Patent Publication Number: US-2013229722-A1

Title: High throughput reflecting microscope objective

Description:
This application claims the benefit of U.S. Provisional Application No. 61/606,151, filed on Mar. 2, 2012, which is incorporated by reference herein in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to microscope objectives. 
     2. Background Art 
     In optics, various optical systems may be used to modify the behavior of light. Such optical systems have optical aberration parameters, including the coma, which is a change in magnification with position on an aperture stop, and the spherical, which is a change in the path length taken by a light ray to the focus for different positions on the aperture stop. In an aplanatic system, the coma and spherical are both zero. Aplanatic systems may be used in various types of optical devices, such as telescopes. 
     In 1905, Karl Schwarzschild published several papers on geometrical optics that dealt with the aberrations encountered in optical systems. In a first paper, Schwarzschild showed how spherical aberrations originate. In a second paper, Schwarzschild demonstrated how a telescope free of aberrations can be formed by combining two mirrors with aspherical surfaces. In a third paper he provided formulas for computing a variety of compound optical systems. 
     In 2005, V. Yu Terebizh published a paper that examined Schwarzschild&#39;s second paper regarding aplanatic telescopes. In his paper, Terebizh indicated that Schwarzschild&#39;s equations held true for arbitrary two-mirror aplanatic systems. These parametric equations from Schwarzschild are commonly approximated by spherical surfaces, which are traditionally easier to manufacture than aspherical surfaces. These approximations, however, are only accurate at smaller aperture sizes and do not maintain the aplanatic condition as accurately as the parametric equations. They suffer from small input aperture diameters because the approximations that they rely on fall apart at larger aperture sizes. They also have relatively large outer diameters relative to the entrance pupil diameter. 
     BRIEF SUMMARY OF THE INVENTION 
     Systems, methods, and apparatuses are described for objectives used in optical systems, substantially as shown in and/or described in connection with at least one of the figures, as set forth more completely in the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES 
       The accompanying drawings, which are incorporated herein and form a part of the specification, illustrate the present invention and, together with the description, further serve to explain the principles of the invention and to enable a person skilled in the pertinent art to make and use the invention. 
         FIG. 1  illustrates a reflective microscope mirror configuration defining first and second curves that may be used in embodiments. 
         FIGS. 2 and 3  show the first and second curves of  FIG. 1 , according to example embodiments. 
         FIG. 4  shows a cross-sectional view of a single-piece objective that contains two powered mirror surfaces defined by equations, according to an example embodiment. 
         FIG. 5  shows a perspective view of the objective of  FIG. 4 , and further shows a position of a mask when a single mask is used, according to an example embodiment. 
         FIG. 6A  shows a perspective view of a mask with two spider arms, according to an example embodiment. 
         FIG. 6B  shows a perspective view of a mask with three spider arms, according to an example embodiment. 
         FIG. 6C  shows a perspective view of a mask with a substantially optically clear material used to support a substantially opaque mask, according to an example embodiment. 
         FIG. 7  shows a cross-sectional view of an objective without a mask allowing a stray light ray to pass through and reflect back, according to an example embodiment. 
         FIG. 8  shows a cross-sectional view of the stray light ray of  FIG. 7  blocked by a mask, according to an example embodiment. 
         FIG. 9  shows a cross-sectional view of an objective including a single mask, according to an example embodiment. 
         FIG. 10  shows an exploded view of an objective that uses the powered surfaces shown in  FIG. 4 , and includes a mechanism for adjusting spacing between the two surfaces, according to an example embodiment. 
         FIG. 11  shows a cut away perspective view of an objective that includes a mechanism for adjusting a spacing between two powered surfaces, according to an example embodiment. 
         FIG. 12  shows a cut away perspective view of an objective that uses a two-element mask of a first type, according to an example embodiment. 
         FIG. 13  shows a cross-sectional view of the objective of  FIG. 12  including the two-element mask of the first type, according to an example embodiment. 
         FIG. 14  shows a cut away perspective view of an objective that uses a two-element mask of a second type, according to an example embodiment. 
         FIG. 15  shows a cross-sectional view of the objective of  FIG. 14  including the two-element mask of the second type, and showing light rays, according to an example embodiment. 
         FIG. 16  shows a cross-sectional view of an objective that uses a single element mask of a third type, according to an example embodiment. 
         FIG. 17  shows a side cross-sectional view of a solid optically clear objective, according to an example embodiment. 
         FIG. 18  shows a side cross-sectional view of the objective of  FIG. 17 , with a mask and with light rays traveling through the objective, according to an example embodiment. 
         FIG. 19  shows a side cross-sectional view of an optically clear two-piece objective, with a mask and with light rays traveling through the objective, according to an example embodiment. 
         FIGS. 20A-20C  show examples of equations that may be used in embodiments to define first and second concave mirror surfaces. 
     
    
    
     The present invention will now be described with reference to the accompanying drawings. In the drawings, like reference numbers indicate identical or functionally similar elements. Additionally, the left-most digit(s) of a reference number identifies the drawing in which the reference number first appears. 
     DETAILED DESCRIPTION OF THE INVENTION 
     Introduction 
     The present specification discloses one or more embodiments that incorporate the features of the invention. The disclosed embodiment(s) merely exemplify the invention. The scope of the invention is not limited to the disclosed embodiment(s). The invention is defined by the claims appended hereto. 
     References in the specification to “one embodiment”, “an embodiment”, “an example embodiment”, etc., indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to effect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described herein. 
     Furthermore, it should be understood that spatial descriptions (e.g. “above”, “below”, “up”, “down”, “left”, “right”, “top”, “bottom”, “vertical”, and “horizontal”, etc) used herein are for purposes of illustration only, and that practical implementations of the structures described herein can be spatially arranged in any orientation or manner. 
     EXAMPLE EMBODIMENTS 
     The example embodiments described herein are provided for illustrative purposes, and are not limiting. Furthermore, their structural and operational embodiments, including modifications, alterations, will become apparent to persons skilled in the relevant art(s) from the teachings herein. 
     An objective is an optical element that gathers light from an object being observed, and focuses the light to form an image. Embodiments are disclosed herein for objectives that may be used in optical systems, such as microscopes and telescopes. In an embodiment, a larger aperture, aplanatic reflective objective is provided that is smaller in overall diameter and cheaper to produce than the tradition Schwarzschild design at the same magnification. 
