Abstract:
An interferometer and method for interferometric analysis are provided. The methodology includes generating first and second light beams from a light source, interacting the first light beam with an object under inspection, forming, from light emanating from the object under inspection in response to the interacting, an image of the light source on an image sensor, projecting the second light beam on the image on the image sensor, the combination of the light emanating from the object under inspection and the second light beam forming a collective image on the image sensor, applying a Fourier transform to the collective image formed on the image sensor, thereby forming a phase image, and isolating a wavefront map of the object under inspection from within the phase image.

Description:
CROSS-REFERENCE TO RELATED APPLICATION  
       [0001]    The benefits of Provisional Application No. 60/935,102 filed Jul. 26, 2007 are claimed under 35 U.S.C. §119(e), and the entire contents of this application are expressly incorporated herein by reference thereto. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates to a technique for analyzing the optical characteristics of an object under consideration. More specifically, the present invention relates to interferometer analysis. 
         [0004]    2. Discussion of Background Information 
         [0005]    Interferometers are used to measure optical characteristics (such as the contours and depth of a surface) of objects such as mirrors and lenses that transmit and/or reflect light. Interferometers typically examine wave fronts of light reflected by or transmitted from an object to generate wave front maps of the object under inspection. One class of interferometers combines an image of the object under inspection with a spatial heterodyne beam to map “fringes” onto the object under inspection and retrieving wavefronts using a Fourier filtering processes. This process is known as SHIFT. 
         [0006]    The fundamentals of SHIFT based interferometry is shown in  FIG. 1 , in which an incoming light beam  110  from a point source  105  and collimating lens  107  is incident on an object under inspection  115 , in this case a mouse-shaped mirror. The reflected light beam  120  is made incident (either by the object  115  under inspection alone or with additional optics such as focusing lens  130 ) on an imaging sensor  125 , such as a CCD. As shown in the rotated view of the imaging sensor  125 , an image of object under inspection  115  is formed on the surface of the imaging device. This image will be combined with an angular heterodyne beam  150  for subsequent interferometric analysis. 
         [0007]    Typical monochromatic or snapshot interferometers have no reference or zero plane, and thus rely upon relative calculations as opposed to absolute calculations. Thus, such an interferometer will be able to conduct a relative measurement of two adjacent points on a surface to detect a discontinuity. However, because there is no reference plane, the interferometer cannot detect in which direction the discontinuity leads. For example, if the object under inspection had the shape of upward stairs, the interferometer could recognize that each step represented a discontinuity relative to adjacent steps, but would not know if any particular step was an upward step or a downward step. 
         [0008]    To overcome this problem, so-called phase shifting interferometers were developed. These interferometers would examine an object under inspection from several different vantage points, often referred to as a push/pull process. At each vantage point, the discontinuity in the object under inspection would present a different wave front to the interferometer. By analyzing the different wave fronts from the different vantage points, the phase shifting interferometers could effectively identify both the discontinuity and its direction. A drawback of these systems, however, was that each of the additional measurements (taken at different points in time) was individually subject to the effects of ambient noise (e.g., vibrations) that would be different from one measurement to the next. 
         [0009]    Efforts have been made to overcome the above drawbacks by creating a hologram that allows for the various measurements to be taken at the same time but at different points in space. The multiple optical observations are then performed electronically from the hologram without the injection of outside temporal noise. However, the examination and analysis is all based on the image of the object under inspection. Also, even though only one snapshot is taken of the object under inspection, the analysis still requires examination of that snapshot from four physical spaces. 
         [0010]    Collections of objects that imitate a larger unitary object present more complex obstacles. One known example of this is the James Webb telescope under construction for orbital deployment to supplement the Hubble program. The diameter of the reflective mirror for this telescope is several times larger than that used by Hubble, thus providing it with much greater capabilities than Hubble. Since a single mirror of this size cannot be launched into space with current space platforms, the mirror is being constructed of smaller abutting hexagonal mirrors that will collectively operate akin to a single large mirror. In such a system, accuracy of alignment of the discrete elements is paramount, but presents challenges. 
         [0011]    Specifically, interferometric measurements generate fringe patterns on the object under inspection.  FIG. 3A  shows such fringe patterns on a hexagonal object under inspection. However, when dealing with adjacent elements, the fringe patterns may be out of alignment such as shown in  FIG. 3B . It is unclear whether fringe line  310  aligns with any of fringe lines  320 ,  330 ,  340 , etc. Known methods for addressing this ambiguity essentially rely upon application of an independent algorithm to the analysis, but this algorithm produces only a “best guess” with questionable accuracy absent providing additional light sources and inducing resulting complications to the instrument. 
