Abstract:
A method of fabricating a catadioptric lens system, the method involving: fabricating a single catadioptric lens element having a bottom surface and an upper surface, the upper surface having a convex portion and a concave portion, both the convex and concave portions sharing a common axis of symmetry; cutting apart the catadioptric lens element to form 2n pie-shaped segments, wherein n is an integer; and reassembling the 2n pie-shaped segments to form the catadioptric lens system with n of the 2n pie-shaped segments being located above a common plane and the rest of the 2n pie-shaped elements being below the common plane.

Description:
This application also claims the benefit of U.S. Provisional Application No. 60/459,493, filed Apr. 1, 2003. 

   TECHNICAL FIELD 
   This invention relates to a method for making catadioptric lens systems for such applications as interferometric confocal microscopy. 
   BACKGROUND OF THE INVENTION 
   A number of different applications of catadioptric imaging systems for far-field and near-field interferometric confocal microscopy have been described such as in U.S. patent applications Ser. No. 10/028,508, filed Dec. 20, 2001 [ZI-38], and No. 10/366,651, filed Feb. 3, 2003 [ZI-43] entitled “Catoptric And Catadioptric Imaging Systems;” U.S. Provisional Patent Application No. 60/447,254, filed Feb. 13, 2003 and U.S. patent application Ser. No. 10/778,371, filed Feb. 13, 2004 [ZI-40] both entitled “Transverse Differential Interferometric Confocal Microscopy,” U.S. Provisional Patent Application No. 60/448,360, filed Feb. 19, 2003 and U.S. patent application Ser. No. 10/782,057, filed Feb. 19, 2004 [ZI-41] both entitled “Longitudinal Differential Interferometric Confocal Microscopy for Surface Profiling;” U.S. Provisional Patent Application No. 60/448,250 and U.S. patent application Ser. No. 10/782,058, filed Feb. 19, 2004 [ZI-42] both entitled “Method and Apparatus for Dark Field Interferometric Confocal Microscopy;” U.S. Provisional Patent Application No. 60/442,982, filed Jan. 28, 2003 and U.S. patent application Ser. No. 10/765,229, filed Jan. 27, 2004 [ZI-45] both entitled “Interferometric Confocal Microscopy Incorporating Pinhole Array Beam-Splitter;” and U.S. Provisional Application No. 60/459,425, filed Apr. 1, 2003 and U.S. patent application Ser. No. 10/816,180, filed Apr. 1, 2004 [ZI-50] both entitled “Joint Measurement Of Fields Of Orthogonally Polarized Beams Scattered/Reflected By An Object In Interferometry.” The above-mentioned patent applications and provisional patent applications are all by Henry A. Hill and the contents are incorporated herein by reference in their entirety. 
   In each of the applications of catadioptric imaging systems for each of the cited U.S. patent applications and U.S. Patent Provisional Patent Applications, tight tolerances are placed on the manufacture of optical elements. In addition to the tolerances normally encountered in designing a diffraction limited imaging system, there are tolerances imposed by the interferometric confocal microscopy applications. The additional tolerances are for example on radii of curvature of certain lens elements with respect to radii of curvature of certain other lens elements and on relative locations of centers of curvature of lens elements. 
   The additional tolerances lead to improved performance of a catadioptric imaging system, e.g., with respect to increasing the average intensity of desired images by a factor of approximately 2 or more and reduced intensity of spurious beams by one or more orders of magnitude, and in addition make it possible to realize interferometric reduction of background fields. The interferometric reduction of background fields leads to a reduction of statistical errors. The increase in intensity of desired images and the reduction of statistical errors lead to an increase in signal-to-noise ratios and to a concomitant increase in through put of a metrology tool using the catadioptric imaging system. The interferometric reduction of background fields further leads to a reduction systematic errors. A consequence of the reduction of systematic errors is a reduction of the computational task required to invert arrays of interference signal values to a multi-dimensional image of an object. 
   SUMMARY OF THE INVENTION 
   In general, in one aspect the invention features a method of fabricating a catadioptric lens system. The method involves: fabricating a single catadioptric lens element having a bottom surface and an upper surface, the upper surface having a convex portion and a concave portion, both the convex and concave portions sharing a common axis of symmetry; cutting apart the catadioptric lens element to form 2n pie-shaped segments, wherein n is an integer; and reassembling the 2n pie-shaped segments to form the catadioptric lens system with n of the 2n pie-shaped segments being located above a common plane and the rest of the 2n pie-shaped elements being below the common plane. 