     An aplanatic system is one where the optical aberrations of spherical and coma are both zero. In V. Yu Terebizh&#39;s 2005 paper, “Two-Mirror Schwarzschild Aplanats. Basic Relations,” Astronomy Letters 31, 129-139, Terebizh examined Karl Schwarzschild&#39;s 1905 paper regarding aplanatic telescopes. Terebizh indicated that Schwarzschild&#39;s equations held true for arbitrary two-mirror aplanatic systems. A Gregorian mirror configuration includes two concave surfaces with an intermediate focus formed between them. Both concave surfaces are positioned between their individual vertexes and the intersection of the two curves. One of the two mirrored surfaces in a Gregorian configuration contains a hole which the light passes through. In embodiments, the mirror surfaces for a Gregorian system are extended beyond the intersection of the two generated reflecting curves, and the resulting larger surface is used to provide a larger input aperture to a microscope objective. In an embodiment, an objective is configured in a shape similar to that of an hourglass (e.g., the shape of two bowls placed bottom to bottom with their centers cut out to let the light pass through). In such an embodiment, the two concave surfaces have openings in their middle areas, and are joined together at the openings, such that the two concave surfaces face away from each other (as opposed to a typical Gregorian system, where the light passes through a hole in only one of the two concave surfaces). 
       FIG. 1  illustrates a configuration of concave mirror surfaces that may be used in embodiments. As shown in  FIG. 1 , the surface configuration includes first and second optical surfaces indicated as first and second concave mirrored surfaces  102  and  104 . In embodiments, first and second optical surfaces (e.g., curved mirrored surfaces) of an objective may be aligned similarly to first and second concave mirrored surfaces  102  and  104  in an objective to focus (e.g., magnify) light passing through.  FIGS. 2 and 3  show first and second concave mirrored surfaces  102  and  104  individually.  FIGS. 1-3  are described as follows to illustrate features of the surface configuration and of first and second optical surfaces  102  and  104 . 
     In  FIG. 1 , concave mirrored surface  102  defines a first concave mirrored surface, and concave mirrored surface  104  defines a second concave mirrored surface. Concave mirrored surfaces  102  and  104  are each concave shapes (e.g., bowl shaped) and are each symmetrical around an axis  106 . Axis  106  is an axis of rotational symmetry that passes through the vertexes of each of concave mirrored surfaces  102  and  104 . First diameter  108  is a diameter of the first concave mirrored surface  102 . As shown in  FIG. 1 , concave mirrored surfaces  102  and  104  are arranged in an hourglass shape. 
     Input light ray  110  is a collimated input ray received at an outermost edge of first concave mirrored surface  102 . Input light ray  112  is input light ray  110  after reflecting off of the outermost edge of first concave mirrored surface  102  towards the center of first concave mirrored surface  102 , through an opening (not shown in  FIG. 1 ) in the centers of first and second concave mirrored surfaces  102  and  104 , onto an outermost edge of second concave mirrored surface  104 . Input light ray  114  is input light ray  112  after reflecting off of the outermost edge of second concave mirrored surface  104  to a focus point  116 . Focus point  116  is a common light ray focus point at an image plane for the surface configuration of  FIG. 1 . The image plane is located outside of the concave shape of second concave mirrored surface  104 . 
     Collimated input light ray  118  is a collimated input light ray received at an edge of first concave mirrored surface  102 . Collimated input light ray  118  is symmetric to input light ray  110  about axis  106 . Line  120  is a line that is colinear with collimated input light ray  118 , and intersects with input light ray  114 . Intersection point  122  is a point of intersection of line  120  and input light ray  114 . Circle  124  is a circle symmetric around axis  106  that defines the effective focal length  126 , F, having a center at focus point  116 . Delta  128  is a distance between the vertex of first concave mirrored surface  102  and the vertex of second concave mirrored surfaces  104 . As shown in  FIG. 1 , the curves of first and second concave mirrored surfaces  102  and  104  overlap with each other, such that the vertex of first concave mirrored surface  102  is located within the concave shape of second concave mirrored surface  104 , and the vertex of second concave mirrored surface  104  is located within the concave shape of first concave mirrored surface  102 . Back focal length  130 , B, is the back focal length or distance from the vertex of second concave mirrored surface  104  (point of second concave mirror surface  104  that intersects axis  106  in  FIG. 1 ) to focus point  116 . Angle  132 , U, is the angle between axis  106  and input light ray  114  after reflecting off of second concave mirrored surface  104 . 
     Referring to  FIG. 2 , an axis  202  is shown that is an axis of rotational symmetry for first concave mirrored surface  102  (which is colinear with axis  106  in  FIG. 1 ). Vertex  204  is a vertex of first concave mirrored surface  102  (a center of, and a point of maximum curvature of first concave mirrored surface  102 ). Axis  206  is an axis defining a first concave mirrored surface direction S1 (e.g., a normal direction to first concave mirrored surface  102  at vertex  204  and collinear with axis  106  in  FIG. 1 ). Axis  208  is an axis defining first concave mirrored surface direction Y1 (e.g., a direction perpendicular to axis  206  from vertex  204 , and tangent to first concave mirrored surface  102 ). Point  210  is a point on first concave mirrored surface  204 . Tangent line  212  is a line tangent to first concave mirrored surface  102  at point  210 . Normal line  214  is a line normal to first concave mirrored surface  102  at point  210 , directed inward from point  210 . 
     Referring to  FIG. 3 , an axis  302  is shown that is an axis of rotational symmetry for second concave mirrored surface  104  (which is colinear with axis  106  in  FIG. 1 ). Vertex  304  is a vertex of second concave mirrored surface  104  (a center of, and a point of maximum curvature of second concave mirrored surface  104 ). Axis  306  is an axis defining second concave mirrored surface direction S2 (e.g., a normal direction to second concave mirrored surface  104  at vertex  304  and collinear with axis  106  in  FIG. 1 ). Axis  308  is an axis defining second concave mirrored surface direction Y2 (e.g., a direction perpendicular to axis  306  from vertex  304 , and tangent to second concave mirrored surface  104  at vertex  304 ). Point  310  is a point on second concave mirrored surface  104 . Tangent line  312  is a line tangent to second concave mirrored surface  104  at point  310 . Normal line  314  is a line normal to second concave mirrored surface  104  at point  310 . 
     Concave mirrored surfaces  102  and  104  may be used in objectives in the surface configuration of  FIG. 1  in embodiments. Examples of such embodiments are described further below. In an embodiment, the curved shapes of concave mirrored surfaces  102  and  104  may be defined by equations. In an embodiment, the surface equations used to meet the aplanatic condition can be written in a parametric from using the parameter t=sin 2  (U/2), where U is the angle a light ray makes with the optical axis (e.g., axis  106  of  FIG. 1 ) at the focus (e.g., focus point  116  of  FIG. 1 ). Examples of equations that may be used in embodiments are shown in  FIGS. 20A-20C .  FIG. 20A  includes equations  2002 - 2032 ,  FIG. 20B  includes equations  2034 - 2046 , and  FIG. 20C  includes equations  2048 - 2054 . Equations  2002 - 2054  are described as follows with reference to  FIGS. 1-3 . The terms used in equations  2002 - 2054  are described with respect to the first equation in which they appear. 
