the lateral shift and the in-plane rotation with the piston and tip, respectively.

## 3. Tiling tolerance analysis

#### 3.1 Compressor focal-spot sensitivity to tiling parameter analysis

A ray-tracing model has been developed to perform tiling tolerance analysis for the full tiledgrating compressor system followed by an *f*/2 parabola for focusing (i.e., 12 gratings grouped in four TGA’s). This model simulates the influence of misalignment on all four TGA’s of a compressor. The spectrum is the OMEGA EP on-shot spectrum. R80 and Strehl ratio for the polychromatic wavelengths were used to evaluate the focal-spot quality. R80 is the radius of a circle inside which 80% of the total focused beam energy is enclosed. Strehl ratio is defined as the ratio of the peak intensity of an aberrated system to that of a diffraction-limited system. The focal-spot specification for OMEGA EP is a 10-*µ*m radius for R80. The sensitivity of R80 and Strehl ratio to tiling parameters was investigated using this model applied to a large range of tiling parameters and for various wavefronts of the incoming beam. Figures 2(a) and 2(b) show the evolution of R80 and Strehl ratio for various amounts of tiling error in the TGA2 central tile (equivalent to moving the two outboard tiles relative to the fixed central tile) in the case of a perfectly flat incoming wavefront. The R80 and the Strehl ratio for the focal spot of this compressor system with perfectly tiled gratings are 2.6-*µ*m and 1, respectively. The simulation results show that the piston is the most sensitive tiling parameter; however, R80 and Strehl ratio change with the longitudinal piston periodically. Therefore the maximum broadening and peak intensity degradation of the focal spot caused by piston can be predicted. When there is 0.5-*λ* optical path difference (OPD) change induced by piston misalignment, the Strehl ratio decreases from 1 to 0.45 and R80 increases from 2.6 *µ*m to 8.2 *µ*m. The focal spot increases approximately linearly with increasing angular misalignment (tip, tilt).

For high-power laser systems, the wavefront of the beam at the input of a compressor is not perfectly flat. The sensitivity of R80 to piston and tilt misalignment was studied for various input-beam model wavefronts. These wavefronts were generated in the ray-tracing model by adding a Zernike phase surface, which mainly includes the low-order aberration terms, such as power, astigmatism, and coma. The coefficients of these terms were generated randomly to ensure that the peak-to-valley variation of the wavefront is less than 2*λ*. As shown in Figs. 3(a) and 3(b), for the same amount of piston and tilt misalignment error, the evolution of R80 is clearly greater for an input beam with flat wavefront compared to that with aberrations. The phase map of Abe 1 is illustrated in Fig. 4(c).

#### 3.2 Focal-spot degradation from all combined tiling errors of a compressor

The performance of a tiled-grating compressor is fundamentally determined by the initial tiling performance and the long-term stability of a TGA. The initial tiling is constrained by the interferometric measurement, which is subject to disturbance caused by turbulence and vibration, and the long-term stability of a TGA is determined by environmental stability, such as temperature and vibration. Taking into account the sensitivity of the tiling interferometer and the mechanical and environmental stability of a TGA, the best achievable tiling accuracy for each tile of each TGA was determined to be approximately ±0.2 *µ*rad, ±0.2 *µ*rad, and ±0.13 *µ*m for tilt plus groove spacing change, tip plus IPR, and piston plus lateral shift, respectively. For the OMEGA EP compressor, there are eight outboard tiles to be aligned to their corresponding central tiles, since there are three independent tiling parameters for each tile, and the total number of independent tiling parameters is 24. It is necessary to understand and predict the combined effect of the tiling errors described above on the focal spot of the tiled-grating compressor. For each outboard tile of a TGA of the compressor, the tilt, tip, and piston were chosen as the independent tiling parameters to perform a Monte Carlo tolerance analysis, i.e., each outboard tile’s position in terms of tilt, tip, and piston was randomly perturbed within the tiling accuracy. This simulation was done in the case of a flat input wavefront and for an aberrated input beam. The mean and standard deviation values of R80 and Strehl ratio were calculated for 500 runs.

