Abstract:
A measuring method and apparatus for measuring the shape of an optical surface using Fizeau interference is disclosed. The states of three surfaces are determined by measuring Fizeau fringes between all possible pairs of said surfaces. In at least one of these measurements means for reversing the coordinate axis in the beam is provided.

Description:
This application is a continuation of application Ser. No. 327,095 filed Mar. 22, 1989, now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a Fizeau interference measuring method and an apparatus therefor, used for measuring a surface shape or the like and, more particularly, a Fizeau interference measuring method and an apparatus therefor, wherein a spherical surface is measured by utilizing interference to test sphericity or the like of an optical spherical surface. In particular, the present invention relates to a Fizeau interference measuring method and an apparatus therefor, wherein a system error of the measuring apparatus can be eliminated so that a surface can be two-dimensionally measured with high precision. 
     2. Related Background Art 
     Strong demand has arisen for high-precision optical elements such as a high-resolution lens for printing semiconductor ICs and an apparatus capable of measuring an optical surface at, e.g., 1/10 or less of a wavelength. Various interferometers employing the principle of interference are commercially available as apparatuses for measuring optical surfaces. In particular, various interferometers capable of performing quantitative measurements with high precision are developed and available along with widespread use of lasers having good coherency and developments of electronics. 
     A difference between two surface shapes is read in each measurement by a conventional interferometer. For this reason, when one of the surfaces is an ideal surface, no problems are presented. Otherwise, several measurements under different conditions must be performed, and measurement results must be analyzed to obtain individual surface shape errors. 
     For example, a Fizeau interferometer is available as an interferometer for measuring a shape error of a spherical surface. 
     FIG. 1A is a schematic diagram showing an optical arrangement of a main part of a Fizeau interferometer. In the Fizeau interferometer shown in FIG. 1, an end surface 43 of a focusing lens 42 serves as a concave spherical surface concentric with a focal point of a laser beam converted into a spherical shape by the lens 42. The end surface 43 therefore serves as a reference spherical surface. 
     A laser beam which passes through a beam splitter 41 from the left side and incident in the form of a plane wave is converted by the focusing lens 4 into a beam in the form of a spherical wave which is focused to one point. Part of the beam having the spherical wave is reflected by the reference spherical surface 43 formed at the end position of the focusing lens 42 and is converted again into a beam of a plane wave by the focusing lens 42. The beam of the plane wave is reflected by the beam splitter 41, and the reflected beam is incident on an observation optical system 45. 
     The remaining laser beam which is kept converted in the form of the spherical wave and focused to one point after passing through the focusing lens 42 is reflected by a spherical surface 44 (a concave surface to be measured in FIG. 1) having the center of radius near the focal point. This reflected beam of the spherical wave is incident on the observation optical system 45 along the same optical path as the beam of the plane wave, thereby forming a two-dimensional interference fringe pattern on a detector surface 46. 
     Measurement values based on the interference fringe pattern include a shape error from an ideal spherical surface as the reference spherical surface 43 and a shape error from an ideal spherical surface of the spherical surface 44. It is impossible to separate these errors by one measurement. For example, as shown in FIG. 1B, the spherical surface 44 is vertically reversed, and the same measurement as described above is performed. In addition, as shown in FIG. 1C, the positions of these two spherical surfaces are then changed and the same measurement as described above is performed. The three measurement results are analyzed to separate and extract the shape error of the spherical surface 44. 
     FIGS. 2A and 2B are views showing a conventional method of plotting coordinates of the measuring light wavefront, the reference spherical surface, and the spherical surface to be measured, all of which are two-dimensionally plotted. 
     Referring to FIGS. 2A and 2B, the coordinates of the measuring light wavefront and the reference spherical surface are represented by positions of the reference spherical surface 43 when viewed from the spherical surface 44 to be measured (FIG. 1), as shown in FIG. 2A. However, the coordinates of the spherical surface 44 to be measured are plotted such that the vertical direction of the spherical surface is set to be a specific direction and this spherical surface is viewed from the front, as shown in FIG. 2A. Therefore, when the vertical direction of the spherical surface is reversed upside down, the resultant coordinates are given, as shown in FIG. 2B. 
     According to the conventional method, the first two of the three measurements are performed under the condition given in FIG. 1B. More specifically, the first measurement is performed such that the vertical direction of the spherical surface 44 is reversed upside down, as shown in FIG. 2B, and the second measurement is performed such that the vertical direction is reversed again to restore the normal state, as shown in FIG. 2A. Measurement values of optical path differences in these two measurements are given as 
     
