Design of null lenses for testing of elliptical surfaces

Transcription

1 Design of null lenses for testing of elliptical surfaces Yeon Soo Kim, Byoung Yoon Kim, and Yun Woo Lee Null lenses are designed for testing the oblate elliptical surface that is the third mirror of the off-axis three-mirror anastigmatic camera used for remote sensing. Modifying the conventional autostigmatic and autocollimation types of null lenses yields a mixed-type design that has a small annular flat mirror and high sensitivity. Detailed analyses of the sensitivity of the mixed-type null lens system with changes in each surface parameter are described Optical Society of America OCIS codes: , , , Introduction As aspheric surfaces are adapted for use in highresolution space telescopes, the need for aspheric mirrors with high surface accuracy is increasing. However, it is not easy to make an accurate aspheric surface because it is difficult to test such a surface. Many test techniques, such as surface profilometry and interferometric null tests, have been proposed. Because profiler measurement accuracy is limited by many factors, interferometric testing methods 1,2 are generally preferred. For a conic surface, the z of revolution is given by ch 2 z 1 1 c 2 1 k h 2, 1 2 (1) where h is the distance from a point on the conic to the optical axis and c is a paraxial curvature. When conic constant k is 1 k 0, the surface is an ellipsoid rotated about its major axis. We can test the surface by using the stigmatic points, which are ideal spots with respect to a pair of specific object and image points. This method is called a stigmatic null test. If k 0, the surface is an oblate ellipsoid rotated about its minor axis. Its stigmatic points are not lined up on the optical axis. In this case, the Y. S. Kim ykim@sunam.kreonet.re.kr is with the Agency for Defence Development, Taejon , South Korea. B. Y. Kim is with the Department of Physics, Korea Institute of Science and Technology, Taejon South Korea. Y. W. Lee is with the Korea Research Institute of Standards and Science, Taejon , South Korea. Received 31 May 2000; revised manuscript received 6 March $ Optical Society of America interferometric methods that use null lenses are more often used than the stigmatic null test. In this paper we present a new type of null lens system for testing an oblate ellipsoid. It is composed of a small annular flat mirror and a biconcave lens. Detailed analyses of the system s wave-front error with respect to the change in the surface parameter of optical elements are described. 2. Design Issues There are two types of null lens, the autostigmatic and the autocollimation types shown in Figs. 1 and 2, respectively. 3 5 The former null lens produces a reference aspheric wave front, which is compared interferometrically with the aspheric surface under test. The latter null lens, however, makes a collimated wave front in combination with the aspheric surface under test; it is twice as sensitive as the former lens because the test beam is reflected twice on the test surface by a reference flat. However, the size of the reference flat should be larger than that of the aspheric surface under test. To eliminate the need for a large reference flat, we propose creating a new null lens system by modifying the autostigmatic and autocollimation types with a small reference flat, as shown in Fig. 3. The reflected wave front from the test surface is collimated after it passes inversely through a null lens. In this case, the inverse ray travels along a different path from that of the incident ray in passing the null lens to arrive at the test surface. After it is reflected from the reference flat, the beam goes backward and retraces the path through which it has just passed. As a result, the beam is reflected twice on the test surface, so it has four times the configuration error as the test surface. Therefore the new null lens has the same sensitivity as the autocollimated type, even if it has a small flat mirror. 1 July 2001 Vol. 40, No. 19 APPLIED OPTICS 3215

2 Fig. 1. Typical autostigmatic-type null lens. Fig. 2. Typical autocollimation-type null lens. Fig. 4. Test mirror configuration and stigmatic points. 3. Aspheric Surface under Test The off-axis three-mirror anastigmatic camera system under development is composed of three mirrors: concave hyperbolic, convex spherical, and concave elliptic mirrors. The field of view of this optical system 6 is 0.1 about the x axis and 3 about the y axis. It has a common optical axis with the three mirrors. The line of sight is 4.5 below the common optical axis, and the aperture stop is located on the secondary mirror. The vertices of both the primary and the tertiary mirrors are coincident, so the distance from the secondary mirror to both the primary and the tertiary mirrors is the same. The tertiary mirror as shown in Fig. 4 has an elliptic surface, a radius of curvature of mm, a conic constant of , and a diameter of 468 mm. The left-hand side of Fig. 4 represents the tertiary mirror configuration to be tested and the size of the parent mirror. 7 The distance D 1 between two stigmatic foci and the distance D 2 between a vertex of the elliptic surface and the plane of foci are obtained from the following equations 1 : D 1 r 2 k, (2) k 1 D 2 r k 1, (3) where r is the radius of curvature and k is the conic constant. We obtain D 1 of mm and D 2 of mm by using the numerical values in Eqs. 2 and 3 for r and k. Stigmatic foci are separated by mm on the major axis, which is perpendicular to the optical axis. 4. Design and Analysis The conventional autostigmatic- and new-type null lenses are designed by use of an optical design tool, Sigma The null lenses are designed to test the surface under test on axis. Both null lenses can measure as much as a semiaperture of 250 mm on an elliptic surface and have a maximum optical path difference OPD of less than 50 at m. Figure 5 shows the designed autostigmatic-type null lens, which is similar to the zoom null lens system of Shafer. 4 It consists of two negative lenses, and there is a separation of 12 mm between lenses for easy alignment. Data on the designed lens are listed in Table 1. The data start from the axial point Fig. 3. New type of null lens. Fig. 5. Designed autostigmatic-type null lens APPLIED OPTICS Vol. 40, No July 2001