One of the assumed input wavefronts used for this sensitivity study, Abe 1 in Fig. 3(a), was included in this Monte Carlo tolerance analysis. Figures 4(a)–4(d) show the wavefronts and far-field focal spots of the compressor system with a flat and the non-flat input wavefront- Abe 1, respectively. In both cases, there was no tiling misalignment. The root-mean-square (rms) and peak-to-valley (p-v) of this non-flat wavefront are 0.29 *λ* and 1.8 *λ*, respectively.

Figures 4(e)–4(h) illustrate how the near field and the far field of a randomly realized tiled-grating compressor system using above-described tiling accuracy degrades for a flat and a non-flat input wavefront. This simulation shows that the effect of tiling error on the overall near-field wavefront and far-field performance of the compressor system depends on the amount of input wavefront error. Given the same type and same value of tiling misalignment for the compressor, the amount of degradation is clearly greater for a perfectly flat input wavefront than that for a non-flat input wavefront. For the latter case, the input wavefront error is the dominant factor that determines the near-field wavefront and the focal-spot quality.

Figures 5(a)–5(d) show the histograms of the far-field performance of 500 randomly realized tiled-grating compressors with the tiling accuracy tilt/tip=0.2 *µ*rad and piston=0.13 *µ*m for the input beams with a flat wavefront and with the non-flat wavefront Abe 1. From this simulation, the performance of the compressor for both cases can be predicted statistically. Table 1 compares the far-field degradation between the two cases. Statically, Strehl ratio and R80 degrade from 1 to 0.9 and from 2.6 *µ*m to 4.2 *µ*m, respectively, for a flat-input beam wavefront. For input-beam wavefront Abe 1, the corresponding Strehl ratio and R80 degradations are from 0.25 to 0.24 and from 7.6 *µ*m to 7.9 *µ*m, respectively. We can conclude that, given the same tiling accuracy, the focal-spot degradation caused by tiling error is greater for a flat input wavefront than for a non-flat input wavefront.

It should be noted that the analysis for the spatial performance of the tiled-grating compressor system did not include the effect of wavefront error of each individual grating tile. We will address this effect in later publications.

#### 3. 3 Effect of tile misalignment on the temporal performance of a compressor

The focus of our tiling tolerance analysis has been on the spatial performance of a tiled-grating compressor. We also developed a numerical model of the tiled compressor that uses Gaussian beamlet ray tracing to analyze the spatiotemporal performance of the compressor. Figure 6(a) compares the far-field pulse (temporal profile of the far-field focal spot) of a compressor with no tiling error (transform limit) with the pulse obtained when the tilt, tip, and piston of each tile are placed at the tolerance limits (0.2 *µ*rad, 0.2 *µ*rad, and 0.13 *µ*m, respectively) in a worst-case realization. Figure 6(b) shows the corresponding temporally integrated far-field beam for the compressor with the tiling errors. The FWHM pulse widths are 521 fs and 543 fs, and the R80 of the far-field focal spots are 2.6 *µ*m and 5.7 *µ*m, respectively, for the two cases. The far-field pulse width remains close to its transform-limited value even when the far-field beam is significantly degraded. More-detailed temporal analysis will be discussed in later publications.

Our current interferometric tiling method detects and sets the differential piston between adjacent tiles within 2*πN*. Partial compensation of the effect of residual piston errors can be achieved by using only a single TGA. We are planning to set the value of *N* to be less than 2 by translating a high-precision mechanical gauge across the bottom gap area of adjacent tiles [6], which will ensure that the residual piston can be corrected by the piston actuators (travel range: 25 *µ*m) of a single TGA. Figures 7(a) and 7(b) show the spatial and temporal far fields when the central tile of each TGA is displaced from its 4*π* boundary by 0.13 *µ*m. Since this displacement is equal to the piston tolerance limit, this example is a worst-case realization of piston errors. Figures 7(c) and 7(d) show the effect of applying the opposite piston on TGA4 to correct for all of the accumulated piston errors. A similar result is obtained when piston compensation is applied using only TGA3 (FWHM=560 fs, R80=3.6 *µ*m). Owing to the angular dispersion within the compressor, there remains a strip of light between adjacent tiles that is not fully compensated [7]. The uncompensated spectral phase within these regions broadens the transform-limited pulse by 39 fs and 35 fs when compensation is applied using only TGA3 or TGA4, respectively.

