Objective The current maximum size of a diffraction grating is a ruled area of 300 mm (groove length) by 400 mm (ruled width). With the latest generation of eight- and ten-meter-class optical telescopes and the desire for resolutions exceeding 100000 for spectroscopic applications, there is a need to produce larger diffraction gratings or assemblies of existing gratings in good alignment. At the same time, descriptions of instrumentation utilizing such large gratings are found in the chirped-pulse-amplification (CPA) scheme. To overcome the current size limit for ruled gratings, a method is required for producing a stable assembly of two or more gratings. The most obvious choice is to build a fixture that can hold individual gratings in a stable configuration relative to one another and relative to the instrument where the gratings are being used. The alignment requirements for mosaic echelle gratings are derived from the image-quality requirements for a high-resolution spectrograph. Several schematic setups have been used to build alignment collimators of the echelle mosaic. Image analysis, as well as the logging of all data, is typically important for achieving the required level of noise reduction. The errors of the mosaic gratings can be analyzed quantitatively, enabling the experiment to succeed more effectively.

Methods The alignments in a mosaic grating are determined by maintaining a reference grating in a static position and moving the adjustable gratings relative to the reference tile. There are five degrees of freedom relative to the reference grating: the tilt Δθ_y, tip Δθ_x, twist Δθ_z, lateral shift Δx, and piston extension Δz. All these alignment errors produce different fringes, so the properly aligned position can be determined by interferometric analysis. In order to increase the precision of the angular-error adjustments in mosaic grating alignment, we use nine-pixel-average algorithm based on Fourier-transform to analyze the interference-fringe frequency patterns with high precision. The algorithm contains several steps. First, a window function is applied to each fringe pattern to reduce edge effects in the Fourier-transform calculations, and the carrier frequency is chosen to position the sidelobes outside the noise spectrum of the interferometer system. Second, we apply an inverse Fourier-transform, followed by phase calculations, to produce a two-dimensional phase reconstruction of the gratings. Finally, any remaining tip/tilt errors can be calculated and resolved. By using the nine-pixel-average algorithm and calculating the centroid coordinates of the sidelobes of the interferogram, we can determine the spatial carrier frequencies with high accuracy. This technique is especially useful when operating in optically “noisy” environments, and it permits a high level of noise rejection. In connection with the interference method for mosaic echelle alignment, we have simulated and analyzed the Fourier-transforms of the different fringes obtained for different angular errors. We also have obtained the calibration coefficients for the angles in alignment and have determined the resolution and the precision tolerance of the Fourier-transform-based fringe-pattern analysis in a co-phasing mosaic-grating-assembly experiment.

Results and Discussion In the zeroth-order diffraction pattern of the mosaic gratings, the angular errors Δθ_x and Δθ_y affect the tilt and the period of the interferogram fringes. Using nine-pixel-average algorithm based on Fourier-transform, we have analyzed the different fringes obtained with different micrometer-head readings, and we have also calculated the calibration coefficients for the angles in alignment system (Fig. 5 and Fig. 6). The calculated angular errors increase linearly as the micrometer-head readings increase, and the linear fitting coefficient is better than 0.999. In addition, we have determined the resolution and precision tolerance of the algorithm. According to the error analysis of calculated misalignment angle, we find that Fourier-transform-based nine-pixel-average algorithm data is in accord with the setting data, and the angular-misalignment error distribution is less than 0.4 μrad (Fig. 8). We have also compared the angular resolution for different fringe algorithms (the peak algorithm and the nine-pixel-average algorithm). We find that the resolution precision of the nine-pixel-average algorithm is better than 0.1 μrad, while the peak algorithm is impractical for use in fringe calculations (Table 2). From this sensitivity study, not only do we find that the nine-pixel-average algorithm shows better performance in the analysis but also we verify its high sensitivity for detecting phase variations. All these results demonstrate the usefulness of the nine-pixel-average algorithm.

Conclusions In the present study, we have devised Fourier-transform-based nine-pixel-average algorithm and have used it to analyze the interference-fringe frequency patterns with high precision. With this technique, we can determine the angle-adjustment coefficients of the system. The alignment angles of the two gratings with respect to each other can be determined with an accuracy of 0.4 μrad, and the resolution precision of the nine-pixel-average algorithm is better than 0.1 μrad. This result is entirely sufficient to guarantee the precision required for mosaic echelle gratings at the telescope, and the work provides a theory for fringe calculations for the co-phasing mosaic alignment of large-sized gratings. We anticipate that in conjunction with future increases in grating size, astronomical spectrographs will be able to attain the significantly increased resolution that is very important for astronomical research.