     Equations  2002 - 2018  are used to calculate various parameters. It is noted that delta Δ is the distance between vertices of concave mirrored surfaces  102  and  104 . 
     With reference to equation  2002 , F is effective focal length  126 , and B is back focal length  130 . 
     With reference to equation  2004 , F1 is the focal length of first concave mirror surface  102  (not shown in  FIG. 1 ). 
     With reference to equation  2006 , D is the input aperture diameter, which is first diameter  108  (the maximum diameter of first concave mirror surface  102 ). 
     With reference to equation  2014 , U is the angle between axis  106  and input light ray  114  reflecting off of second concave mirrored surface  104  to focus point  116 . U may have values that range from 0 degrees to 90 degrees. For instance, U may be incremented by a predetermined amount from 0 degrees to 90 degrees (e.g., in increments of 1 degree, a portion of 1 degree, etc.) to determine values for t in Equation  2014 . 
     Equations  2020 ,  2022 , and  2024  are used to calculate the locations of points of first concave mirrored surface  102  based on a range of values for t (Equation  2024  is used instead of Equation  2022  for values of δ=1, because in such case, Γ of Equation  2010  would be otherwise be infinity). Y1 is a lateral distance for a point on concave mirrored surface  102  from axis  106 , and may be calculated by Equation  2020 . S1 is a distance from the axis  208  to the point resolved in the direction of axis  206 , and may be calculated by Equation  2022  or Equation  2024 . As such, a calculated point resides on concave mirrored surface  102  at the intersection of the distance Y1 from axis  106  and the distance S1 from the axis  208 . A pair of points on either side of axis  106  may be determined for concave mirror surface  102  in this manner (e.g., an first intersection of a first line at distance Y1 on a first side of axis  106  with a second line defined by the perpendicular distance from axis  208  determined by S1, and a second intersection of a third line at distance Y1 on a second side of axis  106  with a line defined by the perpendicular distance from axis  208  determined by S1). 
     Equations  2026  and  2028  are used to calculate the locations of points of second concave mirrored surface  104  based on a range of values for t. Y2 is a lateral distance for a point on concave mirrored surface  104  from axis  106 , and may be calculated by Equation  2026 . S2 is a distance from the axis  308  to the point resolved in the direction of axis  306 , and may be calculated by Equation  2028 . As such, a calculated point resides on concave mirrored surface  104  at the intersection of the distance Y2 from axis  106  and the distance S2 from axis  308 . A pair of points on either side of axis  106  may be determined for concave mirror surface  104  in this manner (e.g., an first intersection of a first line at distance Y2 on a first side of axis  106  with a second line defined by the perpendicular distance from axis  308  determined by S2, and a second intersection of a second line at distance Y2 on a second side of axis  106  with a line defined by the perpendicular distance from axis  208  determined by S2). 
     Equations  2030  and  2032  are used to calculate values of the variable θ used in Equations  2026  and  2028 . 
     Equation  2034  is used to calculate a first derivative of Y1 with respect to t. 
     Equation  2036  is used to calculate a first derivative of S1 with respect to t (when δ≠1). 
     Equation  2038  is used to calculate a first derivative of S1 with respect to t (when δ=1). 
     Equation  2040  is used to calculate a first derivative of Y2 with respect to t (when δ≠1). 
     Equation  2042  is used to calculate a first derivative of Y2 with respect to t (when δ=1). 
     Equation  2044  is used to calculate a first derivative of S2 with respect to t (when δ≠1). 
     Equation  2046  is used to calculate a first derivative of S2 with respect to t (when δ=1). 
     Equation  2048  is used to calculate a first derivative of S1 with respect to Y1 (when δ≠1). For instance, Equation  2048  may determine a slope of tangent line  212  on first concave mirror surface  102  at the point Y1, S1. 
     Equation  2050  is used to calculate a first derivative of S1 with respect to Y1 (when δ=1). For instance, Equation  2050  may determine a slope of tangent line  212  on first concave mirror surface  102  at the point Y1, S1. 
     Equation  2052  is used to calculate a first derivative of S2 with respect to Y2 (when δ≠1). For instance, Equation  2052  may determine a slope of tangent line  312  on second concave mirror surface  104  at the point Y2, S2. 
     Equation  2054  is used to calculate a first derivative of S2 with respect to Y2 (when δ=1). For instance, Equation  2054  may determine a slope of tangent line  312  on second concave mirror surface  104  at the point Y2, S2. 
       FIGS. 4-19  show various embodiments related to objectives configured according to embodiments. For instance, embodiments of objectives described as follows may include first and second concave mirrored surfaces that are shaped and function as described above with respect to  FIGS. 1-3  and one or more of Equations  2002 - 2054  of  FIGS. 20A-20C . For instance, Equations  2020  and  2022  or  2204  may be used to calculate the locations of a plurality of points to define first concave mirrored surface  102 , and Equations  2026  and  2028  may be used to calculate the locations of a plurality of points to define second concave mirrored surface  104 . Examples of such embodiments are described as follows. 
     For instance,  FIG. 4  shows a cross-sectional view of a single-piece objective that includes first and second concave mirror surfaces  404  and  406 , according to an example embodiment. First and second concave mirror surfaces  404  and  406  are examples of first and second concave mirror surfaces  102  and  104  of  FIGS. 1-3 . First and second concave mirror surfaces  404  and  406  may be shaped according to the equations described above. For instance, first concave mirror surface  404  may be defined as a collection of points calculated according to Equations  2020 ,  2022 , and  2204 , and second concave mirror surface  406  may be defined as a collection of points calculated according to Equations  2026  and  2028 . The objective of  FIG. 4  is described as follows. 
     As shown in  FIG. 4 , the objective includes a body  402  and first and second concave mirrored surfaces  404  and  406 . First concave mirrored surface  404  is formed on body  402 , and has a centrally located first opening (at the bottom portion of concave mirrored surface  404  in  FIG. 4 ). Second concave mirrored surface  406  is formed on body  402 , and has a centrally located second opening (at the top portion of concave mirrored surface  404  in  FIG. 4 ). As shown in  FIG. 4 , first and second concave mirrored surfaces  404  and  406  are oriented in opposition to each other—e.g., are concave facing away from each other (e.g., their respective “bowls” face away from each other). First and second concave mirrored surfaces  404  and  406  are coupled together at the first and second openings. For instance, in the embodiment of  FIG. 4 , there is a cylindrical channel and a cone shaped channel (flaring outward towards second concave mirrored surface  406 ) that connect the first opening of first concave mirrored surface  404  to the second opening of second concave mirrored surface  406 . 
