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SECTION I

The collimation of monochromatic light has an important influence on the accuracy in the optical metrology. Therefore, an easy and high-accuracy method to check the quality of monochromatic light is significant. Several collimation testing techniques based on the Talbot effect and moiré technique have been published [1]– [2] [3] [4] [5] [6] [7] [8] [9] [10] [11][12]. For example, in 1971, Silva reported the use of Talbot interferometry with linear gratings for collimation testing [1]. Then, a dual-field grating with self-referencing capability and higher sensitivity was introduced by Kothiyal and Sirohi [5]. Shortly after, a study on collimation testing using spiral gratings was reported by Chang and Su [6]. In their paper, authors also pointed that the resultant moiré fringes of linear gratings are in linear distribution, leading to inconvenience to distinguish the state of collimation clearly. In 1994, Sriram carried out a comparative study on collimation testing with the dual-field linear, spiral and evolute grating [8]. It established that, of these three kinds of grating, the dual field grating exhibits the highest sensitivity for collimation testing. In 2008, the Fourier-transform algorithm (FTA) and the spatial phase-shifting algorithm (SPSA) were jointly proposed to determine the phase distribution of the dual-spiral moiré fringes by Huang and Su [10]. The results showed that the FTA and SPSA are theoretically precise, effective and insensitive to the random noise, but this scheme is not verified by the experiment. In the same year, Shakher and co-workers reported their investigation on collimation testing with circular gratings [3]. Its result revealed that the sensitivity is better than dual-field gratings scheme. However, the quantitative measurement cannot be achieved, making it not suit for the engineering applications.

In summary, those methods above cannot possess two merits below simultaneously:

- • The resultant moiré fringes are convenient to measure qualitatively and quantitatively.
- • The method has a high accuracy.

Herein, a collimation method based on the differential grating is proposed. In this method, the self-image of linear grating is projected on a differential grating. Because of the specifically designed grating pitches, the resultant moiré fringes are divided into two parts (the upper one and the lower one), in which moiré fringes have the same phase distribution in the case of the collimated light. When the collimation is disturbed, the period of resultant moiré fringes is magnified in one part and demagnified in the other part. This design can improve the sensitivity to a great degree and make the result simple to check. According to the experimental result, the divergence or convergence angle of $10^{-7}\ \hbox{rad}$ can be readily measured.

This paper is arranged as follows. In Section 2, the relationship between the divergence or convergence angle and the phase of moiré fringes is analyzed. Next, the simulational and experimental results are given to verify the proposed method. Finally, the results and applications are simply discussed.

SECTION II

The optical arrangement is shown in Fig. 1. A beam of monochromatic light is incident upon the linear grating LG and then projected onto the differential grating DG, which is positioned at the Talbot self-image LG $^{\prime}$ of LG. When the collimating lens is defocused, the plane monochromatic wave becomes spherical (diverging or converging), as shown in Fig. 2. The divergence or convergence angle $\gamma$ of monochromatic light is related to the defocusing distance $\Delta f$ as follows:TeX Source$$\tan\gamma = {h\Delta f \over f(f - \Delta f)}\eqno{\hbox{(1)}}$$ where $h$ is the height above the axis at which the incident ray strikes the collimating lens, and $f$ is the focal length of the collimating lens.

In such a case, the self-image of LG becomes magnified or demagnified (as shown in Fig. 3), depending on whether the collimating lens is in focus or out of focus. The divergence or convergence angle is given asTeX Source$$\tan \gamma = {P^{\prime}_{l} - P_{l} \over Z_{m}}\eqno{\hbox{(2)}}$$ where $Z_{m}$ is the grating separation, and $P_{l}$ and $P_{l}^{\prime}$ represent the pitch of LG and LG $^{\prime}$, respectively. Since $P_{l}$ and $Z_{m}$ are determined by the experiment, the problem now is how to extract the characteristic parameter $P_{l}^{\prime}$.

