Abstract:
The invention features an imaging system for imaging an object point to an image point. The system includes: a beam splitter positioned to receive light rays from the object point and separate each ray into a transmitted portion and a reflected portion, the transmitted portions defining a first set of rays and the reflected portions defining a second set of rays; and a reflecting surface positioned to receive one of the sets of rays from the beam splitter and focus that set of rays towards the image point via the beam splitter. Interferometric techniques may be applied to increase the light throughput of the system.

Description:
[0001]    CROSS-REFERENCE TO RELATED APPLICATIONS  
         [0002]    This application claims priority to U.S. provisional application No. 60/257,833 by Henry A. Hill and filed Dec. 21, 2000, the contents of the provisional application are incorporated herein by reference. 
     
    
     
       BACKGROUND  
         [0003]    This invention relates to optics, and more particularly to catoptric imaging systems.  
           [0004]    Imaging systems of varying sorts have existed for thousands of years. Even after such a long period of development, modern imaging systems still have a similar purpose as their ancient counterparts. Imaging systems gather light from an object point and its vicinity and focus this light into an image at an image point and its vicinity. Light can be focused using refraction and this branch of optics is known as dioptrics. Light can also be focused using reflection and this branch of optics is known as catoptrics. Irregardless of the focusing system used typical imaging systems strive to optimize a few important parameters. For example, many imaging systems are designed to optimize resolution, numerical aperture, and shape of the image plane. Resolution of the imaging system is the smallest distance between features in the object space that can be distinguished in the image plane. Therefore resolution determines the level of detail that can be derived from the image. Numerical aperture relates to the amount of available light that the imaging system collects from the object. For most types of detectors, such as photographic film, charge coupled devices, or even the human eye, a larger numerical aperture increases light intensity and typically yields better images. Lastly, the shape of the image plane can be quite important. A flat plane is typically most useful for detection devices like photographic film or charged coupled devices (CCD). Unfortunately, all of these parameters can be degraded by a host of aberrations.  
         SUMMARY  
         [0005]    The present invention features catoptric optical systems that utilize a beam splitter surface and a reflecting surface. Primary focusing can be achieved with the reflecting surface and therefore longitudinal chromatic aberrations are reduced. The beam splitter is positioned relative to the object point, image point, and the reflective surface such that light rays from the object point which are focused to the image point have been both reflected and transmitted by the beam splitter surface. The combination of a reflection and a transmission for each ray of the beams being focused substantially eliminates first-order variations in the beam intensity due to imperfections in the reflective and transmissive properties of the beam splitter for incident angles deviating from a central design angle. In some embodiments of the system, light transmission may be enhanced by use of interferometric recombination of light reflected and transmitted by the beam splitter. Furthermore, some embodiments of the system may include refractive elements to reduce aberrations.  
           [0006]    In general, in one aspect, the invention features an imaging system for imaging an object point to an image point. The system includes i) a beam splitter positioned to receive light rays from the object point and separate each ray into a transmitted portion and a reflected portion, the transmitted portions defining a first set of rays and the reflected portions defining a second set of rays; and ii) a reflecting surface positioned to receive one of the sets of rays from the beam splitter and focus that set of rays towards the image point via the beam splitter.  
           [0007]    Embodiments of the imaging system may include any of the following features.  
           [0008]    The reflecting surface may be positioned to receive the first set of rays and reflect the first set of rays back to the beam splitter, in which case the beam splitter is positioned to reflect at least a portion of each ray received from the reflecting surface to the image point. Furthermore, the reflecting surface may be substantially concentric with the object point. A center of the reflecting surface may define an object optical axis with the object point, and the beam splitter may be positioned substantially perpendicular to the object optical axis or at an acute angle to the object optical axis (e.g., an acute angle substantially equal to 45 degrees).  
           [0009]    Alternatively, the reflecting surface may be positioned to receive the second set of rays and reflect the second set of rays back to the beam splitter, in which case the beam splitter is positioned to transmit at least a portion of each ray received from the reflecting surface to the image point. Furthermore, the reflecting surface may be substantially concentric with the image point. A center of the reflecting surface may define an image optical axis with the image point, and the beam splitter may be positioned substantially perpendicular to the image optical axis or at an acute angle to the image optical axis (e.g., an acute angle substantially equal to 45 degrees).  
           [0010]    The imaging system may further include a first optic having an internal surface defining the reflecting surface. For example, the internal surface of the first optic may be curved. The first optic may have a flat surface opposite the internal surface, and the beam splitter may be positioned adjacent the flat surface. Furthermore, the system may include a plano-convex optic having a piano surface adjacent one of the object point and the image point and a convex surface contacting the first optic, wherein the interface between the plano-convex optic and the first optic defines a refracting surface.  