     Furthermore, as shown in  FIG. 4 , a central pathway extending from a first end of body  402  (a top end of body  402  in  FIG. 4 ) opposite first concave mirrored surface  404  to a second end of body  402  (a bottom end of body  402  in  FIG. 4 ) opposite second concave mirrored surface  406 . An axis of symmetry of body  402  resides in the central pathway, similar to axis  106  shown in  FIG. 6 . 
     First concave mirrored surface  404  and second concave mirrored surface  406  focus light passing through the central pathway through body  402 . For instance, in the case where collimated input light is received at the first end (e.g., top) of body  402 , the light passes through body  402  to be focused by first and second concave mirrored surfaces  404  and  406  at a focal point  408  (e.g., focus point  116  in  FIG. 1 ) beyond the second end (e.g., bottom) of body  402 . In another embodiment, input light may emanate from focal point  408  outside of the second end (e.g., bottom) of body  402  to be received at the second end of body  402 . The input light passes through body  402  to be collimated (e.g., magnified) by first and second concave mirrored surfaces  404  and  406 , and to be transmitted from the first end (e.g., top) of body  402  as collimated output light. 
     The objective of  FIG. 4  may be manufactured in various ways, in embodiments. For instance, the body of the objective of  FIG. 4  may be manufactured by being machined from a material, by being cast in a mold, by being stamped from a material, etc. Examples of materials included in body  402  of the objective of  FIG. 4  include one or more metals (e.g., an alloy) such as iron, steel (e.g., stainless steel, spring steel), sheet metal, aluminum, copper, brass, glass, a polymer, etc. First and second concave mirror surfaces  404  and  406  may be formed on body  402  in any manner (e.g., vacuum deposition, polishing, plating, etc.), and may be made from any suitable material (e.g., silver, aluminum, gold, dielectric coating, etc.). For instance, in an embodiment, body  402  may have surfaces shaped according to the equations described above, and first and second concave mirror surfaces  404  and  406  may be formed on the shaped surfaces of body  402  to be shaped according to the equations described above. 
     In an embodiment, a mask may be present that at least partially obscures the light passing through the central pathway through body  402 . For instance,  FIG. 5  shows a perspective view of the objective of  FIG. 4 , and further shows a position of a mask  502  in the objective, according to an example embodiment. In the example of  FIG. 5 , mask  502  is a single piece mask, and is positioned at the first end of body  402 . Mask  502  is positioned inside a cylindrical portion of body  402  extending from the first end of body  402 . In the embodiment of  FIG. 5 , the cylindrical portion has smooth inner walls, and is threaded around its perimeter (e.g., to enable the objective to be screwed into a mating optical system component). Mask  502  includes a central obscuration that blocks light from passing through a center of mask  502 , providing the masking function, while enabling light to pass through mask  502  around the central obscuration. An obscuration is an element (e.g., a disk or other shape) that is opaque, and therefore blocks the transmission of light. 
     Mask  502  may be configured in various ways, in embodiments. For instance,  FIGS. 6A-6C  show example embodiments for mask  502 , which are described as follows. In other embodiments, mask  502  may be configured in other ways. 
       FIG. 6A  shows a perspective view of a mask  602  with two arms, according to an example embodiment. As shown in  FIG. 6A , mask  602  is a single-piece mask, and includes a circular central obscuration (e.g., an opaque disk), a ring shaped portion that is separate from and rings the central obscuration, and first and second arms connected between the ring shaped portion and the central obscuration. The first and second arms, which are positioned on opposite sides of the central obscuration, hold the central obscuration in the central position within the ring-shaped portion. Furthermore, the first and second arms enable light to pass through mask  602  between the ring shaped portion and the central obscuration, while the central obscuration blocks light from passing through a center of mask  602 , providing the masking function. 
     Any number of arms (e.g., “support arms”, “spider arms”, etc.) may be included in a mask similar to mask  602  to support a central obscuration. For instance,  FIG. 6B  shows a perspective view of a mask  604  that includes first-third arms, according to an example embodiment. As shown in  FIG. 6B , the first-third arms are connected between the ring shaped portion and the central obscuration. In the example of  FIG. 6B , the first-third arms are spaced 120 degrees apart from each other around the central obscuration, and hold the central obscuration in the central position within the ring-shaped portion. 
       FIG. 6C  shows a perspective view of a mask  606  that has no arms, according to an example embodiment. As shown in  FIG. 6C , mask  606  includes a central obscuration  608  and a substantially optically clear ring shaped portion that rings and supports central obscuration  608 . The optically clear ring shaped portion fulfills the functions of the ring-shaped portion and arms of  FIGS. 6A and 6B , holding central obscuration  608  in the central position. Furthermore, the optically clear ring shaped portion enables light to pass through itself, around central obscuration  608 , while central obscuration  608  blocks light from passing through a center of mask  606 , providing the masking function. 
     The optically clear ring shaped portion of mask  606  may be made from a variety of optically clear/transparent materials, including glass, a clear polymer, a crystal, etc. Central obscuration  608  of mask  606 , mask  602 , mask  604 , and mask  502  may each may be made from a variety of opaque materials, including one or more metals (e.g., an alloy) such as iron, steel (e.g., stainless steel, spring steel), sheet metal, aluminum, copper, brass, glass, a polymer, etc. 
     The presence of a mask in the objective of  FIGS. 4 and 5  enables stray light rays to be prevented from passing through the objective, because such stray light rays can cause poor quality images. For instance,  FIG. 7  shows a cross-sectional view of the objective of  FIG. 4  with no mask present, according to an example embodiment. As shown in  FIG. 7 , a stray light ray  702  is enabled to be received by and pass through the objective to hit a reflective sample surface  704  and be reflected back through the objective as a stray light ray  706 . Stray light ray  702  is an off-axis light ray (a light ray that is not parallel to the axis of symmetry of the objective) that does not interact with first concave mirrored surface  404 . Such stray light rays are undesirable. 
