Measurement of period difference in grating pair based on analysis of grating phase shift

Transcription

1 Measurement of period difference in grating pair based on analysis of grating phase shift Chao Guo, Lijiang Zeng State Key Laboratory of Precision Measurement Technology and Instruments Department of Precision Instruments, Tsinghua University, Beijing , China Tel.: + 86 [10] Fax: +86 [10] guoc@mails.tsinghua.edu.cn Abstract Grating mosaic is effective to manufacture meter-sized large-aperture gratings. In the mosaic progress, it is important to select small-aperture gratings with almost the same mean periods. In this paper, we propose and demonstrate a novel method to measure the period difference in grating pair. The phase shifts of +1st- and 1st-orders diffraction beams are detected while two gratings are translated in grating-vector direction. The relationship between the phase shifts and translation displacement is used for the measurement of grating period difference. The roll and yaw angular deviation of gratings can be easily eliminated by phase compensation. We also use the far-field intensity patterns of gratings to mosaic two gratings in the measurement. For a pair of ~0.674 μm period gratings, the period difference was measured to be nm with a standard deviation of nm. This result coincides with that of optical-diffraction method. Keywords: Grating pair, period difference, heterodyne phase shift, phase compensation 1. Introduction The chirped-pulse-amplification (CPA) system has been a research focus these years. One of the technological difficulties in CPA system is the manufacture of large-aperture diffraction gratings. In order to reduce the production cost for the manufacture of single large grating, two or more small-aperture gratings are mosaicked as a large one. In this progress, it is necessary to select small-aperture gratings with almost the same mean periods. Zuo et al. [1] calculated the expectation of period difference for perfect mosaic: period difference of small gratings should be smaller than nm, with a criterion that the pulse in the far field is broadened less than 10% for the laser of λ = 16 nm. The measurement system for period difference should have high accuracy to meet this demand. Theoretically, we can measure period difference between gratings by measuring the period of each grating. The atomic force microscopes (AFMs) and optical diffractometers are two typical methods to measure the grating period. AFM is an important measurement tool in nanometrology and the estimated relative uncertainty for period measurement reaches 1 ~ [2, 3]. With conventional optical-diffraction method, Song et al. [4] measured the grating period with an uncertainty of nm. These methods have high measurement accuracy, but they have to measure two grating periods one after another to calculate the period difference between two gratings. Supposing the measurement uncertainty of the grating period is, the uncertainty of the period difference will be enlarged as 2. In this paper, we propose a new comparative method for measuring the mean period difference in grating pair directively. The method is based on the analysis of phase shift in +1st- and 1st-orders diffraction beams of gratings when gratings are translated in gratingvector direction. The calculation formula of period difference and experimental setup will be given in the following parts

2 2. Principle We set up the coordinate system as shown in Fig. 1. The x-axis is parallel to the gratingvector direction and the z-axis is perpendicular to the grating plane of G1. A laser beam (Beam1) normally incidents on G1 and the wavelength of the beam is λ. The mth-order diffraction angle is supposed to be θ G1,m. If we translate grating G1 through a displacement L, the displacement components along the x- and z-axes are x and z, respectively. Phase shift δφ G1,m of the mth-order diffraction beam is expressed as 2m 2 δg1, m x 1 cosg1, m z, (1) d1 where d 1 is the mean period of G1. Fig. 1. Definition of coordinate system. Grating G2 is aligned with G1 to form a large-aperture grating and the mean period of G2 is d 2. Another laser beam (Beam2) normally incident on G2. The wavelength of Beam2 is also λ. The phase shift δφ G2,m of the mth-order diffraction beam of G2 is 2m 2 δg2, m x 1 cosg2, m z. (2) d2 By subtracting Eq. (2) from Eq. (1), we get the difference between phase shifts of two gratings for +1st-order (m = +1): 2 d 2 x G1 G2, 1 cos G1, 1 cos G2, 1 z, (3) where the period difference d = d 1 d 2. In Eq. (3), if two gratings are translated along the x-axis, i.e. z = 0, the calculation formula for d can be derived as G1G2, 1 d. (4) 2 x By recoding φ G1 G2,+1 and x during translation, we can get the mean period difference between gratings. The grating periods d 1 and d 2 are regarded as known quantities. They can be measured with optical diffractometer and the measurement errors of d 1 and d 2 introduce negligible uncertainty to d. Eq. (4) shows that for two gratings, φ G1 G2,+1 is proportional to x. Theoretically, if d is small and difficult to measure, we can enlarge x to help identifying φ G1 G2,+1, which means the method can offer high measurement resolution in determining small d. During the measurement of d, the translation stage together with two gratings might rotate around the x- and y-axes which result in roll and yaw angular deviations, respectively (Fig. 2). In Fig. 2(a), two gratings are aligned up-and-down, the roll angular deviation causes z G1 and z G2 nonzero and different from each other. If two gratings are aligned as shown in Fig. 2(b), the yaw deviation causes similar problem. The normal displacement of gratings z G1 and z G2 cannot be ignored in Eq. (3)