       SUMMARY OF THE INVENTION 
       [0012]    According to an embodiment of the invention, a method for interferometric analysis is provided. The method includes generating first and second light beams from a light source; interacting the first light beam with an object under inspection; forming, from light emanating from the object under inspection in response to the interacting, an image of the light source on an image sensor; projecting the second light beam on the image on the image sensor, the combination of the light emanating from the object under inspection and the second light beam forming a collective image on the image sensor; applying a Fourier transform to the collective image formed on the image sensor, thereby forming a phase image; and isolating a wavefront map of the object under inspection from within the phase image. 
         [0013]    The above embodiment may have various features. The second light beam may be at an angle to the first light beam. The angle is preferably sufficiently large such that, within the phase image, the wavefront map of the image does not overlap with any other wavefront image, and sufficiently small such that, within the phase image, the entire wavefront map of the image is within the phase image. The angle is preferably such that, for a diameter of a circle that encloses the object under inspection in pupil space, then the source of the heterodyne beam is positioned in pupil space at a distance of 1.5 diameters from the center of the diameter of the object under inspection. There may be a step of converting, between the generating and the projecting, the second light beam into at least one heterodyne light beam, such that the projecting comprises projecting the heterodyne light beam on the image on the image sensor. The at least one heterodyne beam may be at an angle to the first light beam. There may be a step of determining whether the captured image has sufficient coherency and fringe contract to differentiate fringe patterns from ambient noise. The image of the light source on the image sensor may account for any modifications due to at least intervening optics between the light source and the image sensor. 
         [0014]    According to another embodiment of the invention, a method for interferometric analysis is provided. The method includes generating first and second light beams from a light source, interacting the first light beam with an object under inspection, forming, from light emanating from the object under inspection in response to the interacting, an image of the light source on an image sensor, converting, between the generating and the projecting, the second light beam into a plurality of heterodyne light beams, projecting the heterodyne light beams on the image on the image sensor, the combination of the light emanating from the object under inspection and the heterodyne light beams forming a collective image on the image sensor, applying a Fourier transform to the collective image formed on the image sensor, thereby forming a phase image containing at least a plurality of wavefront maps of the object under inspection, isolating a plurality of wavefront maps of the object under inspection from within the phase image, and generating a final wavefront map of the object under inspection based on the plurality of wavefront maps. 
         [0015]    The above embodiment may have various optional features. Each of the heterodyne light beams may be at an angle to each other and to the first light beam. The angle may be sufficiently large such that, within the phase image, the wavefront map of the image does not overlap with any other wavefront image and sufficiently small such that, within the phase image, the entire wavefront map of the image is within the phase image. The method may include a step of determining whether the captured image has sufficient coherency and fringe contract to differentiate fringe patterns from ambient noise. The image of the light source on the image sensor may account for any modifications due to at least intervening optics between the light source and the image sensor. 
         [0016]    According to still another embodiment of the invention, an interferometer system configured to generate a wavefront map of an object is provided. The system includes a light source, an image sensor, and an optical system. The optical system is configured to direct a first light beam from the light source onto the object, direct light interacting with the object on the image sensor, convert a second light beam from the light source into at least one heterodyne beam, and direct the at least one heterodyne beam onto the image sensor. The image sensor is disposed in the plane in which an image of the light source as modified by the optical system and the object will be in its best focus preferably within plus or minus one depth of focus. The image of the light source and the heterodyne beams form a collective image on the image sensor. 
         [0017]    The above embodiment may have a combination of software and hardware configured to apply a Fourier transform to the collective image formed on the image sensor, thereby forming a phase image, and isolate a wavefront map of the object under inspection from within the phase image. The combination of software and hardware may also be configured to determine whether the captured image has sufficient coherency and fringe contract to differentiate fringe patterns from ambient noise. 
         [0018]    According to yet another embodiment of the invention, an interferometer system configured to generate a wavefront map of an object is provided. The system includes a light source, an image sensor, and an optical system. The optical system is configured to direct a first light beam from the light source onto the object, direct light interacting with the object on the image sensor, convert a second light beam from the light source into a plurality of heterodyne beams, and direct the plurality of heterodyne beams onto the image sensor. The image sensor is disposed in the plane in which an image of the light source as modified by the optical system and the object will be in its best focus preferably within plus or minus one depth of focus. The image of the light source and the heterodyne beams form a collective image on the image sensor. 