       FIG. 8  shows a cross-sectional view of the objective of  FIG. 7  with a mask  804  present such that stray light ray  702  is blocked by mask  804 , according to an example embodiment. In the example of  FIG. 8 , mask  804  is one of mask  602  ( FIG. 6A ) or  604  ( FIG. 6B ). As shown in  FIG. 8 , the central obscuration of mask  804  blocks light ray  702  from entering the objective. Other light rays that pass between the central obscuration and the ring shaped portion (and that are not blocked by a support arm) are enabled to pass through the objective as described above, to be focused at a focal point  806 . 
     In the example of  FIG. 8 , a second objective having a body  808  and a mask  810  is present, for illustrative purposes. The second objective is configured the same as the first objective, has mask  810  positioned similarly to mask  804  in the first objective, is oppositely oriented to the first objective, and is spaced relative to the first objective to have the same focal point  806 . As such, the second end of body  808  of the second objective receives the light that passes through the first objective and through focal point  806 . The received light passes through the second objective, and is converted into collimated light by first and second concave mirrored surfaces of the second objective. The collimated light is transmitted out of the first end of body  808  of the second objective. 
     Thus, the two-objective configuration of  FIG. 8  may be used to filter received light by masking and focusing input collimated light, and re-collimating with a first objective, and masking and re-collimating the masked and focused light. 
       FIG. 9  shows a cross-sectional view of the objective of  FIG. 5  including mask  502 , according to an example embodiment.  FIG. 9  illustrates an inefficiency of a single mask. As shown in  FIG. 9 , an off-axis light ray  902  is received that passes between the central obscuration and ring shaped portion of mask  502  to interact with first concave mirrored surface  404 . Off-axis light ray  902  reflects off of first concave mirrored surface  404  and then off of second concave mirrored surface  406  to pass through the objective and be received at a reflective sample plane/surface  904  at an off-axis focal point  906  (which is a different location from an on-axis focal point  910  corresponding to focus point  116  of  FIG. 1 ). For instance, in order to block on-axis light rays (light rays that are parallel to the axis of symmetry of the objective), mask  502  may have a diameter that is at least the diameter of the first opening hole in first concave mirrored surface  404 . First concave mirrored surface  404  is not fully illuminated by light rays entering at an angle. 
     In an embodiment, an objective may be configured to enable adjustment of the relative positions of first and second concave mirrored surfaces  404  and  406 . For instance, in one embodiment, the objective may include an optional element that enables the concave mirrored surfaces  404  and  406  to move relative to each other perpendicular to the axis of symmetry, allowing for axial alignment between concave mirrored surfaces  404  and  406 . In another embodiment, the objective may include an optional element that enables the two surfaces to be moved closer together or further apart alone the axis of symmetry, enabling the adjustment of spherical aberration. In one situation, this may compensate for the aberration induced by the thickness of an optically clear window used to support, hold, or sandwich a microscope sample. 
     For instance,  FIG. 10  shows an exploded view of an objective that includes the first and second concave mirrored surfaces shown in  FIG. 4 , and includes a mechanism for adjusting spacing between concave mirrored surfaces  404  and  406  (along an axis of symmetry, such as axis  116  of  FIG. 1 ), according to an example embodiment. Such a mechanism may be helpful to relieve manufacturing errors. It also enables the user to add spherical aberration to compensate for the spherical aberration created by rays passing through a window. 
     As shown in  FIG. 10 , the objective includes a mask  1002 , a first (e.g., upper) housing  1004  that internally contains first concave mirrored surface  404 , a vertical slot  1006  in first housing  1004 , a groove  1008  in first housing  1004 , an adjustment ring  1010 , an angled slot  1012  in adjustment ring  1010 , a first pin  1014 , a hole  1016  in adjustment ring  1010 , a second (e.g., lower) housing  1018 , a hole  1020  in second housing  1018 , and a second pin  1022 . As shown in  FIG. 10 , first housing  1004  is cylindrical and includes vertical slot  1006  extending through the cylindrical wall of first housing  1004 , and groove  1008  extends around first housing  1004  adjacent to the edge (e.g. bottom edge in  FIG. 10 ) of first housing  1004 . Adjustment ring  1010  is cylindrical, and angled slot  1012  and hole  1016  each extend through the cylindrical wall of adjustment ring  1010 . Second housing  1018  is cylindrical, and hole  1020  extends through the cylindrical wall of second housing  1018 . 
       FIG. 11  shows a cut away perspective view of the objective of  FIG. 10  assembled together, according to an example embodiment. As shown in  FIG. 11 , second housing  1018  may be positioned in first housing  1004 . Furthermore, adjustment ring  1010  fits around first housing  1004 , such that second pin  1022  may be inserted to extend through angled slot  1012 , vertical slot  1006 , and hole  1020 . Furthermore, first pin  1014  may be inserted to extend through hole  1016  into groove  1008  in first housing  1004 . 
     As such, when adjustment ring  1010  is rotated around first housing  1004 , pin  1022  transfers the rotary motion of adjustment ring  1010  into linear (axial) motion of second housing  1018 . Such motion enables second housing  1018  to be movable relative to first housing  1004  along the axis of symmetry of the objective body to enable adjustment of spherical aberration of the objective. Pin  1022  can slide in slot  1006  during rotation of adjustment  1010  to prevent lower housing  1018  from rotating while allowing the axial movement. Pin  1022  in groove  1008  prevents axial movement of adjustment ring  1010  relative to first housing  1004  while allowing the rotary motion of adjustment ring  1010 . Pin  1014  is in contact with groove  1008  to prevent axial movement of adjustment ring  1010  relative to first housing  1004  while allowing rotary motion. Thus, by twisting/rotating adjustment ring  1010  around the axis of symmetry of the objective, adjustment ring  1010  rotates around first housing  1004 , and second housing  1018  is moved along the axis of symmetry of the objective relative to adjustment ring  1010  and first housing  1004 . Second housing  1018  moves a distance corresponding to the amount that adjustment ring  1010  is rotated around first housing  1004 . 
     Furthermore, in an embodiment, second housing  1018  may be configured to be movable relative to first housing  1004  perpendicularly to the axis of symmetry of the objective to enable axial alignment of first and second concave mirrored surfaces  404  and  406 . For instance, in one example, one or more pins through holes in first housing  1004  may be provided that may hold second housing  1018  in place within first housing  1004 . The pins may be moveable (e.g., by a screwing motion, etc.) in and out of their respective holes in the sides of first housing  1004  to move second housing  1018  laterally with respect to first housing  1004 , thereby moving second housing perpendicularly to the axis of symmetry of the objective, and enabling axial alignment of first and second concave mirrored surfaces  404  and  406 . In embodiments, various other mechanism of  FIGS. 10 and 11  may be used to enable axial and/or lateral adjustment first and second concave mirrored surfaces  404  and  406 , including a threaded adjustment ring rather than the adjustment ring with a slot shown in  FIGS. 10 and 11 , and further types of mechanisms as would be apparent to persons skilled in the relevant art(s) from the teachings herein. 