## 4. Techniques for initial tiling and tiling automation

Interferometry was used to perform initial tiling. A Fourier fringe-pattern analysis (FFA) [8] was used to retrieve the phase of each tile and to calculate the differential tilt, tip, and piston between the central tile and an outboard tile. The differential values were used as feedback to control the tiling actuators to minimize the overall wavefront of the full TGA. For the reported small-scale grating tiling technique [3, 6], the highly smoothed phase information of each tile was used to calculate the differential value (the tile surface was basically treated as a flat plane), far-field measurement with close to diffraction-limited performance was used as a means to verify the fringe analysis results, and the true wavefront of each tile and the wavefront of two tiles after alignment were not recovered. This technique can not be used for large-aperture grating tiling directly owing to the following facts: The typical wavefront error of an OMEGA EP large-aperture grating tile is 0.25 *λ* (p-v), which directly affects the accuracy of the calculated differential tilt/tip and piston, especially the differential piston, depending on the size and location of the tiling beam on the tiles; therefore, highly smoothing the wavefront or treating the tile wavefront as a flat plane will create an inconsistency between the optimum calculated differential value and the minimized wavefront of all tiles after alignment. Furthermore, it is extremely difficult to build a diffraction-limited, far-field imaging system of a large beam (0.37 m × 0.37 m, 0.5 m in diagonal). The aberration in the far-field imaging system creates an inconsistency between tiled wavefront and the measured far field, i.e., what is measured does not correspond to what is tiled. For large-aperture-grating tiling, it is essential to reconstruct the wavefront of individual tiles, to perform tiling with a beam size close to the full aperture of the tiles, therefore obtaining the accurate differential positions between adjacent tiles. The final wavefront of all the tiles after alignment should be used as the tiling criterion without ambiguity. The on-shot beam of OMEGA EP almost covers the whole area (1.41 m) of TGA2 and TGA3 owing to the dispersion from TGA1. Therefore, the wavefront of the three-tiled-tiles needs to be reconstructed and minimized, which requires a nearly full-aperture, near-field tiling technique. Sub-aperture tiling produces a wavefront that is minimized only across gap areas rather than for the whole TGA.

A 0.61-m Fizeau interferometer was built to perform large-aperture grating tiling. Figure 8(a) shows the setup for performing interferometric tiling. A cw laser beam at 1053 nm is up-collimated to 24 in. in diameter. 96% of the beam travels through a transmission flat (TF) to the TGA, and 4% of the beam is retroreflected by the TF to serve as a reference beam. The transmitted beam is incident on the TGA at an angle of 72.5°. The diffracted beam, at an angle of 61.4° relative to the grating normal, is retroreflected by a reflection flat (RF) to form another interference beam. The reflectivity of the RF is 4%. Tiling grating tiles at use angle (72.5°/61.4°) rather than at Littrow angle provides good fringe contrast to reconstruct the wavefront because the intensities of the two interference beams are balanced. If a Littrow angle is used to reconstruct a grating wavefront, the intensity of the test beam is not balanced to that of the reference beam (greater than 96% grating diffraction efficiency versus 4% TF reflectivity). Generally a waveplate is used in the small beam space to manipulate the polarization of the incident beam on the grating, therefore attenuating the test beam to match the intensity of the reference beam. The polarization deviation from the designed parameter of the multilayer dielectric grating distorts the intensity uniformity of the diffracted beam, which significantly degrades the interferogram contrast. The maximum grating surface area that the tiling beam can cover is 0.9 m × 0.43 m owing to the incoming-beam size (0.43 m × 0.43 m) and the diameter of the RF (0.6 m in diagonal).