3 Table 1. Design Data of the Autostigmatic-Type Null Lens Surface Radius mm Thickness mm Material Aspheric a Marginal Ray Height mm l 1 Air Schott BAF50 glass Air Schott BAF50 glass l 2 Mirror cc Air Schott BAF50 glass Air Schott BAF50 glass Air a cc is conic constant. Fig. 6. a OPD and b contour map of the autostigmatic-type null lens. source, and the corresponding ray paths are shown in Fig. 5. Figures 6 a and 6 b show the OPD curve and the contour plot, respectively. The residual wave-front error is less than 100, peak to valley pv. Table 2 shows the OPD error for the displacement of surface parameters such as radius of curvature and lens thickness. The values are obtained when each surface parameter is 0.01 mm off its optimum designed value. All OPD errors have a linear relation to the changes in each surface parameter. Therefore the wave-front errors that are due to fabrication errors in the lens parameters can be nulled by adjustment of the relative positions of lenses and test mirror only if the fabrication errors can be accurately measured. The designed autostigmatic null lens is highly sensitive to its radius of curvature, R1. The fabrication error 0.01 mm of R1 or the measurement error of 0.009% of R1 causes an OPD error of Therefore the performance of this null lens system depends mainly on R1. Figure 7 shows the new null lens setup, which consists of a lens and an annular flat mirror whose diameter is the same as or larger than that of the lens. The flat mirror is located at the position of the axial point source to minimize the portion of the lens that cannot be measured because of the size of the hole in the mirror. Table 3 lists data for the new null lens. Figures 8 a and 8 b show the OPD and the contour plot, respectively. The wave front-error is 100 pv. Table 4 lists the OPD errors of the null lens for surface parameters as for the autostigmatic null lens. The new type of lens shows maximum sensitivity for l 2, which is the distance between the lens and the test mirror. The position or measurement error 0.01 mm of l 2 causes the maximum OPD error 1. Therefore the performance of this null lens system depends mainly on distance l 2. If the surface error of the test mirror is 10, a wave-front distortion of 5 is induced in the autostigmatic null test system because the wave-front error is twice the surface error. But, in the new type, only a wave-front distortion of 2.5 is induced because the wave-front error is four times the surface error. Therefore, in the setup for a null test system Table 2. OPD Error of the Designed Autostigmatic-Type Null Lens a Lens 1 Lens 2 l 1 R1 t 1 R2 d R3 t 2 R4 l 2 Sensitivity OPD a l 1, l 2, d, distances; R1 R4, radii of curvature; t 1, t 2, lens thicknesses. 1 July 2001 Vol. 40, No. 19 APPLIED OPTICS 3217

4 Fig. 7. Designed new-type null lens. that can measure the values of the test mirror with an accuracy of 10 pv, the autostigmatic type of lens has the limitation that its radius of curvature R1 must be fabricated with an accuracy of better than % and l 2 must be determined with an accuracy of 4 m or better. But, in the new type of lens, only l 2 must be determined with an accuracy of 4 m or better. From these results, it can be deduced that the new type is better than the autostigmatic type of system in terms of ease of fabrication and testing. In the new-type null system, a lens that is off center by 0.01 mm will cause a wave-front error of 0.17 pv, and a lens tilt of 1 mrad will cause an error of 5 pv. The Zernike polynomial coefficients of coma induced by the off-center lens and the tilt are 0.03 and 0.9, respectively. The reference mirror tilt of 1 mrad causes a wave-front distortion of 2 pv, and the Zernike polynomial coefficient of coma is 0.3. So careful alignment is needed for testing an oblate ellipsoid with the new type of null lens system. Fig. 8. a OPD and b contour map of the new-type null lens. 5. Conclusions A new null lens system for testing oblate ellipsoids has been proposed. With a lens and a small annular flat mirror, gives a residual wave-front error of less than 50 pv. An investigation of relative sensitivity to changes in lens surface parameters gives the Table 3. Design Data of a New Null Lens Surface Radius mm Thickness Marginal Ray Height mm Material Aspheric a mm l 1 Air Hoya F2 glass l 2 Mirror cc Air Hoya F2 glass Mirror Air Hoya F2 glass Mirror cc Air Hoya F2 glass Air a cc is conic constant. Table 4. OPD Error of the New-Type Null Lens l 1 Lens 1 a R1 t R2 Lens 2 Sensitivity OPD a l 1, distance; R1, R2, radii of curvature; t, lens thickness APPLIED OPTICS Vol. 40, No July 2001

5 result that the mixed type of system is better than the conventional autostigmatic type in terms of fabrication and measurement setup. This new type of system can also be applied to the design of a null lens to measure other aspheric surfaces. References 1. J. C. Wyant, Interferometric testing of aspheric surfaces, in Selected Papers on Optical Shop Metrology, D. Malacara, ed., Vol. MS18 of SPIE Milestone Series Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990, pp J. D. Briers, Interferometric testing of optical systems and components: a review, Opt. Laser Technol. 4, A. Offner, A null corrector for paraboloidal mirrors, Appl. Opt. 2, D. R. Shafer, Zoom null lens, Appl. Opt. 18, D. T. Puryayev, Concept for testing two-mirror optical telescope, Opt. Laser Technol. 28, R. Geyl, Design and fabrication of a three mirror flat field anastigmat for high resolution Earth observation, in Space Optics 1994: Space Instrumentation and Spacecraft Optics, T. M. Dewandre, J. J. Schulte-in-den-Baeumen, and E. Stein, eds., Proc. SPIE 2210, R. Kingslake, Lens Design Fundamentals Academic, London, 1978, p E. Everhart, Null test for Wright telescope mirrors, Appl. Opt. 5, July 2001 Vol. 40, No. 19 APPLIED OPTICS 3219