   Other embodiments include one or more of the following features. Cutting the catadioptric lens element to form the 2n pie-shaped segments is accomplished by cutting along a set of planes each of which contains the common axis. The 2n pie-shaped segments are identically shaped. The parameter n=1 or 2. Each of the four pie-shaped segments is a 90° segment of the single catadioptric lens element. Reassembling involves arranging each of the n pie-shaped segments that are above the common plane to be opposite to and aligned with a corresponding different one of the n pie-shaped segments that are below the common plane. The convex portion is a reflective portion of the catadioptric lens element and the concave portion is a refractive portion of the catadioptric lens element. Reassembling the four pie shaped segments relative to a common plane involves placing two of the four segments are above the plane with their bottom surfaces being substantially parallel to and facing the common plane and placing the other two of the four segments are below the common plane with their bottom surfaces substantially parallel to and facing the common plane. Reassembling also involves orienting the four segments so that each one of the two segments above the common plane are aligned with and adjacent to a corresponding one of the two segments that are below the common plane. Reassembling further involves orienting the two segments that are above the common plane so that they share an axis of symmetry and are radially opposite each other relative to that shared axis of symmetry. 
   In general, in another aspect, the invention features another method of fabricating a catadioptric lens system. The method involves: fabricating a single catadioptric lens element having a bottom surface and an upper surface, the upper surface having a convex portion and a concave portion, both the convex and concave portions sharing a common axis of symmetry; cutting apart the catadioptric lens element to form two identically pie-shaped segments; and reassembling the two pie-shaped segments to form at least part of the catadioptric lens system with one of the two pie-shaped segments being located above a common plane and the other of the two pie-shaped elements being below the common plane, wherein the bottom surfaces of the two pie-shaped elements are facing each other and substantially parallel to the common plane, and wherein the two pie-shaped segments are aligned with each other. 
   In general, in still another aspect, the invention features another method of fabricating a catadioptric lens system. The method involves: fabricating a single catadioptric lens element having a bottom surface and an upper surface, the upper surface having a convex portion and a concave portion, both the convex and concave portions sharing a common axis of rotational symmetry; cutting apart the catadioptric lens element to form four substantially identical segments, wherein cutting involves cutting the catadioptric element along at least one plane that contains the common axis; and reassembling the four segments to form the catadioptric lens system with two of the four segments being located above a common plane and the other two of the four elements being below the common plane, wherein the reassembled four segments have their bottom surfaces substantially parallel to the common plane, and wherein each of the two segments that is above the plane is aligned with and adjacent to a corresponding different one of the two segments that are below the common plane. 
   An advantage of one or more embodiments is a reduction of cost in the manufacture of lens elements for a catadioptric imaging system in interferometric confocal microscopy. 
   Another advantage of one or more embodiments is the improvement of performance of a catadioptric imaging system in interferometric confocal microscopy. 
   Another advantage of one or more embodiments is the increase of the average intensity of desired images by a factor of approximately 2 or more. 
   Another advantage of one or more embodiments is a reduction of intensity of spurious beams by one or more order of magnitudes, 
   Another advantage of one or more embodiments is that it makes it possible to realize interferometric reduction of background fields. 
   Another advantage of one or more embodiments is an increase in signal-to-noise ratios and to a concomitant increase in through put of a metrology tool using a catadioptric imaging system. 
   Another advantage of one or more embodiments is a reduction systematic errors as a consequence of the interferometric reduction of background fields. 
   Another advantage of one or more embodiments is the reduction of the computational task required to invert arrays of interference signal values to a multi-dimensional image of an object wherein the arrays of interference signal values are obtained with an interferometric confocal microscopy system that uses a catadioptric imaging system. 
   The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a schematic drawing of a catoptric imaging system including a reflective surface and a beam splitter. 
       FIG. 2  is a schematic drawing of another catoptric imaging system including a reflective surface and a beam splitter. 
       FIG. 3  is a schematic drawing of a catadioptric imaging system including a reflective surface, a beam splitter, and two refractive surfaces. 
       FIG. 4  is a schematic drawing of a catoptric imaging system including two reflecting surfaces constructed and positioned such that interferometric effects lead to increased light intensity at the image point. 
       FIG. 5  is a schematic drawing of a catadioptric imaging system similar to the imaging system in  FIG. 4  including refractive surfaces that reduce optical aberrations. 
