Talbot interferometry for measuring the small tilt angle variation of an object surface Yoshiaki Nakano and Kazumi Murata

Yoshiaki Nakano is with Hokkaido Institute of Pharma­ceutical Sciences, Physics Department, 7-1 Katsuraoka, Otaru 047-02, Japan; K. Murata is with Hokkaido Univer­sity, Department of Applied Physics, Sapporo 060, Japan. Received 27 March 1986. 0003-6935/86/152475-03$02.00/0. © 1986 Optical Society of America. In measuring the small tilt angle of a surface, an autocolli-

mator is generally used. It is difficult to measure the local variation of tilt angles of a large object surface because the area of measurement is limited. In this Letter we present a simple method for measuring the small tilt angle variation of an object surface. The principle of the method is based on Talbot interferometry.1,2 The use of Talbot interferometry as a method for deflection mapping of phase objects and for measuring the focal length of lenses has been reported by the authors.3-6 The sensitivity of the Talbot interferometry can be easily tuned by varying the crossed angle between two gratings, and the optical system is rather simple.

Figure 1 shows a schematic representation of the optical

Fig. 1. Schematic diagram for measuring small tilt angle variation using Talbot interferometry: (a) g1, grating of the period p; g2, second grating of the same period with g1∙, M, test surface; OS, observing screen, (b) Magnified Talbot image g'1 of the period p' on

the second grating g2 (x axis).

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system. The ξ and x axes are taken in the direction of the periodical amplitude variation of two gratings g1 and g2, respectively. The plane of the second grating g2 is set in the direction perpendicular to the plane of the first grating g1. The test surface M is placed in the middle of two gratings. The optical path length along the optical axis from the grat­ing g1 through the test surface M to the grating g2 is set to be equal with the Talbot length kp2/λ, where k is an integer, p is the period of the grating g1 and λ is the wavelength of the incident light.

A monochromatic plane wave illuminates the grating g1 and undergoes diffraction. After being reflected by the test surface, which is tilted by angle δ from the optical axis, the wave propagates to the grating g2. Therefore, the grating g1 produces the Talbot image g'1 on the second grating g2. If the tilt angle of the test surface is π/4, the pitch of the Talbot image g'1 is the same as grating g1. However, as the tilt angle is usually not equal to 7r/4, we obtain the magnified Talbot image. If the tilt angle of the surface deviates from 7r/4 by an amount δ', the pitch of the magnified Talbot image g1 on a screen at the plane x axis is given by p' = p/cos2δ' = p/sin2δ, as seen from Fig. 1(b).

Next we obtain the moire fringe pattern by superimposing the Talbot image g'1 on the second grating g2, which is the same as the first grating g1 but is crossed by a small angle θ. As seen from Fig. 2, the resulting moire fringes are observed with an inclination angle α to the x axis. The pitch ratio between the Talbot image g'1 and the second grating g2 is given by7

Let the inclination angle of the moire fringe in this case be given by α'. The tilt angle δ + ∆δ of the test surface is represented by

Accordingly, from Eqs. (2) and (4) the small angle variation ∆δ of the test surface is given by

When the crossed angle θ is fixed, the small angle variation ∆δ can be obtained by measuring the tilt angle α and α' of the moire fringe.

In the experiment, an expanded monochromatic plane wave produced by a He-Ne laser (λ = 632.8 nm) and a collimator illuminates the first grating g1, test surface M, and second grating g2 successively. We used two Ronchi-type gratings, the pitches of which are p = 0.22 mm. The optical path length along the optical axis from g1 through M to g2 is the first Talbot image length (2p2/λ = 153 mm). The crossed angle θ between the two gratings is set by 1°. We performed an experiment by changing the tilt of the plane mirror surface of 35- × 45-mm size. The moire fringe pat­terns were observed on a screen attached to the grating g2. Figure 3 shows an example of the moire fringe patterns in the experiments. Figure 3(a) shows the fringes with inclination angle α = 12° for the test surface at arbitrary tilt, and Fig.

Therefore, the tilt angle δ from the optical axis of the test surface is represented by

Hence, if the test surface is slightly tilted further by a small amount ∆δ, the pitch p" of the Talbot image g"1 on the plane x axis is given by

Fig. 3. Relationship between the tilt angle δ of the test surface and the inclination angle α of the moire fringes.

Fig. 4. Moire fringe patterns observed before (a) and after (b) the Fig. 2. Moire fringes generated by two gratings of unequal period. variation of the tilt angle: (a) α = 12°; (b) α' = 25°.

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3(b) shows the one with α' = 25° observed after the slight tilt variation ∆δ of the test surface. From Eq. (5), the small variation of the tilt angle of the test surface is given by ∆δ = 1.2°. In this case the shift amount of the inclination angle of moire fringes was ∆α = α' - α = 13°, which is ~10 times as much as the small variation ∆δ. Figure 4 shows the theoreti­cal curves giving the relation between the inclination angle α of the moire fringe and the tilt angle δ of the test surface. The appropriate region for measuring variation of tilt angles of the test surface can be found by these curves. For exam­ple, for a crossed angle θ = 1°, when the tilt angle of the test surface is 40°, this system is most sensitive to the tilt angle.

To summarize: It has been shown that Talbot interferom-etry can be applied for measuring the small variation of the tilt angle of the test surface. This method has the following merits:

(1) The optical system for the measurement is simple having a light source and two gratings. A precise optical arrangement is not necessary.

(2) As the second grating is placed in the Talbot image plane of the first grating, we can obtain good contrast of the moire fringe patterns.

(3) It is especially effective for measuring a large surface having a local tilt variation.

This method enables a real time and automatic measure­ments of small tilt angle variation by digital Talbot interfer-ometry8 based on computer processing.

References 1. A. W. Lohmann and D. E. Silva, "An Interferometry Based on the

Talbot Effect," Opt. Commun. 2, 413 (1971). 2. S. Yokozeki and T. Suzuki, "Shearing Interferometer Using the

Grating as the Beam Splitter," Appl. Opt. 10, 1575 (1971). 3. Y. Nakano, "A Deflection Mapping of Phase Objects Using Dou­

ble Gratings," Kogaku 11, 286 (1982) (in Japanese). 4. Y. Nakano and K. Murata, "Measurements of Phase Objects

Using the Talbot Effect and Moire Techniques," Appl. Opt. 23, 2296 (1984).

5. Y. Nakano and K. Murata, "Measurement of the Focal Length of a Lens Using a Moire Technique," in Conference Digest, Thir­teenth Congress of the International Commission for Optics (Publisher, Location, 1984), p. 308.

6. Y. Nakano and K. Murata, "Talbot Interferometry for Measuring the Focal Length of a Lens," Appl. Opt. 24, 3162 (1985).

7. D. W. Swift, "A Simple Moire Fringe Technique for Magnifica­tion Checking," J. Phys. E 7, 164 (1974).

8. M. Takeda and S. Kobayashi, "Lateral Aberration Measure­ments with a Digital Talbot Interferometer," Appl. Opt. 23, 1760 (1984).

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