Sensing a wave front by use of a diffraction grating

Transcription

1 Sensing a wave front by use of a diffraction grating Nazario Bautista-Elivar, Carlos Ignacio Robledo-Sánchez, Alberto Cordero-Dávila, and Alejandro Cornejo-Rodríguez A method is presented to sense the wave front at the exit of an optical surface. This method uses a set of diffracted rays generated when a He Ne laser impinges on a rectangular diffraction grating. The grating was placed near the curvature center of the surface to be tested. After they are reflected in the test surface, the diffracted rays have the information of the slopes of the wave front, like in the Hartmann test. The Hartmann pattern was registered near the curvature center and captured with a CCD camera. The slopes for each ray are measured from the experimental pattern, and they are compared with the ideal ones simulated in a computer. The evaluation was carried out by use of Seidel polynomials to obtain the information of the aberrations of a mirror 53 cm in diameter Optical Society of America OCIS codes: , Introduction The basic idea for the evaluation of an optical surface by use of the Hartmann test 1 is the measurement of the slopes of the wave front in two orthogonal directions. This sampling is carried out by placement of a screen with holes, known as the Hartmann screen, in front of the surface under test. The evaluation of the wave front in this test is carried out by measurement of the transverse ray aberrations from the socalled Hartmanngram. In the arrangement of the Hartmann test, the screen has the same dimensions as the surface under test. Over the years, different arrays of holes have been used in the design of the screen. The experimental Hartmanngram is observed and registered when the surface is illuminated with a point light source, which is located near the curvature center of the optical surface. The observed Hartmanngram is located near the curvature center but outside the caustic. Because we use a screen with two-dimensional distributed holes, the N. Bautista-Elivar nelivar@fismat1.fcfm.buap.mx, C. I. Robledo-Sánchez crobledo@fcfm.buap.mx and A. Cordero-Dávila are with the Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Apartado Postal 1152, Puebla, Puebla, México. A. Cornejo-Rodríguez is with the Instituto Nacional de Astrofisica, Óptica y Electrónica, C.P , Puebla, México. Received 1 October 2002; revised manuscript received 19 February $ Optical Society of America information obtained in the Hartmanngram is also two dimensional. In contrast, in the traditional Ronchi test, 2 for example, the grating is one dimensional and has a relatively small size. The Ronchi ruling is placed next to the light source, and both are located near the curvature center. The grating normally used has a low frequency, approximately some hundreds of lines per inch. From the Ronchigram we obtain information about the transverse aberration of each ray. Whereas a real image is obtained in a Hartmanngram, the Ronchigram is a virtual image. To obtain bidimensional information in the Ronchi test, it is necessary to use a grating with perpendicular lines. Another method used to sense the wave front coming from a surface under test is the use of the Hartmann Shack sensor, 3 which uses a fixed array of small circular holes containing microlenses and is placed at the exit of the optical system to sample the wave front. A method that uses the trace of a laser ray 4 can also be used. This last method consists of a scanner with a plane mirror added to follow the progressive direction of a laser ray and a second mirror used to hit the tested surface; in this way, spots formed in the image plane are recorded, and the position of each centroid is calculated. The method proposed in this paper is a modified Hartmann test in which a set of rays diffracted by a rectangular diffraction grating are projected on the surface under test to create the Hartmann screen. In this array the rays diffracted by the grating, which is localized at the curvature center and is illuminated 1 July 2003 Vol. 42, No. 19 APPLIED OPTICS 3737