           [0011]    More generally, the imaging system may include a refracting surface positioned between the object point and the beam splitter to receive the light rays from the object point. For example, the refracting surface may be substantially concentric with the object point. Alternatively, or in addition, the system may include a refracting surface positioned between the beam splitter and the image point to receive the light rays focused by the reflecting surface. For example, the refracting surface may substantially concentric with the image point.  
           [0012]    The system may also include a second optic adjacent the first optic, wherein the beam splitter is positioned at an interface between the first and second reflective optics. Furthermore, the system may include a plano-convex optic having a piano surface adjacent one of the object point and the image point and a convex surface contacting the second optic, wherein the interface between the plano-convex optic and the second optic defines a refracting surface. Moreover, the system may further include another plano-convex optic having a piano surface adjacent the other of the object point and the image point and a convex surface contacting the first optic, wherein the interface between the plano-convex optic and the first optic defines another refracting surface.  
           [0013]    The second optic may be an optical flat. Alternatively, the second optic may have an internal surface defining a second reflecting surface, and wherein the first reflecting surface is positioned to receive the first set of rays and the second reflecting surface is positioned to received the second set of rays and focus them towards the image point via the beam splitter. Furthermore, the beam splitter may be positioned to interferometrically recombine the first set of rays received from the first reflecting surface and the second set of rays received from the second reflecting surface. The first reflecting surface may be concentric with the object point and the second reflecting surface may be concentric with the image point.  
           [0014]    In general, in another aspect, the invention features an imaging system including a first imaging subsystem for imaging an object point to a first image point; and a second imaging subsystem for imaging the first image point to a second image point. The first imaging subsystem second includes i) a first beam splitter positioned to receive light rays from the object point and separate each ray into a transmitted portion and a reflected portion, wherein the transmitted portions define a first set of rays and the reflected portions defining a second set of rays; and ii) a first reflecting surface positioned to receive one of the sets of rays from the first beam splitter and focus that set of rays towards the first image point via the first beam splitter. The second imaging subsystem includes (i) a second beam splitter positioned to receive light rays from the first image point and separate each ray into a transmitted portion and a reflected portion, wherein the transmitted portions define a first set of rays and the reflected portions defining a second set of rays; and (ii) a second reflecting surface positioned to receive one of the sets of rays from the second beam splitter and focus that set of rays towards the second image point via the second beam splitter.  
           [0015]    In general, in another aspect, the invention features an imaging method for imaging an object point to an image point. The method including: receiving light rays from the object point and separating each ray into a transmitted portion and a reflected portion, the transmitted portions defining a first set of rays and the reflected portions defining a second set of rays; and receiving one of the sets of rays from the beam splitter and reflecting that set of rays to focus them towards the image point via the beam splitter.  
           [0016]    Embodiments of the inventions may include any of the following advantages.  
           [0017]    They may have a large numerical aperture in the object space. They may have substantially no longitudinal chromatic aberrations. They may have an image plane whose location is substantially independent of the spectral region used in image formation. They may use a large spectral range for image formation. They may have a flat image plane. They may have a reduced set of optical aberrations. They may have a magnification which is less than, greater than, or equal to one.  
           [0018]    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 DRAWINGS  
       [0019]    [0019]FIG. 1 is a schematic drawing of a catoptric imaging system including a reflective surface and a beam splitter.  
         [0020]    [0020]FIG. 2 is a schematic drawing of another catoptric imaging system including a reflective surface and a beam splitter.  
         [0021]    [0021]FIG. 3 is a schematic drawing of a catadioptric imaging system including a reflective surface, a beam splitter, and two refractive surfaces.  
         [0022]    [0022]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.  
         [0023]    [0023]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.  
         [0024]    [0024]FIG. 6 is a schematic drawing of a catadioptric imaging system similar to the imaging system in FIG. 5 except that the optical axes are not coextensive.  
         [0025]    [0025]FIG. 7 is a schematic drawing of a composite catadioptric imaging system that includes imaging systems similar to those in FIGS. 5 and 6.  
         [0026]    [0026]FIG. 8 is a schematic drawing of a composite catadioptric imaging system that inlcudes imaging systems similar to those in FIG. 5.  
         [0027]    [0027]FIG. 9 is a schematic drawing of a composite catadioptric imaging system that includes imaging systems similar to those in FIG. 5. 
     