       FIG. 6  is a schematic drawing of another catadioptric imaging system similar to the imaging system in  FIG. 5  that generates two image points that are spatially separated in the transverse direction to the optical axis. 
       FIG. 7  is a schematic drawing of another catadioptric imaging system similar to the imaging system in  FIG. 5  that generates two image points that are spatially separated in the longitudinal direction relative to the optical axis. 
       FIG. 8  is a schematic drawing of a catadioptric imaging system similar to the imaging system in  FIG. 4  but including refractive surfaces that are Fresnel mirrors. 
       FIG. 9  is a perspective drawing of a catadioptric imaging system. 
       FIG. 10  is a perspective drawing of a catadioptric imaging system with the elements separated for purposes of illustration. 
   

   DETAILED DESCRIPTION 
   Referring to  FIG. 1 , a catoptric imaging system  100  includes an object point  160 , an image point  162 , a beam splitter  150 , a curved reflective surface  132 , and light transmitting elements  130  and  140 . Light emanating from the object point  160  passes through the light transmitting element  130  and is incident on the beam splitter  150 . The beam splitter  150  reflects and transmits portions of the incident light beams. In the presently described embodiment, the portion of light that is initially transmitted is ignored and it is omitted from  FIG. 1 . The reflected portion is shown in  FIG. 1  and is incident onto the reflective surface  132 . The surface  132  is constructed such that each light ray emanating from the object point  160  that is reflected from the beam splitter  150  and incident onto the surface  132  is reflected to the image point  162  after being transmitted by the beam splitter  150 . In other words, light emanating from the object point  160  is focused onto the image point  162  by the following path: i) light is emanated from the object point  160 ; ii) reflected by beam splitter  150 ; iii) reflected by reflective surface  132 ; iv) transmitted by the beam splitter  150 ; and v) converges onto the image point  162 . 
   Because reflecting surface  132  causes the focusing of the rays to the image point, and not refraction by media  130  and  140 , the image plane is independent of the spectral region used in image formation (provided that media  130  and  140  do not substantially differ in index). In other words, there is no longitudinal chromatic aberration. Accordingly, a large spectral range can be used for image formation. 
   The index of refraction of medium  130  impacts the numerical aperture of the system. In particular, the numerical aperture of system  100  scales linearly with the index of refraction of the medium  130 . Although by no means limiting, the rest of this discussion assumes that the indices of refraction for elements  130  and  140  (and their analogs in other embodiments) are substantially the same. 
   In one embodiment, the features of system  100  are achieved with the following design. Given the object point  160  and the image point  162 , beam splitter  150  is positioned to lie in the plane defined by points that are equidistant from the object and image points. Furthermore, reflective surface  132  is designed to be concentric with the image point  162 . As a result of this construction, a light ray emanating from the object point at an angle φ is incident on the beam splitter at some point P with an angle of incidence of φ. By design light is incident onto surface  132  at a normal angle of incidence and therefore such light rays are reflected through 180 degrees. Furthermore, after reflection from surface  132 , the light is incident on the beam splitter at the same point P with angle of incidence of φ and after transmission by the beam splitter  150  the light ray is incident on the image point with angle of incidence of φ. 
   As described above, the light incident on the image point is both reflected and transmitted by the beam splitter surface. Therefore, the light reaching image point  162  is proportional to R(φ)T(φ), where R and T are the reflection and transmission coefficients of beam splitter  150 , respectively. Both of these coefficients are typically dependent on the angle of incidence. Using techniques known in the art, beam splitter  150  is designed such that for some angle φ′ beam splitter  150  is ideal. That is, for some angle φ′, R(φ′)≅T(φ′)≅0.5. As the angle of incidence differs from φ′, the coefficients will often demonstrate non-ideal beam splitter behavior. Specifically, the behavior deviates from the ideal by some δ(φ), and R(φ)=0.5+δ(φ−φ′) and T(φ)=1−R(φ)=0.5−δ(φ−φ′) where δ(0)=0. Because the light rays incident on image point  162  as shown in  FIG. 1  are both reflected and transmitted, then T(φ)R(φ)=0.25−δ(φ−φ′) 2 . Thus even though the beam splitter may deviate from an ideal beam splitter with some deviation δ(φ), the non-ideal behavior will only impact the light intensity to second order in δ(φ). 