The LG and DG are illustrated in Fig. 4(a) and (b), respectively. The DG is comprised of two sets of line gratings. The upper line grating has the pitch $P_{du}$ while the lower grating has the pitch $P_{dd}$. $P_{dd}$ is closely equal to $P_{du}$ plus an additional term $\Delta P(>\ 0)$, that is, $P_{dd} = P_{du} + \Delta P$. The LG consists of one set of line grating with pitch $P_{l}$. Specifically, $P_{l}$ is set as: $P_{l} = P_{dd}\ P_{du}/(P_{dd} + P_{du})$. For simplicity, let the amplitude transmittances of LG, $\hbox{DG}_{\rm u}$, $\hbox{DG}_{\rm d}$, and $\hbox{LG}^{\prime}$ be written asTeX Source$$t_{i} (x) = {1 \over 2} + {1 \over 2}\cos \left({2\pi \over P_{i}}x + \theta_{0} \right)\eqno{\hbox{(3)}}$$ where $P_{i}$ represents the pitch of LG, $\hbox{DG}_{\rm u}$, $\hbox{DG}_{\rm d}$ or LG $^{\prime}$ when replaced by $P_{l}$, $P_{du}$, $P_{dd}$ or $P_{l}^{\prime}$. When the LG is projected onto the DG, the equivalent amplitude transmittance generated can be given as follows:

- for the $\hbox{DG}_{\rm u}$TeX Source$$\eqalignno{t_{u} (x) =&\, t_{l}^{\prime} (x) t_{du} (x)\cr \noalign{\vskip3pt} =&\, {1 \over 4} + {1 \over 4}\cos \left({2\pi \over P_{l}^{\prime}}x + \theta_{0}\right) + {1 \over 4}\cos \left({2\pi \over P_{du}}x + \theta_{0}\right) \cr\noalign{\vskip3pt}&+ {1 \over 8}\cos \left[\left({2\pi \over P_{du}} - {2\pi \over P_{l}^{\prime}}\right)x\right]+ {1 \over 8}\cos \left[\left({2\pi \over P_{du}} + {2\pi \over P_{l}^{\prime}}\right)x + 2\theta_{0}\right]&\hbox{(4)}}$$
- for the $\hbox{DG}_{\rm d}$TeX Source$$\eqalignno{t_{d} (x) =&\, t_{l}^{\prime} (x) t_{dd} (x)\cr \noalign{\vskip3pt} =&\, {1 \over 4} + {1 \over 4}\cos \left({2\pi \over P_{l}^{\prime}}x + \theta_{0}\right) + {1 \over 4}\cos \left({2\pi \over P_{dd}}x + \theta_{0}\right)\cr\noalign{\vskip3pt} & + {1 \over 8}\cos \left[\left({2\pi \over P_{l}^{\prime}} - {2\pi \over P_{dd}}\right)x\right] + {1 \over 8}\cos \left[\left({2\pi \over P_{dd}} + {2\pi \over P_{l}^{\prime}}\right)x + 2\theta_{0}\right].&\hbox{(5)}}$$

The difference-frequency components in Eqs. (4) and (5) contain the information of collimation. Therefore, the phases of difference-frequency components are extracted by the use of FTA [10], [12], [13]. The difference between them can be given as follows:TeX Source$$\Delta \varphi = \varphi_{u} - \varphi_{d} = 2\pi x\left({1 \over P_{dd}} + {1 \over P_{du}} - {2 \over P_{l}^{\prime}}\right).\eqno{\hbox{(6)}}$$

From Eq. (6), $P_{l}^{\prime}$ can be deduced asTeX Source$$P_{l}^{\prime} = {4\pi \over 2\pi (1/P_{dd} + 1/P_{du}) - d (\Delta \varphi)/dx}.\eqno{\hbox{(7)}}$$