     As described above, a mask used to filter light through an objective may have a single piece. In further embodiments, a mask used to filter light through an objective may have multiple pieces. Examples of types of two-element masks are shown in  FIGS. 12-16 . For instance,  FIG. 12  shows a cut away perspective view of an objective that uses a two-element mask, according to an example embodiment. As shown in  FIG. 12 , the objective includes a body  1202  that is generally similar to body  402  of the objective of  FIG. 4 , and further includes a first mask portion  1204  (also referred to as a “first sub-mask” or a “first mask element”) and a second mask portion  1206  (also referred to as a “second sub-mask” or a “second mask element”). 
     First mask portion  1204  has a central obscuration positioned on the axis of symmetry of body  1202  and is positioned at the first end of body  1202 . First mask portion  1204  is positioned inside a cylindrical portion of body  1202  extending from the first end of body  402  (similar to the placement of mask  502  in  FIG. 5 ). In the embodiment of  FIG. 12 , the cylindrical portion has smooth inner walls, and is threaded around its perimeter (e.g., to enable the objective to be screwed into a mating optical system component). First mask portion  1204  includes a central obscuration that blocks light from passing through a center of first mask portion  1204 , providing a portion of the masking function, while enabling light to pass through first mask portion  1204  around the central obscuration. First mask portion  1204  may be configured in various ways, in embodiments, including as shown for masks  602 ,  604 , and  606  of  FIGS. 6A-6C . Note that the first mask portion  1204  can be substantially smaller than mask  502  shown in  FIG. 5  and mask  802  shown in  FIG. 8  because the two mask elements of  FIG. 12  can be configured to block stray light rays together similarly to (e.g., equal to, less than, or greater than) a single mask. 
     Second mask portion  1206  is separate from first mask portion  1204 , and has an annular obscuration positioned in a channel in body  1202  between the first opening and the second opening of first and second concave mirrored surfaces  404  and  406 . The annular obscuration obscures light in a perimeter ring shaped area, but allows light to pass through a central region (e.g., in an opposite fashion to a central obscuration). As shown in the embodiment of  FIG. 4 , first and second concave mirrored surfaces  404  and  406  are coupled together by a cylindrical channel. In the embodiment of  FIG. 12 , first and second concave mirrored surfaces  404  and  406  are coupled together by a cylindrical channel that includes a first cylindrical channel portion (also referred to as a “first sub-channel”) adjacent to the first opening of first concave mirrored surface  404 , and a second cylindrical channel portion (also referred to as a “second sub-channel”) between the first cylindrical channel portion and the second opening of second concave mirrored surface  404 . The second cylindrical channel portion has a diameter that is greater than the diameter of the first cylindrical channel portion. Second mask portion  1206  is positioned in the second cylindrical channel portion. Second mask portion  1206  is generally ring shaped (e.g., similar to a washer) having an outer ring with a central opening. The inner edges of the outer ring extend inwardly from the walls of the second cylindrical portion further than the rim of the first cylindrical portion, and provide an annular obscuration to light passing through the central pathway of the objective. 
     Note that in one embodiment, the inner edge of the outer ring may be perpendicular to the top and bottom surfaces of the outer ring. In another embodiment, as shown in  FIG. 12 , the inner edge of the outer ring may be not be perpendicular to the top and bottom surfaces of the outer ring, and instead may be angled (an angle that is not  90  degrees, acute or obtuse) with regard to the top surface. An acute angle with respect to the top surface of the outer ring, as shown in  FIG. 12 , may be used to enable additional light rays to travel through the central opening of second mask portion  1206  without being blocked. 
     For instance,  FIG. 13  shows a cross-sectional view of the objective of  FIG. 12  including the two-element mask, according to an example embodiment.  FIG. 13  illustrates examples of light rays that pass through the objective or are filtered by the two-element mask. In  FIG. 13 , first concave mirrored surface  404  is indicated as first concave mirrored surface  1302 . A first ray  1304  parallel to the axis of symmetry is shown being received at the first end of the objective. Light ray  1304  passes through first mask portion  1204 , does not interact with first concave mirrored surface  1302 , and is blocked (filtered) by second mask portion  1206 . A second light ray  1306  parallel to the axis of symmetry is shown being received at the first end of the objective. Light ray  1306  passes through first mask portion  1204 , does not interact with first concave mirrored surface  1302 , and is blocked (filtered) by second mask portion  1206 . A first off-axis light ray  1308  is shown being received at the first end of the objective. Off-axis light ray  1308  passes through first mask portion  1204 , interacts with first concave mirrored surface  1302 , and shows that all of first concave mirror surface  1302  is illuminated. A second off-axis light ray  1310  is shown being received at the first end of the objective. Off-axis light ray  1310  passes through first mask portion  1204 , interacts with first concave mirrored surface  1302 , and passes through to the focal plane near the focal point of the objective without interacting with second mask portion  1206 . Additional light rays parallel to the axis of symmetry (collimated light rays) received by the objective are shown that passes through first mask portion  1204 , interact with first concave mirrored surface  1302 , pass through second mask portion  1206 , interact with the second concave mirrored surface, and are thereby focused to the focal point in the focal plane of the objective, as desired. 
     As described above, any number of arms may be used in first mask portion  1204  to support the central obscuration, including one or more arms. Two arms may provide better support that one arm. Furthermore, two arms still appear as two arms in the aperture plane upon reflection from the sample, but three arms appear to become  6  arms. As such, two arms may be optimal for support and limiting the obscured area in some embodiments, although in other embodiments, other numbers of arms may be used. 
       FIG. 14  shows a cut away perspective view of an objective that uses another type of two-element mask, according to an example embodiment. As shown in  FIG. 14 , the objective includes a body  1402  that is generally similar to body  1202  of the objective of  FIG. 12 , and further includes a first mask portion  1404  (also referred to as a “first sub-mask” or a “first mask element”) and a second mask portion  1406  (also referred to as a “second sub-mask” or a “second mask element”). As shown in  FIG. 14 , the objective further includes first and second concave mirrored surfaces  404  and  406  (labeled as  1410  in  FIG. 14 ). 
     First mask portion  1404  is shaped the same and positioned similarly to second mask portion  1206  of  FIG. 12 , providing an annular obscuration in the channel in body  1402  (e.g., the second cylindrical channel portion) between the first opening and the second opening of first and second concave mirrored surfaces  404  and  1410 . 