         w.sub.1 (x,y)=w.sub.R (x,y)+w.sub.S (-x,y)                 (a) 
    
     
         w.sub.2 (x,y)=w.sub.R (x,y)+w.sub.S (x,-y)                 (b) 
    
     where x and y are incident positions of object optical paths on the spherical surface 44. The right-hand term w R  (x,y) in each of equations (a) and (b) represents a contribution amount of the corresponding optical path difference due to the shape error of the reference spherical surface 43, and the right-hand term w S  (-x,y) or w S  (x,-y) represents a contribution amount corresponding optical path difference due to the shape error of the spherical surface 44 itself. In the measurements by the Fizeau interferometer under the conditions shown in FIGS. 1A and 1B, even if a laser beam of a spherical wave incident on the reference spherical surface 43 slightly includes aberration by an optical system for receiving this laser beam, the aberration can be canceled since it acts on both the wavefront reflected by the reference spherical surface 43 and the wavefront reflected by the spherical surface 44 with equal magnitudes. Therefore, the influence of aberration is negligible, which has been theoretically proved. This is why the right-hand sides of equations (a) and (b) consist of only shape errors of the reference spherical surface 43 and the spherical surface 44. This is a great advantage in the Fizeau interferometer as compared with other interferometers such as a Twyman interferometer. It is impossible to completely extract the contribution amount caused by the shape error of the spherical surface 44 by only equations (a) and (b). Therefore, the additional measurement is performed while the spherical surface 44 is located at a focal point of the measuring light beam. 
     In this case, the interference fringe pattern is not associated with the shape of the spherical surface 44 but only with the shape of the reference spherical surface 43. If aberration is slightly included in the laser beam of the spherical wave incident on the reference spherical surface 43, symmetrical components cancel each other during interference. However, asymmetrical components are doubled to adversely affect the interference fringe pattern. This can be empirically confirmed and can be theoretically explained. If the measurement value in this case is defined as w 3  (x,y), the following equation can be established: 
     
         w.sub.3 (x,y)=w.sub.0 (x,y)+W.sub.R (x,y)                  (c) 
    
     The first term w 0  (x,y) of the right-hand side represents a contribution amount by the asymmetrical components of aberration. 
     Since the term w 0  (x,y) represents the asymmetrical aberrational component, w 0  (x,y) and w 0  (-x,-y) which is obtained by rotating w 0  (x,y) through 180° about the origin satisfy the following relation: 
     
         w.sub.0 (-x,-y)=-w.sub.0 (x,y)                             (d) 
    
     In order to obtain the contribution amount w S  (-x,y) caused by the shape error of the spherical surface 44 on the basis of the three measurement values represented by equations (a), (b), and (c), w 2  (-x,-y) and w 3  (-x,-y) obtained by rotating the coordinates of the measurement values w 2  (x,y) and w 3  (x,y) shown in equations (b) and (c) through 180°, respectively, must be obtained. As a result, the contribution amount w S  (-x,y) of the shape error of the spherical surface 44 is obtained by using the measurement values w 1  (x,y) and w 3  (x,y) shown in equations (a) and (c) and the obtained w 2  (-x,-y) and w 3  (-x,-y) as follows: 
     
         w.sub.S (-x,y)=1/2{w.sub.1 (x,y)+w.sub.2 (-x,-y)}-1/2{w.sub.3 (x,y)+w.sub.3 (-x,-y)}                                                  (e) 
    
     In general, in order to measure sphericity of a spherical surface with high precision of 1/10 or less of the wavelength by an interferometer, three interference measurements in different conditions must be inevitably performed by any interferometer to extract the shape error of the object spherical surface since an ideal spherical surface serving as a comparison reference does not exist. In this case, when aberration of the optical system constituting the interferometer is mixed in the interference fringe pattern, it adversely affects the interference fringe pattern equally or greater than the shape error of the object spherical surface. It is, therefore, apparent that reliability of the measurement result is greatly degraded. 
     Of the three measurements performed by the conventional Fizeau interferometer, the first two measurements are performed under the condition shown in FIG. 1B. This condition does not cause aberration of the optical system to adversely affect the interference fringe pattern. However, since the condition in FIG. 1C is employed in the third measurement, the asymmetrical aberrational components of the optical system are doubled and mixed in the interference fringe pattern. As a result, the measurement reliability is degraded as a whole. 
     In order to enhance the advantage of the measurement by the Fizeau interferometer, it is preferable to perform all three measurements under the condition shown in FIG. 1B to separate and extract the shape error of the object spherical surface. In order to achieve this, a conventional three-surface alignment method (e.g., D. Malacara, Optical Shop Testing (John Wiley &amp; Sons, 1978, PP. 41-42) known as a method of testing optical planes with high precision can be applied to measurements of a spherical surface. 
     The three-surface alignment method is practiced as follows. Three optical surfaces A, B, and C are combined into pairs of surfaces A and B, A and C, and B and C. Each pair of surfaces are located to oppose each other to cause optical interference, and shape errors of these surfaces A, B, and C are separated and extracted by three measurements. If coordinates of the three optical surfaces are plotted, as shown in FIG. 2A, measurement values w 1  (x,y), w 2  (x,y), and w 3  (x,y) of the optical path differences obtained by the three measurements are given as follows: 
     