   Furthermore, this embodiment has an object point image that is diffraction limited. Although other points in the object plane may not be diffraction limited, there does exist a planar disc centered on the object point and parallel with the beam splitter  150  whose image is also a flat disc of the same radius. In other words, the image plane is flat and the magnification is 1. 
   Element  130  and surface  132  may be made in a number of ways. Transmitting element  130  and the reflecting surface  132  may be made from a solid light-transmitting medium (e.g. fused silica). In this case, the solid light-transmitting medium can be shaped to have one side that is to match the shape of the beam splitter  150  and another side whose shape matches the desired shape for reflecting surface  132 . By suitably depositing a reflecting film onto the curved surface, the reflecting surface  132  is formed. This could be accomplished using any of the well-known techniques in the art for forming reflecting films. The reflecting film is not applied within some neighborhood of the object point  160  (not shown). Instead the surface near the object point would be constructed to allow light rays to enter into the imaging system. For example, an antireflection coating may be applied to surface  132  in the vicinity of object point  160 . Such an aperture allows light rays from the object point to enter into the imaging system. 
   In another embodiment, light-transmitting element  130  may be a hollow region of vacuum or filled with a light transmitting gas or fluid. In such embodiments, the reflective surface  132  may be formed onto some mechanically supporting substrate (not shown) and its external surface is either intrinsically reflective (e.g. a polished metal surface) or is made reflective by application of a reflective film. Furthermore, an aperture is formed in the vicinity of the object point  160  such that light can enter the imaging system (not shown). 
   In other embodiments, the reflecting surface  132  may be a non-smooth and/or discontinuous surface. For example, the reflecting surface may be formed by an array of flat reflecting surfaces positioned to be substantially concentric with the image point  162  so as to provide the same optical function as the surface  132  in FIG.  1 . Furthermore reflecting surface  132  may have deviations from a concentric shape (e.g. elliptical or parabolic). Such deviations may be useful in correcting for higher order aberrations. 
   In some embodiments of system  100 , element  130  is a high-index material and element  130  and beam splitter  150  are positioned such that element  130  contacts object point  160  to thereby maximize the numerical aperture of the imaging system. This is a non-limiting case, however, and in other embodiments the object point need not contact element  130 . Similarly, element  140  need not contact image point  160 . Moreover, in subsequently described embodiments, the object point and/or the image point need not contact an element of the imaging system, although, depending on the embodiment, this may be preferable to maximize numerical aperture. 
   Although not intended to be limiting in any way, as a theoretical curiosity it is noteworthy to point out that imaging system  100  functions equivalently to a pair of planar elements each having opposite indices of refraction (i.e., one element having a positive index +n, and the other element having a negative index—−n). In particular, refraction at the interface between two such elements causes light rays emitted from the object point to bend and focus to the image point. This can be seen from a trivial application of Snell&#39;s law of refraction. Such bending and focusing is effectively achieved in system  100  by the initial reflection from beam splitter  150  and the subsequent reflection by reflecting surface  132 . A similar effect is also present in the subsequently described embodiments. 
   From the design of imaging system  100 , it is clear that light that initially is transmitted by the beam splitter is ignored and only the reflected component is used. Other imaging systems can be designed such that the initially transmitted component is utilized and the reflected component is discarded. Referring to  FIG. 2 , a catoptric imaging system  200  includes an object point  260 , an image point  262 , a beam splitter  250 , a curved reflective surface  242 , and light transmitting media  230  and  240 . The embodiment of  FIG. 2  is similar to that of  FIG. 1  except that in the embodiment of  FIG. 2 , reflecting surface  242  is positioned to receive light transmitted by the beam splitter surface, whereas the reflecting surface in  FIG. 1  is positioned to receive light reflected by the beam splitter surface. In an embodiment of system  200 , the reflecting surface  242  is concentric with object point  160 . As is the case with the embodiment in  FIG. 1 , the intensity of incident light imaged to image point  262  is proportional to T(φ)R(φ)=0.25−δ(φ−φ′) 2 . Thus the image point light intensity has no first order deviations due to non-ideal beam splitter behavior. Furthermore, as described with reference to  FIG. 1  a transparent window or an apertures in surface  242  allows access to the image point  262  for light emanating from object point  132 . 
   In the embodiments of  FIGS. 1 and 2 , although the object point is diffraction limited, the points in the vicinity of the object point may not be. Such points may suffer from certain optical aberrations. Such aberrations may be corrected for a large part of the object plane by introducing refractive surfaces. 