To perform FFA, an interferogram with carrier frequency was generated by tilting the TGA. This interferogram was Fourier transformed; one of the first harmonic signals was chosen, filtered, and inverse-Fourier-transformed back to the spatial domain to obtain the wavefront information of the tiles. During the tiling process, two adjacent tiles, such as left and central tiles, were tiled first for their full apertures, then the TGA was translated for tiling the right and central tiles. Finally the TGA was translated for measuring the wavefront of all three tiles, including the full central tile, and approximately half of each outboard tile. Figures 8(b) and 8(c) show the interferograms of two tiles and three tiles, respectively.

In FFA, the reference and test beams do not follow the same optical paths in the interferometer, and the wavefront error of the interferometer induces a detrimental error that needs to be calibrated. The calibration for this non-common path error was performed by measuring the wavefront of a reference flat that was tilted and tipped to generate the same fringes (number and orientation) as the TGA produced when positioned for tiling. This measured wavefront represents the interferometer background error and was subtracted from the interferometric measurement for tiling.

The tiling algorithm and process are described in the flow diagram shown in Fig. 9. To achieve tile-to-tile alignment, the angular misalignment is evaluated and removed first; then the differential piston is removed. The tilt and tip angle of each tile relative to the TF was calculated, the true wavefront of each tile was recovered by using FFA first; then the differential angle between the two tiles was calculated; and finally the differential value was fed back to tilt/tip actuator control system, which drove an edge tile in the opposite direction until the differential values were within predetermined tolerance of 0.2 *µ*rad.

For piston removal, the diffraction effect of tile gaps and the wavefront error of each individual tile affect the accuracy of calculating the differential piston by fringe analysis, causing the optimum differential piston to not necessarily correspond to the minimized wavefront of the tiles after alignment. Removing the piston using only a small section of the tiles near the gap will further decrease the accuracy of piston calculation. The goal of tiling is to align the wavefront of the tiles; therefore, we choose to use the measured wavefront of all tiles to close the piston removal control loop. After angular misalignment is removed, the piston of the edge tile is transversed through a distance corresponding to a change in OPD of 1 *λ* (*λ*=1053 nm) with step size equivalent to 1/8-*λ* OPD change. After eight steps, the position providing minimum wavefront error of the tiles should be obtained. The edge tile is then placed at this position. Higher resolution could be achieved by reducing the step size to 1/16 *λ* if necessary, depending on the sensitivity of the tiling interferometer. The relation between differential piston change and optical-path-length change can be described by the following equation:

ΔOPD=2 × ~Δpiston × (cos*θ*
_{in}+cos*θ*
_{diff})

where *θ*
_{in}=72.5° and *θ* diff=61.4°, the former being the incident angle and the latter being the diffraction angle. In order to generate a 1-*λ* OPD change (*λ*=1053 nm), a 0.64-*λ* differential piston change needs to be induced. We drove the piston actuators of one of the edge tiles of a TGA over 676 nm with a step size of 85 nm to validate the piston removal process. The rms wavefront of the two tiles was recorded; Strehl and R80 of the corresponding far-field focal spots were calculated. These parameters were normalized to their peak values. Figure 10 shows that these three parameters change periodically with the induced piston. This result indicates that the rms wavefront of the tiles, Strehl ratio or R80, can be used as criteria for closing the tiling control loop interferometrically where rms wavefront provides the highest sensitivity.

Figure 11 shows the automatic tiling process. Angular misalignment was removed after six steps; then the piston was stepped through 1 *λ* with a step size of 1/8 *λ*; finally, minimized wavefront (rms wavefront=0.056 *λ*) of the two tiles was achieved by returning to the converged piston position.

This automatic tiling interferometry is used only for initial tiling owing to the fact that the resolution of tiling interferometry with the large aperture beam is limited by the environmental effects, such as turbulence and vibration, which could induce more noise into the tile position feedback to the tiling actuators than that from the capacitive sensors used for monitoring tile positions. The tiled position of each TGA is maintained by closing the tiling actuator control loops with the detected tile position changes through capacitive sensors across the tile gaps.