2 Fig. 2. Liquid-crystal display from a slide projector used as a diffraction grating. Fig. 1. Experimental setup for sensing the wave front by use of a set of diffracted rays. ST is the surface under test; DG is the diffraction grating plane, illuminated by the He Ne laser. At the plane, containing point P p, the complete observed Hartmanngram is registered. by the He Ne laser, hit on certain points of the tested surface, as if they were rays selected by the holes of a Hartmann screen. Hence in this scheme the screen does not exist over the surface tested see Fig. 2. To compare this method with the traditional Hartmann test, we evaluated a hyperbolic mirror with a conic constant of , a diameter of 53 cm, and an F number of 1.4. We discuss the advantages and problems of this method and also reach a resolution. 2. Ray-Tracing Method In the Hartmann test the evaluation of an optical surface requires the measurements of the difference between the experimental transverse ray aberrations and their ideal positions in the Hartmanngram observation plane. The experimental arrangement of Fig. 1 shows the parameters necessary to carry out the ray-tracing algorithm to obtain the ideal positions for each ray; it is also shows the diffraction grating DG placed at the point P,,, which is illuminated directly by a He Ne laser. A diffracted ray hits the surface test ST at a point P 0 X 0, Y 0, Z 0 and is reflected to the point P p X p, Y p, Z p. This last point is registered with a CCD camera. The observed pattern with the CCD camera becomes a Hartmanngram; the difference in this case is that no screen over the surface is used, as is required in the traditional Hartmann test. To obtain the ideal Hartmanngram for points P p j, we considered a ray that originated at point P, hits the surface at point P 0, and is reflected to point P p. The coordinates X p, Y p of a ray are obtained with the following equations: X p X 0 z Z p S 2x S 2z, (1) Y p Y 0 z Z p S 2y, (2) S 2z where S 2x, S 2y, and S 2z are the components of the reflected vector S 2 from the tested surface. S 2 is related to S 1 and the normal unit vector N by the reflection law 5 S 2 S 1 2 S 1 N N, (3) where S 1 and S 2 are the incident and reflected rays, respectively. We calculated the components of S 1 with X 0, Y 0, and Z 0. N is the normal unit vector in point P 0 of the conical surface. This normal is obtained from the following expression: z x, z y, 1 N z x 2 z y 2 1 2, (4) 1 where the derivatives of the sagitta z are obtained from the equation 6 z 1 k 1 R R2 K 1 X 2 0 Y (5) In Eq. 5, R is the paraxial radius of a conical surface, and K is its conic constant. For the usual flat Hartmann screen, a correction for the coordinates of point P 0 must be applied. 7 In addition, the defects of the image pattern in the observation plane will diminish if the screen is close to the surface under test and increase in the contrary case. However, for the present method, as is shown in the arrangement in Figs. 1 and 2, such coordinate correction is not necessary because the spots are on the surface, and because the screen does not exist the rays are not obstructed. Morales and Malacara 8 have described a method to calculate the optimal diameter of the holes, taking into account the diffraction and geometric theories. For the method shown in Fig. 1, the diffracted spots on the surface have a Gaussian transverse profile, and therefore they have a smaller size than those obtained when the rays are obstructed with the Hart APPLIED OPTICS Vol. 42, No July 2003

3 Table 1. Functions U j and Their Derivatives Table 2. Coordinates of Points P and P p j U x 0, y 0 U x 0 U y 0 Aberrations 0 x Tilt x 1 y Tilt y 2 x y 0 2x 0 2y Defocus 3 x y 0 2x 0 6y Astigmatism 4 y x 2 0 y 2 0 2xy x y 0 Coma 5 y 2 0 x x 0 x 2 0 y 2 0 4y 0 x 2 0 y 2 0 Spherical mann screen with circular holes and close to the optical surface. For a Gaussian profile, the spot size on the Hartmanngram is obtained by means of the Fraunhofer diffraction theory 9 and is equal to R, where R is the radius of curvature of the surface, is the transverse diameter of the beam, and is the wavelength of the He Ne laser used. For a circular aperture, the same theory gives a transverse size of 2.44 R. Therefore, with the Gaussian profile of the spots projected on the surface, spots produced at the Hartmanngram were 2.44 times smaller than those produced by a circular aperture in the classical Hartmann screen. This means that the spots with a Gaussian profile allowed us to reach a better resolution. 