3 Fig. 2. Effects of (a) roll and (b) yaw angular deviations of gratings during translation. In order to modify Eq. (3) and (4), we put forward a phase-compensation method to eliminate the roll and yaw angular deviation. Considering the effects of z G1 and z G2, we express the phase difference between phase shifts of two gratings for +1st- and 1st-orders diffraction beams as: d 2 G1 G2, 1 2 x 1 cosg1, 1 zg1 1 cosg2, 1 z G2, (5) d 2 G1 G2, 1 2 x 1 cosg1, 1 zg1 1 cosg2, 1 z G2. (6) For normal instance, cosθ G1,+1 = cosθ G1, 1, cosθ G2,+1 = cosθ G2, 1. By subtracting Eq. (6) from Eq. (5), the items containing z G1 and z G2 are eliminated which means the roll and yaw angular deviation of gratings have no effect on ( φ G1 G2,+1 φ G1 G2, 1 ). Then we have an improved formula for the calculation of d: G1G2, 1 G1 G2, 1 d, (7) 2 2x G1G2, 1 G1 G2, 1 where the item ( φ G1 G2,+1 ) in Eq. (4) is replaced by. By phase 2 compensation, the influence of the roll and yaw angular deviations are eliminated. 3. Experimental setup and results In order to measure the period difference between gratings G1 and G2, we set up the measurement system as shown in Fig. 3. In this system, gratings G1 and G2 are aligned upand-down as shown in Fig. 2(a) and the grating periods are ~0.674 μm. The system consists of two parts: the far-field-intensity-pattern-monitoring system is expressed by the dash-dot line, which is used for the relative attitude adjustment of gratings; the phase-differencemeasurement system is expressed by the continuous line and the dashed line, which is used for the measurement of phase differences between diffraction beams

4 In the far-field-intensity-pattern-monitoring system, the laser beam is expanded and collimated to a beam of ~15 mm in diameter. Then, the beam incidents onto the gap of gratings. Grating G2 is Littrow mounted. The 1st-order diffraction beams of G1 and G2 are reflected by two beam splitters BS5 and BS6 and focused together with the 0th-order diffraction beams by a long-focal-length (700 mm) lens (L). Far-field intensity patterns of two orders are magnified by a 10 microscope objective (MO) for detection with a chargecoupled device (CCD). In the experiment, two 32 mm 50 mm gratings are mounted on two holders with angular adjustment (Model 8807 New Focus Inc.). Rotation angles of G1 and G2 around the x- and y-axes can be controlled step by step with a resolution of ~0.7 μrad. Fig. 3. Experimental setup for measuring the period difference. In the heterodyne-phase-shift-measurement systems, two laser beams are modulated by acoustic-optical modulators AOM1 and AOM2 with modulation frequencies 70MHz and 70.1MHz, respectively. Then the beam illustrated by dashed line, which is named as Lower Beam, is adjusted by mirrors M1 and M2 to be lower than the beam illustrated by continuous line which is named as Upper Beam. It is noticeable that both incident beams are in the same y-z plane and we cannot clearly illustrate these two beams with a top-view figure as Fig. 3, so we translate the Lower Beam aside to help identifying two beams. The Upper Beam normally incidents on grating G1 and the Lower Beam normally incidents on grating G2. There are two interferometers which are composed by M4-BS3 and M5-BS4 to measure the heterodyne phases of +1st- and 1st-orders diffraction beams of G1 and G2. In the experiment, after translating two gratings along the x-axis through x = 28.1 mm, we realize the scanning of measurement areas on gratings. The mean period difference G1G2, 1 G1 G2, 1 between two measurement areas is obtained by linearly fitting 2 and x according to Eq. (7)

5 Figure 4 shows three heterodyne-phase curves in the experiment. Both φ G1 G2,+1 and φ G1 G2, 1 are affected by the roll angular deviation. After phase compensation, effect of roll angular deviation is eliminated. The mean d for nine times measurements is nm with a standard deviation of nm. In comparison, we measured the period of each grating with conventional optical-diffraction method and the mean period difference is nm with a standard deviation of nm. This result agrees with that measured by our method. Fig. 4. Phase compensation effect of roll angular deviation. 4. Conclusion We proposed a novel method for measuring the mean period difference in grating pair based on the analysis of grating phase shift. Phase differences between two +1st and 1storders diffraction beams of grating pair were used for phase compensation, which could eliminate the roll and yaw angular deviation during translation. By fitting the compensated phase difference, we could get the period difference in grating pair. Our measurement result of period difference coincides with the result of optical-diffraction method. Our method is a comparative measurement and the measurement resolution can be easily improved by enlarging the translation distance. 5. Acknowledgements This work was supported by the National Natural Science Foundation of China under Project No References 1. Y. Zuo, X. Wei, Q. Zhu, H. Liu, X. Wang, Z. Huang, Y. Guo, C. Ying. Acta Physica Sinica. 2007, vol. 56, pp F. Meli, R. Thalmann. Meas. Sci. Technol. 1998, vol. 9, pp G. Dai, F. Pohlenz, T. Dziomba, M. Xu, A. Diener, L. Koenders, H. U. Danzebrink. Meas. Sci. Technol. 2007, vol. 18, pp W.Y. Song, K.Y. Jung, B.H. O, B.C. Park. J. Korean Phys. Soc. 2004, vol. 45, pp