   Referring to  FIG. 3 , a catadioptric imaging system  300  includes an object point  360 , an image point  362 , a beam splitter  350 , a curved reflective surface  332 , a plano-concave-convex element  330 , a plano-concave element  340 , and plano-convex elements  320  and  380 . The common center of curvature for surface  322  of element  320  is the object point  360 . The common center of curvature for surface  344 , surface  332  of element  330 , and surface  382  of element  380  is image point  362 . Element  320  and element  330  are formed such that the radius of curvature of surface  322  of element  320  is substantially the same as the radius of curvature of surface  334  of element  330 . Element  340  and element  380  are formed such that the radius of curvature of surface  344  of element  340  is substantially the same as the radius of curvature of surface  382  of element  380 . Surfaces  322  and  344  are preferably coated with an antireflection coating. 
   The refracting surfaces in system  300  provide additional degrees of freedom that can be used to reduce optical aberrations in the image field. In particular, any of the index of refraction of elements  320 ,  380 ,  340  and the radius of curvature of surface elements  334 ,  344 ,  332  may be varied to reduce such aberrations. For example, optical ray tracing methods may be used to calculate the amplitude of the various aberrations as functions of such variables and in this way particular values of the parameters can be found that minimize the aberrations. Such optimizations may also take into account other design criteria such as magnification, planarity of the image field, numerical aperture, optical absorption and other material limitations. Notably, for example, the numerical aperture of system  300  scales with the index of refraction of the element  320 . Thus, by use of a high index material, the numerical aperture can be improved. Moreover, an optimization may fix the indices of refraction for elements  320 ,  330 ,  340 , and  380  simply because specific materials are to be used for these elements. 
   In some embodiments, element  380  or element  320  may be excluded. Elements  380  or  320  may be replaced by a void to be filled with a gas, liquid or vacuum. In some embodiments only one refractive surface may be used. In such cases, the index of refraction of element  380  or  320  matches the index of elements  330  and  340  such that interface  322 / 334  or  344 / 382  is no longer a refractive surface. Use of a void provides access to the image point or object point. Such access may be useful, for example, to position a detector near the image point. 
   As described above, the light intensity at the image point for imaging system  100 ,  200 , and  300  are proportional to T(φ)R(φ)=0.25−δ 2 . Even in the ideal case, where δ=0, only 25% of the available light reaches the image point. 
   Referring to  FIG. 4 , a catoptric imaging system  400  includes an object point  460 , an image point  462 , a beam splitter  450 , a curved reflective surface  432 , a curved reflective surface  442  and plano-convex elements  430  and  440 . The reflective surface  442  is constructed such that light rays emanating from the object point  460  are focused to the image point  462  by following the path: i) the light emanates from the object point; ii) is transmitted by the beam splitter  450 ; iii) is reflected by surface  432 ; iv) is reflected by the beam splitter  450 ; v) is incident onto the image point  462 . This can be accomplished by designing curved surface  442  to be concentric with the object point  460 . Similarly the reflective surface  432  is constructed such that light rays emanating from the object point are focused to image point  462  by following the path: i) the light emanates from the object point; ii) is reflected by the beam splitter  450 ; iii) is reflected by surface  432 ; iv) is transmitted by beam splitter  450 ; and v) is incident onto the image point  462 . This can be accomplished by designing curved surface  432  to be concentric with the image point  462 . 
   In the embodiment described for  FIG. 4 , both the initially reflected and initially transmitted beams from the beam splitter are used. A beam is split by beam splitter  450  into two portions that are then reflected by surfaces  432  and  442 , respectively, back to the same point on the beam splitter. Generally, the two portions recombine interferometrically to produce two new beams. One beam is directed to the image point  462  and the other is directed to the object point  460 . The intensities of the respective beams depend on the difference in optical path length for the beam portions reflected from surfaces  432  and  442 .  FIG. 4  labels the two optical paths for the portions as OPL 1  and OPL 2 . The optical path lengths for the portions corresponding to each ray are matched such that the two beams interfere constructively to direct all of the optical energy to the image point. Thus, the concentric curved surfaces  442  and  432  are positioned and shaped to agree to within a small fraction of a wavelength. Nonetheless, even where the optical path lengths are not exactly matched for all rays, the transmission to the image point can be enhanced relative to the earlier embodiments where transmission is limited to 25%. 