Substituted Eq. (7) into Eqs. (1) and (2). The small $\gamma$ and $\Delta f$ can be expressed as TeX Source$$\eqalignno{\gamma =&\, {4\pi \over Z_{m}\left[2\pi (1/P_{dd} + 1/P_{du}) - d(\Delta \varphi)/dx\right]} - {P_{l} \over Z_{m}}&\hbox{(8)}\cr\noalign{\vskip3pt} \Delta f =&\, {f^{2} \over h}\left\{{4\pi \over Z_{m}\left[2\pi (1/P_{dd} + 1/P_{du}) - d(\Delta \varphi)/dx\right]} - {P_{l} \over Z_{m}}\right\}.&\hbox{(9)}}$$

For the collimated light, $1/P_{l}^{\prime} = 1/P_{l} = (P_{dd} + P_{du})/2$, then $\Delta \varphi = 0$, $\gamma = 0$, and $\Delta f = 0$. The two sets of moiré fringes are aligned (as shown in Fig. 4(c)) and the collimation of the beam is perfect.

For the divergent light, $1/P_{l}^{\prime}\ <\ 1/P_{l}$, then $\gamma\ <\ 0$ and $\Delta f\ <\ 0$. The lower moiré fringes are closer and two sets of moiré fringes are staggered, as shown in Fig. 4(d). Because the light source is in focus, the collimated lens CL should be moved away from the light source.

For the convergent light, $1/P_{l}^{\prime}\ >\ 1/P_{l}$, then $\gamma\ <\ 0$ and $\Delta f\ <\ 0$. As shown in Fig. 4(e), the upper moiré fringes are closer and two sets of moiré fringes are also staggered. Because the light source is out of focus, the collimated lens CL should be moved toward the light source.

SECTION III

The numerical simulation is performed based on the theoretical model mentioned above. The pitches of DG and LG in the simulation are, respectively $4\ \mu\hbox{m}\ (P_{du})$, $6\ \mu\hbox{m}\ (P_{du})$. and $4.8\ \mu\hbox{m}\ (P_{l})$. A beam of monochromatic light with wavelength 632.8 nm is expanded by a 4× microscope objective and a pinhole of 6.6 $\mu\hbox{m}$ diameter. Simultaneously, the separation between the two gratings $(Z_{m})$ is set as 7.428 mm.

In the simulation, the random noise as 80% of the fringe patterns amplitude is added into the fringe patterns to calculate the divergence or convergence angle $\gamma$. To examine the accuracy of the proposed method, four groups of measurement are carried out with defocusing distances −500, 500, −50 and 50 $\mu\hbox{m}$, respectively. The mean values $(\bar{\gamma})$, standard deviations $(\sigma)$ and error percentages of the calculated $\gamma$ are shown in Table 1 while the corresponding moiré fringe patterns and phase distributions are given in Figs. 5–8. In the first group, the error percentage and standard deviation of $\gamma$ is 1.99% and 1.167e-008 rad, respectively. In the second group, the error percentage and standard deviation of $\gamma$ is 2.18% and 1.241e-008 rad, respectively. In the third group, the error percentage and standard deviation of $\gamma$ is 0.45% and 0.138e-008 rad, respectively. In the fourth group, the error percentage and standard deviation of $\gamma$ is 0.15% and 0.155e-008 rad, respectively.

SECTION IV

The experimental setup is shown in Fig. 9. A He–Ne laser (25 LHR 151) with wavelength 632.8 nm and output power 5 mW, respectively, is used as the illumination source. The focal length and magnification of the collimation lens are, respectively 50 mm and 4×, and the diameter of pinhole is 6.6 $\mu\hbox{m}$. The CCD camera combined with a computer is used to capture the moiré fringe patterns. There is one differential grating used as DG, whose pitches are, respectively, $4\ \mu \hbox{m}\ (P_{du})$ and $6\ \mu\hbox{m}\ (P_{dd})$, and one line grating used as LG, whose pitch is $4.8\ \mu\hbox{m}\ (P_{l})$. They are designed by us and fabricated by the Institute of Microelectronics of Chinese Academy of Sciences (Beijing, China). In the experiment, the collimating lens is mounted upon a precision translation stage to translate the lens along the optic axis. The translation stage is driven by a micrometer screw with a reading precision of 1 $\mu\hbox{m}$. The lens position is adjusted to provide a collimated beam. The grating separation can be adjusted to set the required value of the Talbot plane at $Z_{m}$. The captured moiré fringe patterns are processed by a computer using FTA.