3. Evaluation of the Aberration Function To obtain the aberration function W x 0, y 0, we follow the method reported in Ref. 9. To do this, we expand the aberration function into a linear combination of K polynomials U j X 0, Y 0 : k W X 0, Y 0 w j U j X 0, Y 0, (6) j 1 where U j X 0, Y 0 are the Seidel polynomials and w j are the coefficients associated with each one of these polynomials. To apply the method, it is necessary to build the functions F j in the following way: F j U j, U j X 0 Y 0. (7) Point X 0 cm Y 0 cm Z 0 cm P P p Experimental Setup and Results The scheme for the experimental setup is shown in Fig. 1. We used a liquid-crystal display coming from a slide projector, Epson, Model ELP-3500, as a diffraction grating, located at point P. The experimental Hartmanngram is acquired by a CCD camera, with a size of m pixel, Hitachi, Model KP- 160 at point P p. Because it is important to know the coordinates of points P and P p, with respect to the system centered on the surface under test, in order to know the coordinates of points P and P 0, with respect to the reference system, it is necessary to know the location of the optic axis Z. To do this last step, we had to locate the position of point O. InO we placed a point light source, and the image produced by the mirror of this source O was placed on the same point. The coordinates of points, P and P p, from Fig. 1, are shown in Table 2, and the parameters of the surface are as follows: radius of curvature, 153 cm; conic constant, ; and diameter, 53 cm. AHe Ne laser beam with 20 mw and nm impinges on the diffraction grating. The diffraction grating produced 170 diffracted rays that impinge over the tested mirror, located at Z cm and shown in Fig. 3. In this figure we can observe the diffracted rays impinging on the mirror. The reflected spots from the mirror are shown in the Hartmanngram of Fig. 4; this Hartmanngram was specifically registered at a distance Z cm, before the focus and outside the area of the caustic. The next step of the process is the reading of each one of the spots coordinates in the Hartmanngram. To read their coordinates, we made a program that saves in a computer file the data of the coordinates for The functions F j are calculated by U j X 0 and U j Y 0 and with the N data of the surface under test X 0, Y 0. The functions U j X 0, Y 0 are shown in Table 1. With the functions F j we can construct the next linear expansion of the transverse aberrations T: k T j 1 w j F j. (8) T contains N data written as X P X exp, Y P Y exp j, where X P and Y P are the coordinates obtained from the theoretical calculations. Equations 1 and 2 and X exp, Y exp are the coordinates obtained from the experimental Hartmanngram. The coefficients of aberration w j were obtained by means of a meansquare fitting. 10 Fig. 3. Set of rays that impinge the surface under test. 1 July 2003 Vol. 42, No. 19 APPLIED OPTICS 3739

4 Fig. 4. Set of diffracted rays that produce the Hartmanngram. Fig. 5. Diagram that shows the angular resolution in two tests T 1 and T 2, are tangent to TS. each spot. The evaluation was done by use of 170 beams with a transverse diameter of 2 mm. For comparison, we evaluated the mirror with the traditional Hartmann test. In this test we used a 86-hole Hartmann screen, with a diameter of 7 mm. The aberration coefficients obtained from both tests are shown in Table 3. As can be observed, the coefficient of spherical aberration is bigger; this aberration is typical of aspheric surfaces. The rms value obtained indicates that the difference between the ideal mirror and the real data is acceptable. For this mirror we obtained an error of 1.4. Although the rms is close for the two methods, the different results explain which method gives the best information. An important parameter in this type of test is the angular resolution. This parameter was measured considering the Rayleigh criterion, 8 which tells us how two spots can be resolved. It says that the resolution is determined when a second spot overlaps half of the diameter, at most, of the first one see Fig. 5. This means that the size of the minimum spot that is possible to measure will determine the