   The matched concentric curved surfaces  442  and  432  may be constructed using known techniques for fabricating precision surfaces. For example, a master set of reflecting surfaces  432  and  442  are constructed using high precision techniques for grinding spherical surfaces in conjunction with high precision metrology techniques. From the master set, replication techniques are employed to mass-produce copies of the surfaces. Such methods are commonly used to produce diffraction gratings. Furthermore, if there is some uncertainty in the resulting structures, testing can be used to retain only those copies that enhance transmission. Such testing may include the light transmission properties and surface profile measurements. 
   Similar to the discussion of imaging system  300 , the object point of imaging system  400  is diffraction limited, but points in the vicinity of the object point may be distorted by aberrations. By the use of refractive surfaces it is possible to make these aberrations substantially zero for points in the object plane displaced from the object point. Referring to  FIG. 5 , an catadioptric imaging system  500  includes an object point  560 , an image point  562 , a beam splitter  550 , a curved reflective surface  532  and  542 , plano-concave-convex light transmitting elements  530  and  540 , and plano-convex elements  520  and  580 . Element  520  and element  530  are formed such that the radius of curvature of surface  522  of element  520  is substantially the same as the radius of curvature of surface  534  of element  530 . Element  540  and element  580  are formed such that the radius of curvature of surface  544  of element  540  is substantially the same as the radius of curvature of surface  582  of element  580 . In the described embodiment, the common center of curvature for surface  522  of element  520 , for surface  534  of element  530 , and for surface  542  of element  540  is the object point  560 . Furthermore in the described embodiment the common center of curvature for surface  544  of element  540 , for surface  532  of element  530 , and for surface  582  of element  580  is the image point  562 . Surfaces  522  and  544  are preferably coated with an antireflection coating. Furthermore, similar to the imaging system  400  of  FIG. 4 , the surfaces  542  and  532  are constructed such that light rays which are split by the beam splitter  550  recombine at a common point on beam splitter  550  and interfere constructively to enhance the light transmission to the image point  562 . 
   In some embodiments, element  580  is composed of air. This allows for optical detection devices like CCD&#39;s to be positioned easily near the image point. The radii of curvature r 522 , r 534 , and r 544  of the refractive surfaces  522 ,  534 , and  544 , respectively, are chosen to minimize certain optical aberrations. Non-limiting examples of radii of curvature are shown in Table 1 for several different combinations of refractive materials with r 532 =r 542 =50 mm where r 532  and r 542  are the radii of curvature of surfaces  532  and  542 , respectively. It is assumed that element  580  is air. Results of geometrical ray traces through systems employing the combination of refractive materials listed in Table 1 show that the images formed by the first embodiment are diffraction limited for an object field of 0.5 mm with an object space numerical aperture equal to 0.77 times the index of refraction of element  520  where n 520 , n 530 , and n 540  are the refractive indices of elements  520 ,  530 , and  540 , respectively. 
   In additional embodiments, the reflective surfaces in, for example, the embodiments of any of  FIG. 4  or  5 , may be reconfigured to produce an imaging system that images the object point to two spatially separated image points. The two image points may be displaced relative to each other along the optical axis, in a plane orthogonal to the optical axis, or a combination of both. Such embodiments may also be used in “reverse” to image two spatially separated object points to a common image point. The reconfiguration of the reflective surfaces may include, for example, adjusting their relative positions and/or changing their radius of curvature. 
   Referring to  FIG. 6 , a catadioptric imaging system  1000  is shown that is similar to system  400  of  FIG. 5 . System  1000  includes an object point  1060 , spatially separated image points  1062  and  1064 , a beam splitter  1050 , curved reflective 
                                       TABLE 1                   Element   n 520     n 530 , n 540     r 522 , r 534     r 544         Lens 520   530, 540   (633 nm)   (633 nm)   (mm)   (mm)                   GaP a     Fused Silica   3.3079   1.4570   8.467   17.500       BSO b     Fused Silica   2.5500   1.4570   5.551   12.270       YSZ c     Fused Silica   2.1517   1.4570   3.000    6.720       YAG d     Fused Silica   1.8328   1.4570   2.997   16.030                 a GaP: Gallium phosphide         b BSO: Bismuth silicon oxide, Bi 12 SiO 20           c YSZ: Ytterbium stabilized zirconia, ZrO 2 : 12% Y 2 O 3           d YAG: Yttrium aluminum garnet, Y 3 Al 5 O 12              
surfaces  1032  and  1042 , plano-convex-concave light transmitting elements  1030  and  1040 , and plano-convex elements  1020  and  1080 . Element  1020  and element  1030  are formed such that the radius of curvature of surface  1022  of element  1020  is substantially the same as the radius of curvature of surface  1034  of element  1030 . Beam splitter  1050  is oriented normal to an optical axis  1002  connecting object point  1060  to image point  1062 . As in the embodiment of  FIG. 5 , the center of curvature of reflective surface  1042  coincides with object point  1060 . Thus, a first set of rays  1092  corresponding to those rays from object point  1060  transmitted by beam splitter  1050  reflect from curved surface  1042  and then reflect from beam splitter  1050  to focus onto image point  1062 .