In the experiment, the grating separation is adjusted to about 7.428 mm. When the LG and DG are adjusted to be parallel, two sets of linear moiré fringes are obtained. After that, the position of collimation is approached from either side (in focus or out of focus position) by the movement of the micrometer screw attached to the translation stage. This process continues until the two sets of moiré fringes are aligned, after which collimation is achieved. As soon as collimation is achieved, its position relative to the micrometer screw is recorded. The micrometer screw is rotated in the same or opposite direction until the collimation is just disturbed, and the position of the lens is again recorded. The difference between the two values is the defocusing distance. Four sets of measurements are recorded. To improve the accuracy in phase extraction, the captured patterns are cut with integral number of moiré fringe period, as shown in Figs. 10–13. The mean values $(\bar{\gamma})$, standard deviations $(\sigma)$ and error percentages of the $\gamma$ are listed in Table 2. It can be seen that the error percentages are below 2.55%.

SECTION V

The above simulational and experimental results indicate that the proposed method can achieve a measured accuracy of $10^{-7}\ \hbox{rad}$ level. By comparing Tables 1 with 2, its obvious that the experiment results are in good agreement with the simulation results, but the error ranges of $\gamma$ in the experiment are larger than that in the simulation, as shown in Fig. 14. This is mainly caused by three reasons as follows.

- The grating pitchs in the simulation are uniform, but not in the experiment. Because of the machining errors, the inconsistency of grating pitches influences the calculation of phase difference between two sets of fringes.
- The phase distribution of moiré fringes correlates closely with the relative position between the LG and the DG. The relative angle between the LG and DG will lead to the tilt moiré fringes, which has an influence on the high-accuracy measurement because the extracted phase of tilt moiré fringes may inevitably lead to some errors [14], [15].
- The noise in the experiment is more irregular than that in the simulation. In the simulation, only a randomly distributed Gaussian white noise is added, but the noise in practice is more complex and diverse, such as the speckle noise, CCD noise and environmental disturbance.

In practice, the smaller divergence or convergence angle $(\Phi_{min})$ can be detected, the more sensitive the system is. The $\Phi_{min}$ is therefore a measure of sensitivity. The grating pitches in earlier studies are at the sub-millimeter level and good results are obtained with a long grating separation (at the decimeters or meters level). This is because the smallest detectable divergence or convergence angle increases with the decrease of grating pitch and grating separation, such as the experiment done by C. Shakher and co-workers. In their experiment, the results were obtained using the circular gratings with a pitch of 0.2 mm at several Talbot planes (250, 500, 750, 1000, 1250, 1500, and 1750 mm) [3]. And the relationships of the smallest detectable divergence or convergence angle with the grating pitch and grating separation are, respectively shown in Fig. 15(a) and (b). Therefore, it's a challenge to achieve high-precision results by adopting short grating pitch and grating separation. The proposed method in this paper is hopeful to solve these deficiencies. Comparatively, in our experiment, the gratings with pitches of 4, 4.8 and 6 $\mu\hbox{m}$ are used while the grating separation is only 7.428 mm. And the divergence or convergence angles of $10^{-7}\ \hbox{rad}$ are readily achievable.

SECTION VI

In this paper, a novel collimation testing method based on the differential grating is proposed. Based on the relationship between the divergence or convergence angle and the phase of moiré fringes, it's feasible to determine whether the collimating light is collimated or not. The experimental results verify that this method has a detectable measurement capability of $10^{-7}\ \hbox{rad}$, which is promising for the practical applications of mini measurement devices and lithography tools [16]– [17] [18][19].

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