This tiling process was used for tiling a TGA consisting of three tiles. Figure 12(a) shows the minimized wavefront of three tiled full-size OMEGA EP gratings. The measured surface area of the TGA is 90 cm × 38 cm; the rms wavefront is 0.047 *λ*; and the dashed lines indicate the gaps crossing the tiles. Figure 12(b) shows that one single far-field spot with a Strehl ratio of 0.92 is achieved from the tiled-grating assembly.

## 5. Conclusion and perspectives

We have reported on an interferometric near-field method and its implementation for large-aperture grating tiling. A thorough tiling tolerance analysis has been conducted for input beams with various wavefronts. We have discussed how the focal-spot performance, in terms of R80 and Strehl ratio, degrades with increasing tiling error. We conclude that tiling error has greater impact on a compressor system having a flat input wavefront than that having an input beam with aberrations. The statistical performance of a compressor can be predicted for an identified tiling accuracy with the developed ray-tracing model. Temporal analysis shows that a close to transform-limited pulse width could be achieved for the achievable tiling accuracy. Pulse broadening induced by accumulated piston error from one or all three TGA’s can be corrected by inducing the same amount, opposite piston on one of the tiles of the last TGA. An automatic tiling process has been developed and used to tile three full-size gratings for OMEGA EP. The tiling process for the TGA’s in the real system of OMEGA EP will be similar to what has been described in Sec. 4. A Fizeau interferometer is presently being built in the grating compressor chamber to align each individual TGA of the compressor in a vacuum environment separately. After four TGA’s are tiled interferometrically, they will be rotated to their corresponding use angles for compressor alignment to obtain the best pulse compression. The long-time stability of the tiling will be maintained by compensating the temporal drift of shift and in-plane rotation with piston and tip. The detailed tiling process and results will be reported in later publications.

## Acknowledgment

We are grateful for valuable support from D. Hassert, A. L. Rigatti, R. Jungquist, S. F. B. Morse, and S. J. Loucks. Special thanks go to J. Price for his excellent mechanical design work on the TGA’s. This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-92SF19460, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.

## References and links

**1. **J. A. Britten, W. A. Molander, A. M. Komashko, and C. P. Barty, “Multilayer dielectric gratings for petawatt-class laser systems,” Proc. SPIE **5273**, 1–7 (2004). [CrossRef]

**2. **T. Zhang, M. Yonemura, and Y. Kato, “An array-grating compressor for high-power chirped-pulse amplification lasers,” Opt. Commun. **145**, 367–376 (1998). [CrossRef]

**3. **T. J. Kessler, J. Bunkenburg, H. Huang, A. Kozlov, and D. D. Meyerhofer, “Demonstration of coherent addition of multiple gratings for high-energy chirped-pulse-amplified lasers,” Opt. Lett. **29**, 635–637 (2004). [CrossRef]

**4. **A. Cotel, M. Castaing, P. Pichon, and C. Le Blanc, “Phased-array grating compression for high-energy chirped pulse amplification lasers,” Opt. Express **15**, 2742–2752 (2007). [CrossRef]

**5. ** J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, “OMEGA EP: High-energy petawatt capability for the OMEGA laser facility,” J. Phys. IV France **133**, 75–80 (2006). [CrossRef]

**6. **
“Demonstration of real-time, phase-locked alignment of tiled gratings for chirped-pulse-amplified lasers,” *LLE Review Quarterly Report*100, 242–251, Laboratory for Laser Energetics, University of Rochester, Rochester, NY, LLE Document No. DOE/SF/19460-578, NTIS Order No. PB2006-106672 (2004). Copies may be obtained from the National Technical Information Service, Springfield, VA 22161.

**7. **J. Bunkenburg, T. J. Kessler, W. Skulski, and H. Huang, “Phase-locked control of tiled-grating assemblies for chirped-pulse-amplified lasers using a Mach-Zehnder interferometer,” Opt. Lett. **31**, 1561–1563 (2006). [CrossRef]

**8. **M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**, 156–160 (1982). [CrossRef]