 
   However, in contrast to the embodiment of  FIG. 5 , the center of curvature  1063  of reflective surface  1032  is displaced from image point  1062  by an amount δy 1  along a direction normal to optical axis  1002 , which corresponds to reflective surface  1032  being displaced by the amount δy 1  along the direction normal to optical axis  1002 . As a result, a second set of rays  1094  corresponding to those rays from object point  1060  reflected by beam splitter  1050  reflect from curved surface  1032  and then transmit through beam splitter  1050  to focus onto image point  1064 , which is displaced from center of curvature  1063  by an amount δy 2 =δy 1  along the direction normal to optical axis  1002 . Thus, in system  1000  image points  1062  and  1064  are displaced from one another by an amount 2δy 1  along the direction normal to optical axis  1002 . 
   Additional elements  1020  and  1080  provide refracting surfaces selected minimize aberrations as described above. For simplicity, the effects of any such refraction are not shown in  FIG. 6  with respect to the path of rays  1092  and  1094 . 
   In another similar embodiment shown in  FIG. 7 , the center of curvature of one of the reflective surfaces is displaced along the optical axis. 
   Referring to  FIG. 7 , a catadioptric imaging system  1100  includes an object point  1160 , spatially separated image points  1162  and  1164 , a beam splitter  1150 , curved reflective surfaces  1132  and  1142 , plano-convex-concave light transmitting elements  1130  and  1140 , and plano-convex elements  1120  and  1180 . Element  1120  and element  1130  are formed such that the radius of curvature of surface  1122  of element  1120  is substantially the same as the radius of curvature of surface  1134  of element  1130 . Beam splitter  1150  is oriented normal to an optical axis  1102  connecting object point  1160  to image point  1162 . As in the embodiment of  FIG. 4 , the center of curvature of reflective surface  1142  coincides with object point  1160 . Thus, a first set of rays  1192  corresponding to those rays from object point  1160  transmitted by beam splitter  1150  reflect from curved surface  1142  and then reflect from beam splitter  1150  to focus onto image point  1162 . 
   However, in contrast to the embodiment of  FIG. 5 , the center of curvature  1163  of reflective surface  1132  is displaced from image point  1162  by an amount δz 1  along optical axis  1102 , which corresponds to reflective surface  1132  being displaced by the amount δz 1  along optical axis  1102 . As a result, a second set of rays  1194  corresponding to those rays from object point  1160  reflected by beam splitter  1150  reflect from curved surface  1132  and then transmit through beam splitter  1150  to focus onto image point  1164 , which is displaced from center of curvature  1163  by an amount δz 2  along optical axis  1102 . The amounts δz 1  and δz 2  are related to one another by the spherical lens formula 1/s 1 +1/s 2 =2/R, where R is the radius of curvature of reflective surface  1132 , s 1 =R−δz 1 , and s 2 =R+δz 2 . Thus, in system  1100  image points  1162  and  1164  are displaced from one another by an amount δz 1 +δz 2  along optical axis  1102 . 
   Additional elements  1120  and  1180  provide refracting surfaces selected minimize aberrations as described above. For simplicity, the effects of any such refraction are not shown in  FIG. 7  with respect to the path of rays  1192  and  1194 . 
   In further embodiments, the reflective surface may be displaced both by an amount δy 1  along a direction normal to the optical axis and by an amount δz 1  along optical axis  1102 . In such embodiments, the longitudinal displacement of the second image point is the same, however, the transverse displacement further includes a magnification factor M=s 2 /s 1 , in which case δy 2 =Mδy 1 . 
   In yet further embodiments, the other of the reflective surfaces may be displaced, or both surfaces may be displaced. Furthermore, the radius of curvature of one or both of the reflective surfaces may be modified, which have a similar effect as that of the longitudinal displacement described with reference to  FIG. 7 . 