minimum local slope of the wave front. Considering the relationships W x, y x T R, x x, and W x W x, T R, which is the angle subtended by the spot, we find that W x, where r R. The angle is determined by the radius r of the spot according to the Rayleigh approach and the distance R. x corresponds to the Table 3. Aberrations in Seidel Polynomial Coefficients of the Aberrations in Seidel Polynomials Coefficients for the Classical Hartmann Test units of Coefficients for the Hartmanngram Test with Diffraction Grating units of Tilt y Tilt x Defocus Astigmatism Coma Spherical rms transverse diameter of the ray that impinges on the optical surface. In the traditional test, x corresponds to the diameter of a hole of the Hartmann screen. The calculations done are applied by use of a traditional screen with a hole diameter of 7 mm, compared with the transverse diameter of 2 mm of a diffracted ray with the grating. The results of the resolution for each test are the following: For the classical Hartmann test spot diameter of 7 mm, the minimum resolution is 0.8 m; for the modified Hartmann test with the diffraction grating spot diameter of2mm, the minimum resolution is 0.2 m. We observe from these data that a better resolution is reached with the use of the diffraction grating. 5. Conclusion We presented a method for sensing the wave front from an optical surface, following the methodology of the traditional Hartmann test. The innovation consists of replacing the Hartmann screen by the spots generated by a diffraction grating, located close to the center of curvature of the surface. With the introduction of the diffraction grating, the experimental arrangement is more simple than the traditional Hartmann test. As each diffracted spot that senses the surface test has a Gaussian profile, with this type of spot we obtain a smaller spot size in the observation plane, and we can reach a resolution better than the one reached with the traditional Hartmann screen. With this method we can have a bigger dynamic range, making it useful for the measurement of extremely deformed surfaces. For this reason, the method can even be implemented during the initial polishing process. We say that the experimental setup used in this study of putting the diffraction ruling at the curvature center of the surface is as valid as the well-known Ronchi test and the real pattern obtained through the array used in this paper is as valid as the Hartmann test. For comparison, we evaluated a hyperbolic mirror with a diameter of 53 cm and a radius of curvature of 150 cm. The rms obtained with this method was 4% smaller than the value obtained with the traditional Hartmann test 3740 APPLIED OPTICS Vol. 42, No July 2003

5 that uses a screen, thereby ensuring that this method is a reliable test. In this paper we used as diffraction grating, a liquid-crystal display from a projector, Epson, Model ELP-3500 ; with this element, 170 diffracted rays were produced inside the diameter of the mirror. This method has the disadvantage of requiring the design of an appropriate diffraction grating in agreement with the characteristics of the surface under test. References 1. I. Ghozeil, Hartmann and other screen tests, in Optical Shop Testing, D. Malacara, ed. Wiley, New York, 1992, pp A. Cornejo-Rodríguez, Ronchi test, in Optical Shop Testing, D. Malacara, ed. Wiley, New York, 1992, pp R. G. Lane and M. Tallon, Wave-front reconstruction using a Shack Hartmann sensor, Appl. Opt. 31, R. Navarro and E. Moreno-Barriuso, Laser ray-tracing method for optical testing, Opt. Lett. 24, E. Hecht, Optics Addison-Wesley Iberoamericana, Madrid, 2000, Chap. 4, pp D. Malacara, Appendix 1. An optical surface and its characteristics, in Optical Shop Testing Wiley, New York, 1992, pp E. Luna-Aguilar, A. Cornejo-Rodríguez, and A. Cordero Dávila, Prueba nula de Ronchi Hartmann, Rev. Mex. Fís. 38, A. Morales and D. Malacara, Geometrical parameters in the Hartmann test of aspherical mirrors, Appl. Opt. 22, J. W. Goodmann, Introduction to Fourier Optics McGraw-Hill, New York, 1996, Chap. 2, pp C. Robledo-Sánchez, G. Camacho-Basilio, A. Jaramillo-Nuñez, and David Gale, Aberration extraction in the Hartmann test by use of spatial filters, Appl. Opt. 38, July 2003 Vol. 42, No. 19 APPLIED OPTICS 3741