Development of shape measuring system using a line sensor in a lateral shearing interferometer

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1 Development of shape measuring system using a line sensor in a lateral shearing interferometer Takashi NOMURA*a, Kazuhide KAMIYA*a, Akiko NAGATA*a, Hatsuzo TASHIRO **b, Seiichi OKUDA ***c a Toyama Prefectural University, b Toyama University, c Computer Engineering &Consulting LTD., * Kurokawa, Kosugi-machi, Toyama , Japan, ** Gofuku, Toyama, Toyama , Japan, *** Shibuya, Tokyo , Japan 1. Introduction Mirrors with spherical or aspherical shape are manufactured with ultraprecision lathes. In order to increase the accuracy of the mirrors and the efficiency of the manufacturing process, it is desired that the shapes of the mirrors be measured on the lathes themselves either after or during processing. In a noncommon-path interferometer such as the Fizeau interferometer, the fringes obtained are affected by mechanical vibrations and air turbulence, because the wavefront under test and the reference wavefront traverse in different spaces. Therefore, it is difficult to obtain interference fringes by the noncommon-path interferometer under such measurement conditions.[1] On the other hand, in a common-path interferometer, the fringes obtained by the interferometer are slightly affected by air turbulence, because the wavefront under test and the reference wavefront travel in the same space. A lateral shearing interferometer can be constructed as a common-path interferometer. Therefore, the shapes of the mirror can be measured by an interferometer constructed as a common-path one mounted on the lathe.[2] When the mirror under processing is measured by the interferometer, a cutting tool and a tool post obstruct a part of the measurement area of the surface under test. It is difficult to analyze the entire area of the surface due to defective fringes. To solve the problem, we propose a shape measuring system using a line sensor in a lateral shearing interferometer. 2. Lateral Shearing Interferometer Figure 1 shows a lateral shearing interferometer. A laser beam is collimated by lenses L1 and L2. The collimated beam is diffracted by a zone plate, and the diffracted beam progresses perpendicularly onto the mirror surface under test. The beam reflects off the surface and is nominally collimated by the zone plate. When the surface of the mirror has errors, the wavefront of the reflected beam is deformed. The wavefront is incident upon the optical parallel glass plate and reflects off the front and back surfaces of that plate. The two reflected wavefronts are laterally sheared with respect to each other. Interference fringes are obtained from the area where the two wavefronts in the lateral shearing interferometer overlap, as shown in Fig. 2. When the mirror under processing is measured by the

2 interferometer, a cutting tool and a tool post obstruct a part of the measurement area of the surface under test. 3. Principle A sensor is used to capture the interference fringes in every direction of shear. Because the surface under test rotates, some interference fringes are captured on the line. The line includes the central axis of the rotation. Shape error of the entire surface is obtained by analyzing some interference fringes. The wavefront under test is expressed as W θ ( x, y), where ( x, y) are coordinates of a measurement point and θ is the angle of rotation. When this wavefront is sheared in the x-direction by an amount s, the sheared wavefront at the same points is W θ ( x-s, y). The optical path difference W θ ( x, y) between the original wavefront and the sheared one is W θ ( x-s, y)-w θ ( x, y). When s is small, the optical path difference WG, W ( x, y) W ( xs, y) W ( x, y) ( s n x G G G, (1) where λ is the wavelength of the light source, and n is the order of the interference fringe. The interference fringes obtained by the lateral shearing interferometer represent the derivative of the shape error in the sheared direction of the wavefront under test. By integrating the phase distribution, the shape error of the mirror under test in the sheared direction is obtained. When the fringes on the x axis are captured by the line sensor, the wavefront is 1 x WG, x W, x( x, y) dx C s, G G. (2) x0 To obtain three-dimensional shape error, the constant C θ should be determined. Because the measurement values at the central axis of rotation are the same at all rotation angles, C θ is defined as zero. As a consequence, the shape error of the mirror under test is reconstructed. 4. Simulation We performed a simulation to evaluate the validity of the principle. A shape under test which was constructed by Zernike coefficients of the 4th item was used in the simulation as shown in Fig. 3. In the proposed method, some interference fringes obtained by the line sensor are used. Some fringes are analyzed in the radial direction, and these analyzed data are composed. The sampling number of interference fringes obtained by the line sensor is required to be more than twice the number of undulations in the circumferential direction at least. In this simulation, in order to reconstruct the wavefront correctly, a sampling angle of 1 degrees is used. The number of pixels of the line sensor is 512. The wavelength of the laser is 633 nm, and the shear ratio is 0.2 in the simulation. The result of the simulation is shown in Fig. 4. The result agreed well with the original shape under test. The difference between the original shape and reconstructed one was 1 %.

3 5. Shape measuring system A shape measuring system using a line sensor in a lateral shearing interferometer was developed as shown in Fig. 5. The optical arrangement is shown in Fig. 1. The mirror under test is attached to the spindle with an encoder. The angle is detected by the encoder. A personal computer counts the signals from the encoder, and sends a trigger signal to a CCD camera and a video grabber board. Interference fringes are obtained by the CCD camera with an aperture stop and are analyzed by the computer. In the developed measuring system, the angular simulation of the encoder is 2π/5000 rad, and the number of pixels of the line sensor is 512. The wavelength of the laser is 633 nm, and the shear ratio is 0.2 in the simulation. 6. Conclusion We proposed a shape measuring system using a line sensor in a lateral shearing interferometer. A shape under test constructed by Zernike coefficients of the 4th item was used in the simulation. A sampling angle of 1 degree was used in the simulation. The result obtained by the proposed method agreed well with the original shape under test. The difference between the original shape and reconstructed one was 1 %. A shape measuring system using a line sensor in a lateral shearing interferometer was developed. [1] T. Nomura, K. Yoshikawa, H. Tashiro, K. Takeuchi, N. Ozawa, Y. Okazaki, M. Suzuki, F. Kobayashi and M. Usuki, On-machine shape measurement of workpiece surface with Fizeau interferometer, Precision Engineering, 14, pp , [2] T. Nomura, K Kamiya, H. Miyashiro, S. Okuda, H. Tashiro and K. Yoshikawa, Shape measurements of mirror surfaces with a lateral shearing interferometer during machine running, Precision Engineering, 22, pp , Keywords: on-machine shape measurement, common path interferometer, lateral shearing interferometer, shape measurements of a spherical surface

4 Mirror under test M1 L1 He-Ne laser BS ZP Spindle CCD camera M2 L2 Detector Encoder AS L3 OP Trigger Pulse Computer L : Lens BS : Beam splitter M : Mirror AS : Aperture stop ZP : Zone plate OP : Optical parallel glass plate Fig. 1 Optical arrangement of a shape measuring system using a line sensor in a lateral shearing interferometer ΔW θ(x,y) Wθ (x-s,y) S y Wθ (x,y) x Line sensor Shadow of tool post Interference fringes Fig. 2 Fringes obtained by the lateral shearing interferometer on the lathe

5 2.0π (rad ) 1.28π (rad ) Fig. 3 Shape under test used in the simulation Fig. 4 Result of the simulation Fig. 5 Photograph of the developed interferometer