    
       [0028]    Like reference symbols in the various drawings indicate like elements.  
       DETAILED DESCRIPTION  
       [0029]    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 .  
         [0030]    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.  
         [0031]    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.  
         [0032]    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.  
         [0033]    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 nonideal 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 nonideal behavior will only impact the light intensity to second order in δ(φ).  
         [0034]    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.  
         [0035]    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.  
         [0036]    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).  
         [0037]    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.  
         [0038]    In preferred 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.  
         [0039]    Although not intended to limit the invention 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.  
         [0040]    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 a preferred 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 .  
         [0041]    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.  
         [0042]    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.  
         [0043]    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.  
         [0044]    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 . In the preferred embodiment this is 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 . In the preferred embodiment this is accomplished by designing curved surface  432  to be concentric with the image point  462 .  
         [0045]    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 . In preferred embodiments, 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%.  
         [0046]    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.  
         [0047]    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 preferred 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 preferred 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 .  
         [0048]    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 m 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 .  
                                                             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                                                  
 
         [0049]    where n 520 , n 530 , and n 540  are the refractive indices of elements  520 ,  530 , and  540 , respectively.  
         [0050]    Referring to FIG. 6, a catadioptric imaging system  600  includes an object point  660 , an image point  662 , a beam splitter  650 , a curved reflective surface  632  and  642 , plano-concave-convex light transmitting elements  630  and  640 , and plano-convex elements  620  and  680 , and optical axes  603  and  605 . Optical axis  603  is the axis between the object point  660  and the center of spherical surface  642 . Optical axis  605  is the axis between the image point  662  and the center of the curved reflective surface  632 . This embodiment is analogous to the embodiment of imaging system  500 . The main difference is that the corresponding optical axes in FIG. 5 (not shown) are coextensive. For imaging system  600 , the optical axes intersect at some nonzero angle. (FIG. 6 is shown with optical axes nearly orthogonal but this is not a limiting example; other angles of intersection are possible). Other than the non-zero angle between the axes, the description of this embodiment is identical to the description of imaging system  500  except with  100  added to each elemental number in the description. Although imaging system  600  is described in analogy with imaging system  500 , the non coextensive optical axes of imaging system  600  can also be applied in analogous fashion to imaging systems  100 ,  200 ,  300 , and  400 .  
         [0051]    The imaging system  600  does have object planes and image planes which are no longer parallel. They are at the same angle as the angle between the optical axes  603  and  605 . In some embodiments this may be an advantage. Furthermore, the angle of incidence of the central ray (a ray coincident with the optical axes) is no longer 90 degrees (as in systems  100 - 500 ). As shown in FIG. 6, the angle of incidence of the central ray is about 45 degrees. As discussed above, the angle of incidence to the beam splitter influences the performance of the beam splitter. Therefore, the angle between the optical axes also influences overall imaging system performance and as such the angle between the optical axes is a design choice which impacts the design and optimization of the beam splitter. Furthermore, in imaging system  600  the central ray is not obscured by refractive elements  620  or  680 .  
         [0052]    Any of the imaging systems described above can be chained together to form composite imaging systems. Referring to FIG. 7, a composite catadioptric imaging system  700  includes an image point  762 , object point  760 , and catadioptric imaging subsystems  720  and  740 . The image point of subsystem  720  is set to coincide the object point of subsystem  740 . The catadioptric imaging subsystem  720  is similar to catadioptric imaging system  500  and the catadioptric imaging subsystem  740  is similar to the catadioptric imaging system  600 . Additional embodiments include any other permutation. For example, referring to FIG. 8, a composite catadioptric imaging system  800  includes an image point  862 , object point  860 , and catadioptric imaging subsystems  820  and  840 . Both imaging subsystems are similar to imaging system  500 . Referring to FIG. 9, a composite catadioptric imaging system  900  includes of an image point  962 , object point  960 , and catadioptric imaging subsystems  920  and  940 . Both imaging subsystems are similar to imaging system  500  wherein the analogous element  580  of the first subsystem is identical to element  520  of the second system. The magnification of imaging system  900  is one independent of the index of refraction of element  580  of the first subsystem. In addition, both the image point  962  and object point  960  are both accessible. This may be useful, for example, to allow a sample stage to access the object point  960  or to allow optical detectors access to image point  962 . Other embodiments include permutations with more than two imaging systems ( 100  to  900 ) chained together.  
         [0053]    A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.