         w.sub.1 (x,y)=w.sub.A (x,y)+w.sub.B (-x,y)                 (1) 
    
     
         w.sub.2 (x,y)=w.sub.A (x,y)+w.sub.C (-x,y)                 (2) 
    
     
         w.sub.3 (x,y)=w.sub.B (x,y)+w.sub.C (-x,y)                 (3) 
    
     The sign of the x-coordinate of each second term is opposite to that of each first term in the right-hand side because the y-axis is reversed so that the surface represented by the second term opposes that represented by the first term. The term w B  (-x,y) of the right-hand side in equation (1) is not generally equal to the term w B  (x,y) of the right-hand side. In order to separate and extract the shape errors of the individual surfaces by using the above linear equations with three unknowns, x=0 must be established, i.e., only y-coordinates on the y-axis are considered since the sign of the x-coordinate does not then pose any problem. 
     In order to obtain shape errors of all the surfaces, the surfaces are slightly rotated, and similar three measurements must be repeated. This repetition is the problem of the three-surface alignment method. 
     In the measurement of sphericity by the Fizeau interferometer, the same measurements as in the three-surface alignment method of the optical surfaces can be performed for the three spherical surfaces A, B, and C. In this case, the spherical surfaces A and B are concave reference spherical surfaces integral with the focusing lens, and the spherical surface C is any spherical surface. These spherical surfaces A to C are arranged, as shown in FIGS. 1A and 1B. In the three measurements, the reference surface 43 and the surface 44 to be measured are A and B, A and C, and B and C, respectively. The coordinates of the reference surface 43 when viewed from the front are defined as in FIG. 2A, and the coordinates of the surface 44 are defined as in FIG. 2B. The optical path differences obtained by the three measurements have the same relationships with those defined by equations (1), (2), and (3). 
     Therefore, the resultant shape errors are one-dimensional, i.e., along the y-axis of each surface in the three measurements. Therefore, the above attempt is not free from the drawback of the three-surface alignment method. 
     SUMMARY OF THE INVENTION 
     The present invention has been made in consideration of the conventional problems described above, and has as its first object to provide a Fizeau interference measuring method and an apparatus therefor, wherein two-dimensional shape errors can be easily tested with high precision. 
     In order to achieve the above object of the present invention, when a Fizeau interferometer and the principle of three-surface alignment method are utilized to test a spherical surface condition such as sphericity of an optical surface, a plane mirror is arranged in part of the interferometer to reverse coordinates within a light wavefront, and at least one of the three measurements is performed through the plane mirror, thereby measuring the two-dimensional surface state of the entire spherical surface with high precision. 
     The above and other objects, features, and advantages of the present invention will be apparent from the detailed description of preferred embodiments in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1A is a schematic view showing an arrangement of a conventional Fizeau interferometer; 
     FIGS. 1B and 1C are views showing optical arrangements of the Fizeau interferometer shown in FIG. 1; 
     FIGS. 2A and 2B are views showing coordinate systems; 
     FIG. 3 is a view for explaining a conventional measurement for flatness of a surface; 
     FIG. 4 is a schematic view showing a Fizeau interference measuring apparatus according to a first embodiment of the present invention; 
     FIG. 5 is a schematic view showing a main part of the apparatus shown in FIG. 4; and 
     FIG. 6 is a schematic view showing a Fizeau interference measuring apparatus according to a second embodiment of the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the following preferred embodiments, as previously described, a movable plane mirror is located near a focal plane of the light beam output from an interfering means, any two of the three object spherical surfaces are selected and arranged to oppose each other so as to obtain a common center of curvature. The light beams from the two spherical surfaces are superposed to interfere with each other. The optical path length measurements for different combinations of two surfaces are repeated three times. In this case, at least one of the three measurements is performed by using a beam deflected laterally by the plane mirror located near the center of curvature. Therefore, the two-dimensional surface condition of the entire spherical surface as an object can be measured. 