     Second mask portion  1406  is separate from the first mask portion  1404 , and includes a central obscuration  1406  positioned on the axis of symmetry of body  1402  between the channel and the second end of body  1402  (e.g., within a space formed within second concave mirror surface  1410 ). Central obscuration  1406  is held in position by at least one arm  1408  (two arms are shown in  FIG. 14 ; a third arm is not visible) extending through a hole  1412  in second concave mirrored surface  1410 . 
     In an embodiment, the arms (e.g., arm  1408 ) may be flexed and then allowed to expand into place, to hold second mask portion  1406  in position. For instance, second mask portion  1406  may be made of a thin piece of spring steel that enables the arm(s)  1408  to flex before being extended through hole(s)  1412  and to return to an un-flexed shape after being extended through hole(s)  1412 . In another embodiment the arms of second mask portion  1406  may be angled and attached to the second end of body  1402 . 
     The position of second mask portion  1406  enables greater blocking of stray light rays while lessening a need to position the mask as accurately, relative to the two-element mask of  FIG. 12  and the single element mask of  FIG. 9 . For instance,  FIG. 15  shows a cross-sectional view of the objective of  FIG. 14  with first and second concave mirror surfaces  404  and  1410 , and showing light rays passing there through, according to an example embodiment. In an embodiment, first mask portion  1404  may be removed/not used, leaving second mask portion  1406  as a single element mask to block stray light rays, although maintaining both first and second mask portions  1404  and  1406  may be desired to enable a wider range of angles of stray light rays to be filtered than having second mask portion  1406  alone. For instance,  FIG. 16  shows a cross-sectional view of the objective of  FIGS. 14 and 15  with second mask portion  1406  present (first mask portion  1404  is not present), and showing light rays passing through the objective, according to an example embodiment. 
     First and second mask portions  1204  and  1206 , mask  804 , mask  810 , mask  1002 , and first and second mask portions  1404  and  1406  may each be made from one or more metals (e.g., an alloy) such as iron, steel (e.g., stainless steel, spring steel), sheet metal, aluminum, copper, brass, glass, a polymer, etc. A mask may include a first mask portion made from a substantially optically clear window with a substantially opaque obscuration, eliminating the need for support arms for the obscured area. 
     In embodiments, such as those described above with respect to  FIGS. 4-16 , objectives that have a solid body surrounding an interior space through which light passes are provided. In further embodiments, objectives that have a transparent solid body through which the light passes are provided. For instance,  FIG. 17  shows a side cross-sectional view of a solid optically clear objective, according to an example embodiment. In the embodiment of  FIG. 17 , first and second concave mirrored surfaces  404  and  406  defined by the disclosed equations are formed on a substantially optically clear material, and light rays traveled inside of the material of the objective. 
     As shown in  FIG. 17 , the objective includes a body  1702 , a first surface  1704 , a second surface  1706 , and an apex  1708 . Body  1702  is made of a transparent, optically clear material, such as glass, a transparent polymer, a transparent epoxy, a crystal, etc. Body  1702  is shaped in an hourglass shape, similar to the interior of the objective of  FIG. 4 . Body  1702  has a first (e.g., upper) portion that has first surface  1704  as a surrounding perimeter surface, has a second (e.g., middle portion) that has second surface  1706  has a perimeter surface, and a third (e.g., bottom portion) that has apex  1708  at a tip (at the second end of the objective, at the focal plane). The first portion of body  1702  has a flat surface at which light may be received by and/or transmitted from the objective, depending on the particular application. First surface  1704  is shaped according to the corresponding equations (e.g., Equations  2020 ,  2022 ,  2024 ) described above, and therefore may have first concave mirror surface  404  formed thereon. Second surface  1706  is shaped according to the corresponding equations (e.g., Equations  2026 ,  2028 ) described above, and therefore may have second concave mirror surface  406  formed thereon. The third portion of the objective is conical shaped, with the wide end of the cone coupled to the second portion of the objective, and the pointed end of the third portion, which may or may not be flattened, is where apex  1708  is located. Subjects to be sampled (e.g., viewed) may be positioned adjacent to apex  1708 , which is the focal point for the objective. In an embodiment, apex  1708  of the objective reflects light rays that enter the first portion (and pass through the second and third portions of the objective) by total internal reflection, which allows for Attenuated Total Reflection (ATR) sampling. 
       FIG. 18  shows the side cross-sectional view of the objective of  FIG. 17 , with light rays traveling through the objective, according to an example embodiment. As shown in  FIG. 18 , collimated light rays may enter the flat surface of the first portion of the objective, may pass through the first portion of the objective, reflecting off of first concave mirror surface  404  into the second portion of the objective, and reflecting off of second concave mirror surface  406  through the second portion into the third portion of the objective to apex  1708 . The light may reflect from apex  1708  back through the second portion, to be reflected from the second concave mirror surface  406  to the first portion of the objective, to be reflected from the first concave mirror surface  404  to collimated and transmitted from the flat surface at the first end of the objective. 
     In embodiments, a mask may be used with the objective of  FIG. 17  to filter light rays. For instance, in an embodiment, as shown in  FIG. 18 , a mask  1802  may be positioned on the flat surface at the first end of the objective to filter some light rays (e.g., collimated light rays close to the axis of symmetry and/or off-axis light rays) from entering the objective in a similar fashion as mask  502  of  FIG. 5 . For instance, mask  1802  may be a circular or disk-shaped obscuration (no arms or outer ring shaped portion needed to be present), or may have other shape, and may be made of a metal, an opaque polymer, an opaque filling, and/or other material disclosed herein or otherwise known. In another embodiment, surfaces  1704  and  1706  may not be mirrored, but may rely on the light rays reflecting by total internal reflection. If the angle with which the light ray strikes the surface is greater than the critical angle for the material the body is made from, the ray will reflect off of the surface with close to 100% efficiency. 
     In another example of a transparent objective with a mask, a two-piece transparent object may be used to include a mask in the transparent objective interior. For instance,  FIG. 19  shows a side cross-sectional view of a two-piece transparent objective, with a mask between the two pieces and with light rays traveling through the objective, according to an example embodiment. The objective has a body, a first surface  1904 , a second surface  1906 , and an apex  1914 . The objective of  FIG. 19  is similar to the objective of  FIG. 17 , having first-third portions, with first surface  1904  of the first portion shaped according to the corresponding equations described above, second surface  1906  of the second portion shaped according to the corresponding equations described above, and a conical third portion having apex  1914  at a tip (adjacent to a focal point in a focal plane/sample area). However, the body is separated into two pieces. The first and second portions form a first piece  1906  of body  1902 , and the third portion forms a second piece  1908  of body  1902 . First and second pieces  1906  and  1908  are made of a transparent material, as described elsewhere herein or otherwise known. A flat interface  1910  is present between the second portion and the third portion where surfaces of the first and second pieces may be coupled together. 