   In additional embodiments of the catoptric systems described herein, one or both of the reflective surfaces in any of the embodiments described above, may be a Fresnel mirror. As defined above, a Fresnel mirror is a reflecting surface formed by multiple curved facets each having a common center of curvature. 
   Referring to  FIG. 8 , for example, a catadioptric imaging system  1200  includes an object point  1260 , image point  1262 , a beam splitter  1250 , curved reflective surfaces  1232  and  1242 , and plano-convex-concave light transmitting elements  1230  and  1240 . System  1200  is similar to that of  FIG. 4 , except both of the reflective surfaces are Fresnel mirrors. In particular, reflective surface  1232  includes curved facets  1232   a ,  1232   b , and  1232   c , which each have a common center of curvature at image point  1262 . Facets  1232   b  and  1232   c  may be fabricated, for example, as an outer annular section of a lens having a surface with the same radius of curvature as facet  1232   a . Similarly, reflective surface  1242  includes curved facets  1242   a ,  1242   b , and  1242   c , which each have a common center of curvature at object point  1260 . Furthermore, facets  1242   b  and  1242   c  may be fabricated, for example, as an outer annular section of a lens having a surface with the same radius of curvature as facet  1242   a.    
   Referring still to  FIG. 8 , implementing the Fresnel mirrors allows oblique rays emerging from object point  1260 , such as rays  1261 , to be imaged to image point  1262  in addition to less oblique rays such as rays  1263 . In contrast, oblique rays  1261  would not be imaged to the image point by the system if it only included central facets  1232   a  and  1242   a  (as indicated by the dashed lines extending facets  1232   a  and  1242   a ). Thus, implementing the Fresnel mirrors increases the numerical aperture and working distance of the system. 
   In each of the embodiments, the requirements for matched elements with respect to tolerances on radii of curvature, thickness of plano-convex elements, thickness of a plano-concave-convex element, and lateral shears of elements are typically associated with respect to a pair of elements or a set of four elements that have pie-sections as apertures such as shown in perspective drawing in  FIG. 9  for catadioptric imaging system  600 . System  600  comprises elements  630 ,  632 ,  640 , and  642  and each of the four elements represents a 45 degree pie-section. Elements  630 ,  632 ,  640 , and  642  are constructed by cutting a single element, such as element  530  shown in  FIG. 5 , into four sections. In general, that starting element is a catadioptric lens element that includes a planar bottom surface and an upper surface having a convex reflective portion and a concave refractive portion, with both the convex and concave portions sharing a common axis of symmetry (typically, they are spherical or substantially spherical surfaces). As a consequence of the way they are produced, the elements  630 ,  632 ,  640 , and  642  have the same radii of curvature and thickness of the plano-convex-concave dimension to the same accuracy that the surface  532  shown in  FIG. 5  can be manufactured, e.g., λ/10. 
     FIG. 10  shows the catadioptric system of  FIG. 9  with the elements separated in order to display the features more clearly. 
   The use of matched pie-sections is of particular value in ellipsometric interferometric applications of the catadioptric imaging system such as described in the above-mentioned U.S. Provisional Application entitled “Joint Measurement Of Fields Of Orthogonally Polarized Beams Scattered/Reflected By An Object In Interferometry.” The pie-sections may comprise sections with angles less than 45 degrees. 
   The relative radii of curvature of elements  630 ,  632 ,  640 , and  642  may be modified by a fraction λ or of the order of λ with the deposition of a thin layer on the respective concave or convex surfaces. Also the thickness of the plano-convex-concave dimension of elements  630 ,  632 ,  640 , and  642  may be modified by a fraction λ or of the order of λ with the deposition of a thin layer on the respective plano surfaces. The addition of the thin layers would serve for example the purpose of introducing a π/2 or π phase shift in a measurement beam. 
   In catadioptric imaging system comprising pie-sections such as shown in  FIG. 6 , the construction method described herein easily accommodates the introduction of lateral shears of elements  630 ,  632 ,  640 , and  642  as desired in an end use application. 
   The use of matched pie-sections of a catadioptric imaging system also has the additional advantage of permitting two or more different matched pie-sections having different properties, e.g., numerical apertures, different π/2 or π phase shifts in a measurement beam, and/or different operating wavelengths. 
   Other embodiments are within the following claims.