     FIG. 4 is a schematic view showing the main part of a Fizeau interferometer according to an embodiment of the present invention. The interferometer includes a beam splitter 1 and a focusing lens 2. A laser beam from the beam splitter 1 is focused at a point P. A reference spherical surface 3 corresponds to the end lens of the focusing lens 2 and has a concave surface concentric with the point P. A spherical surface 4 to be measured is located such that its center of curvature almost coincides with the point P, i.e., the spherical surfaces 3 and 4 have a common center of curvature. The reference spherical surface 3 and the spherical surface 4 to be measured are held by holders 3a and 4b, respectively. The interferometer also includes an observation optical system 5, a detector surface 6 for detecting an interference fringe pattern, and a plane mirror 7. The plane mirror 7 is supported to be pivotal about a point 7a and is pivoted to a position indicated by a dotted line, i.e., the position of the point P by a driving means (not shown). An arithmetic unit 8 calculates surface precision by using an output signal from the detector surface 6. 
     A laser beam of a plane wave incident through the beam splitter 1 from the left is converted by the focusing lens 2 into a laser beam of a spherical wave focused to one point. Part of the beam of the spherical wave is reflected by the reference spherical surface 3 located at the end of the focusing lens 2 and is converted again into a beam of a plane wave by the focusing lens 2. The beam of the plane wave is reflected by the beam splitter 1, and the reflected beam is incident on the observation optical system 5. The remaining laser beam of the spherical wave focused to one point through the focusing lens 2 is reflected by the spherical surface 4 (concave surface in FIG. 4) which is held by the holder 4a and has the center of curvature near the focal point. The reflected beam is incident on the observation optical system 5 along the same path as the beam of the plane wave, thereby forming a two-dimensional interference fringe pattern on the detector surface 6. 
     In this embodiment, the beam splitter 1 and the focusing lens 2 constitute part of the interfering means. 
     In this embodiment, three interference measurements are performed in combinations of spherical surfaces A and B, A and C, and B and C by the Fizeau interferometer, and sphericity of each spherical surface is obtained on the basis of the measurement values. In this case, each of the spherical surfaces A and B included in the above three spherical surfaces is the concave reference surface 3 integral with the focusing lens 2, as shown in FIG. 4. The plane mirror 7 shown in FIG. 4 is pivoted from the position indicated by the solid line to the position indicated by the dotted line. As shown in FIG. 5, the plane mirror 7 is located at or near the position P where the measuring beam is focused at this position by the focusing lens 2 having the reference spherical surface 3. The measuring beam is laterally deflected by the plane mirror 7, and the concave spherical surface 4 is held by the holder 4b so that its center of curvature almost coincides with the focal point of the measuring beam. Under this condition, at least one of the three measurements is performed. In the measurement under the condition shown in FIG. 5, the coordinates within the light wavefront are reversed by the plane mirror from the state shown in FIG. 2A. 
     In this case, the deflection angle by the plane mirror 7 can be arbitrarily set within the range in which the measuring beam after reflection does not interfere with the measuring beam before reflection. The spherical surface 4 is arranged upside down with respect to the reference spherical surface 3. 
     Under these conditions, assume that a measurement value obtained by setting the spherical surface A as the spherical surface 3 and the spherical surface B as the spherical surface 4 is defined as w 1  (x,y), that a measurement value obtained by setting the spherical surface A as the spherical surface 3 and the spherical surface C as the spherical surface 4 is defined as w 2  (x,y), that a measurement value obtained by setting the spherical surface B as the spherical surface 3 and the spherical surface C as the spherical surface 4 is defined as w 3  (x,y). Under these assumptions, the following equations are established: 
     
         w.sub.1 (x,y)=w.sub.A (x,y)+w.sub.B (x,y)                  (4) 
    
     
         w.sub.2 (x,y)=w.sub.A (x,y)+w.sub.C (x,y)                  (5) 
    
     
         w.sub.3 (x,y)=w.sub.B (x,y)+w.sub.C (x,y)                  (6) 
    