     Furthermore, a mask  1912  may be formed in the objective, as shown in  FIG. 19 . 
     Mask  1912  is positioned between the first and second pieces at flat interface  1910 . For instance, a depression may be formed in the surface of the first piece and/or the second piece at flat interface  1910  in which mask  1912  may reside. Mask  1912  may is positioned to filter some light rays passing through body  1902  in a similar fashion as second mask portion  1406  of  FIGS. 14-16 . For instance, mask  1912  may be a circular or disk-shaped obscuration (no arms or outer ring shaped portion needed to be present), or may have other shape, and may be made of a metal, an opaque polymer, an opaque filling, and/or other material disclosed herein or otherwise known. 
     As such, various embodiments for objectives, including single piece objectives, multi-piece objectives, hollow objectives, transparent objectives, adjustable objectives, and objectives with or without masks. Such objectives may be used in various applications to provide for focusing, magnifying, and filtering of light. 
     For instance, Attenuated Total Reflection (ATR) is a spectroscopic technique that requires the light rays to reflect off of an interface between two materials at an angle greater than the critical angle. In an embodiment, a low cost, monolithic ATR microscope objective could be made using the equations described herein rather than the spherically approximated microscope objective available today. Both ends of the objective may be flat, and as the input beam would be collimated the flat ends would not add spherical or coma. Distortion may increase, as well as chromatic aberration, but this should be minimal due to the small maximum input angle. 
     In an embodiment, a dome shaped ATR crystal may be placed at the focus to achieve ATR microscopy as well. The dome may be part of the objective, or may be removable to allow a two mode objective (e.g., a normal mode and an ATR mode). 
     In an embodiment, a microscope system that includes three identically configured objectives (used as objective, condenser, and detector optics) using symmetry may minimize optical aberrations. 
     Embodiments of objectives may be used in space born application, as the objectives do not change due to vibration. This is an advantage in a high-vibration environment, such as in military analytical instrumentation on the battlefield. 
     In an embodiment, a “grazing angle” microscope objective can be formed by making the aperture large enough that the light rays form a maximum angle of 85 degrees with the sample. 
     A larger entrance aperture leads to more energy through the objective. This may be important to scientists who measure chemical composition using an FTIR (Fourier transform infrared spectroscopy) microscope because more throughput enables a higher signal to noise ratio, and a higher signal to noise ratio enables smaller chemical quantities to be detected. 
     Embodiments enable a narrower microscope objective, which enables more objectives to fit on a rotary turret. 
     Embodiments enable a monolithic design, which enables objectives to be more rugged, durable, and less likely to fall out of an aligned state. 
     A monolithic design also enables improved thermal stability because the entire objective may be made of one material rather than multiple materials (glass-aluminum or steel-brass), which may have different coefficients of thermal expansion (CTE). 
     In embodiments, masks may be used to block out light rays that would otherwise hit the reflective sample surface and bounce back or would transmit through both objectives in an objective/condenser pair. An example of such a mask embodiment is an annular mask between the first and second concave mirror surfaces, and a smaller round mask in front of it (e.g., as shown in  FIGS. 12 and 13 ). Such a mask provides a maximum blocking of stray light rays while reducing vignetting at field angles greater than 0 degrees. 
     An example advantage of embodiments is that a relatively large amount of energy that can be transferred through the objectives when compared to a comparable spherically approximated Schwarzschild objective. 
     Another example advantage is that some objective embodiments may be manufactured from a single piece of material. This provides a cost savings because current conventional implementations require alignment of the primary and secondary mirrors, which is a labor intensive process. In embodiments where both the primary and secondary mirrors are part of the same element, there is no need to align them relative to each other. 
     Another example advantage is that monolithic fabrication enables a more robust objective that is capable of withstanding greater vibratory forces that a traditional Schwarzschild design. 
     Furthermore, computerized light ray tracing and optimization is enabled because the surface slopes and normal vectors can be calculated precisely, unlike in the numerical techniques. The second derivative may be taken, which yields the instantaneous curvature of the concave mirror surface at that point in space. A numerical method requires two calculated points to determine the slope, and requires three calculated points to determine the curvature. Because each point is itself an approximation and the slope obtained would also be an approximation, the use of the shown parametric equations provides a great advantage. The advantage extends beyond specifying the surface accurately: It increases the accuracy of any optical models used in the design process which require accurate slopes, normals, and curvatures. 
     Still further, in embodiments, because the profile of the objective is narrower than that of traditional reflecting objectives, more objectives can fit on a nosepiece turret. Currently only two of the traditional objectives fit on a turret, with any more causing interference between the outer diameters. 
     The equations disclosed herein to determine the shape of the first and second concave mirrored surfaces give exact values for the points that define their surfaces. The other techniques that are available are either approximations that are not as accurate or use standard conic surfaces that do not correct for coma except in certain limited cases. By using the equations disclosed herein, not only can the points of the concave mirror surfaces be determined accurately, but also the slopes and normals to those points, which enables accurate trace light rays to be generated. 
     In some cases, the two-element mask of  FIG. 12  may be better than the one stage mask (of  FIG. 5 ) at blocking stray light rays from traveling through the objective because all of concave mirror surface  404  is enabled to be used. When a single element mask is used, the mask may be made larger because light rays entering the objective at an angle should be blocked. As such, the mask should be larger than the first opening in the first concave mirror surface. The first mask portion for the two-element mask can be substantially smaller than the first opening in first concave mirror surface  404 , and therefore block out fewer of the usable light rays. 
     The two-element mask of  FIG. 14  has an advantage of better stray light blocking while at the same time allowing for less accurate placement. It also helps to protect the inner surface of the objective. 
     The single element mask of  FIG. 16  enables adequate blocking of stray light rays by placing a centrally obscured mask after first concave mirror surface  404 . 
     By separating the first and second concave mirror surfaces  404  and  406  is different pieces of an objective (e.g., as in  FIGS. 10 and 11 ), compensation for the blurring of the image created when the microscope sample is placed on or sandwiched between two optically clear windows is enabled. The ability to eliminate this blurring by adding an amount of spherical aberration equal and opposite to the spherical aberration induced by the window is a substantial advantage over a fixed system. 
     Conclusion 
     While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not limitation. It will be apparent to persons skilled in the relevant art that various changes in form and detail can be made therein without departing from the spirit and scope of the invention. Thus, the breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.