     If all the three measurements are performed in the optical arrangement in FIG. 5, amounts w A  (x,y), w B  (x,y), and w C  (x,y) representing the shape errors of the respective spherical surfaces are easily calculated using the measurement values w 1  (x,y), w 2  (x,y), and w 3  (x,y) as follows: 
     
         w.sub.A (x,y)=1/2{w.sub.1 (w,y)+w.sub.2 (x,y)-w.sub.3 (x,y)}(7) 
    
     
         w.sub.B (x,y)=1/2{w.sub.1 (x,y)-w.sub.2 (x,y)+w.sub.3 (x,y)}(8) 
    
     
         w.sub.C (x,y)=1/2{-w.sub.1 (x,y)+w.sub.2 (x,y)+w.sub.3 (x,y)}(9) 
    
     In the above embodiments, any two of the three spherical surfaces A, B, and C are selected, and the three measurements are performed in different combinations of the spherical surfaces. However, at least one measurement must be performed in the state of FIG. 5, while two other measurements may be performed by the optical arrangement not using the plane mirror shown in FIG. 4. 
     If the measurement values in the second and third measurements are defined as w 2  &#39;(w,y) and w 3  &#39;(x,y), the following equations are established in place of equations (5) and (6): 
     
         w.sub.2 &#39;(x,y)=w.sub.A (x,y)+w.sub.C (-x,y)                (5)&#39; 
    
     
         w.sub.3 &#39;(x,y)=w.sub.B (x,y)+w.sub.C (-x,y)                (6)&#39; 
    
     In this case, amounts representing the spherical errors of the respective spherical surfaces are obtained in place of equations (7), (8), and (9) as follows: 
     
         w.sub.A (x,y)=1/2{w.sub.1 (x,y)+w.sub.2 &#39;(x,y)-w.sub.3 &#39;(x,y)}(7)&#39; 
    
     
         w.sub.B (x,y)=1/2{w.sub.1 (x,y)-w.sub.2 &#39;(x,y)+w.sub.3 &#39;(x,y)}(8)&#39; 
    
     
         w.sub.C (-x,y)=1/2{-w.sub.1 (x,y)+w.sub.2 &#39;(x,y)+w.sub.3 &#39;(x,y)}(9)&#39; 
    
     FIG. 6 is a schematic view showing the main part according to another embodiment of the present invention. This embodiment exemplifies a case wherein all three measurements are performed through a plane mirror 7 located near the center of curvature of the object spherical surface. The same reference numerals as in FIG. 4 denote the same parts in FIG. 6. 
     Referring to FIG. 6, the plane mirror 7 is movable in a direction indicated by, e.g., an arrow 7b to cope with changes in focal points of beams output from a reference spherical surface 3 in accordance with changes in radii of curvature of the object spherical surfaces. 
     The principle of measurement and the measuring method of this embodiment are basically the same as those of the embodiment shown in FIG. 4. The shape errors of the respective spherical surfaces are obtained by equations (7), (8), and (9). 
     According to the above embodiments of the present invention, at least one measurement is performed through the plane mirror in the Fizeau interferometer to simultaneously obtain the two-dimensional shape errors of the all spherical surfaces. When an object to be measured is a spherical surface, there is a position where the measuring beam is focused at one point. This point is utilized to laterally reflect the measuring beam by the plane mirror 7 located at the focal point. The beams from the two spherical surfaces are caused to interfere with each other through the plane mirror, and the sign of the x-coordinate can be changed to be positive without mixing unnecessary aberrational components. Therefore, the principle of three-surface alignment method can be applied to the entire spherical surface. 
     According to each embodiment of the present invention, aberrational components of the optical system and other unnecessary components are not included in the measurement values w 1  (x,y), w 2  (x,y), and w 3  (x,y) or the measurement values w 1  (x,y), w 2  &#39;(x,y), and w 3  &#39;(x,y). Therefore, highly reliable measurements of sphericity can be performed with high precision. 
     In each embodiment described above, if two spherical surfaces which serve as the reference surfaces are concave surface, the remaining surface may be a convex surface. In this case, the arrangement shown in FIG. 5 is utilized only when the two concave surfaces oppose each other. Other two measurements are performed in the arrangement of FIG. 4 such that the radius of curvature of the concave surface is set to be equal to that of the convex surface. 
     According to the present invention, there is provided an interference measuring apparatus wherein when a Fizeau interferometer and the principle of three-surface alignment are used to measure a surface state such as sphericity of an optical surface, at least one of the three measurements is performed by using a movable plane mirror located in part of the interferometer, thereby eliminating system errors and measuring the two-dimensional surface state of the entire spherical surface with high precision.