         [0019]    The above embodiment may have a combination of software and hardware configured to apply a Fourier transform to the collective image formed on the image sensor, thereby forming a phase image containing at least a plurality of wavefront maps of the object under inspection, isolate a plurality of wavefront maps of the object under inspection from within the phase image, and generate a final wavefront map of the object under inspection based on the plurality of wavefront maps. The combination of software and hardware may also be configured to determine whether the captured image has sufficient coherency and fringe contract to differentiate fringe patterns from ambient noise. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0020]    The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
           [0021]    The present invention is further described in the detailed description which follows, in reference to the noted plurality of drawings by way of non-limiting examples of certain embodiments of the present invention, in which like numerals represent like elements throughout the several views of the drawings, and wherein: 
           [0022]      FIG. 1  illustrates an overview of interferometric methodology of the prior art; 
           [0023]      FIGS. 2A and 2B  illustrate an overview of interferometric methodology of an embodiment of the invention; 
           [0024]      FIGS. 3A and 3B  show uniform and non-uniform wave front maps of a hexagonal object under inspection; 
           [0025]      FIGS. 4A-4D  show the images formed on the image sensor of the prior art and the embodiments of  FIG. 2A , respectively; and 
           [0026]      FIG. 5  is a flowchart showing the processing steps of the image formed on the image sensor in an embodiment of the invention. 
           [0027]      FIGS. 6A and 6B  show a phase image and a wave front map (phase data omitted for clarity) respectively based on the presence of a single heterodyne beam. 
           [0028]      FIG. 6C  shows a phase image (phase data omitted for clarity) based on the presence of two heterodyne beams. 
           [0029]      FIG. 7  shows a preferred mathematical relationship between the parameters of the test beam and image sensor. 
           [0030]      FIG. 8  shows a preferred spatial relationship in pupil space between the image of the object under inspection and the source of the heterodyne beam. 
           [0031]      FIG. 9A  shows an image of the object under inspection in pupil space. 
           [0032]      FIG. 9B  shows as example of the diffraction limited image (in logarithmic scale) formed by the mask in  FIG. 9A . 
           [0033]      FIG. 9C  shows the OTF mathematically stretched in order to highlight the silhouette of the OTF location. 
           [0034]      FIGS. 10A-10C  show the effect of the addition of a heterodyne beam to the views shown in  FIGS. 9A-9C . 
           [0035]      FIG. 11  shows an embodiment of an interferometer according to the invention. 
           [0036]      FIG. 12A  shows a test beam pathway in the embodiment of  FIG. 11 . 
           [0037]      FIGS. 12B and 12C  shows heterodyne beam pathways in the embodiment of  FIG. 11 . 
           [0038]      FIG. 13  shows another embodiment of an interferometer according to the invention. 
           [0039]      FIGS. 14A-14E  show various views of an interferometric analysis of a circular mirror. 
           [0040]      FIG. 15  is a blow up of  FIG. 14C . 
           [0041]      FIG. 16  shows another embodiment of an interferometer according to the invention. 
       
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
       [0042]    The particulars shown herein are by way of example and for purposes of illustrative discussion of the embodiments of the present invention only and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the present invention. In this regard, no attempt is made to show structural details of the present invention in more detail than is necessary for the fundamental understanding of the present invention, the description taken with the drawings making apparent to those skilled in the art how the several forms of the present invention may be embodied in practice. 
         [0043]      FIGS. 2A and 2B  shows an embodiment of an interferometer  200  for interferometric analysis of an object under inspection  205 . (“Object” as used herein refers to both the singular individual objects and the plural, such as collections of objects.) A point light source  210  emanates spherical waves of light  215  toward a collimating lens  220 , which converts light  215  into planar (flat) waves  225 . Planar light waves  225  then interact (transmits and/or reflects with object under inspection  205 ) before the resulting light waves  230  move toward image sensor  240 . Object under inspection  205  may have any shape, and be reflective, transparent and/or translucent. In  FIGS. 2A and 2B , object under inspection  205  is a mouse-shaped mirror for illustrative purposes, such that the interaction would be reflection and the resulting light  230  would be reflected light. 
         [0044]    The resulting light  230  is ultimately incident on image sensor  240 . Image sensor  240  is positioned in the image plane of the interferometer  200 , i.e., the plane in which an image of light source  210  as modified by the intervening optical elements and the object under inspection  205  will be in its best focus preferably within plus or minus one depth of focus. Object under inspection  205  may be able to form such an image on its own (e.g., a concave mirror), but most likely additional optics  235  upstream and/or downstream of the object under inspection  205  will be needed (only downstream elements are shown in  FIGS. 2A and 2B ). The resulting image  245  represents the light point source  210  as modified by object under inspection  205  and the optics of the interferometer. 
         [0045]    The resulting image  245  formed on image sensor  240  represents a significant difference between the above embodiment and the prior art. As noted above, prior art systems rely upon an image of the object under inspection, whether in two dimensions on an image sensor or in a three-dimensional hologram. In the noted embodiment, it is image  245  of the point light source  210  which will be processed for interferometric analysis. 
         [0046]    The distinction is illustrated in  FIGS. 4A and 4B .  FIG. 4A  shows the image that a prior art system would form on the image sensor for a mouse-shaped mirror.  FIG. 4B  shows how the embodiment of  FIG. 2  forms a different image of light source  210  for the same mouse-shaped mirror. (Image  4 B is effectively equal to the square of the absolute value of the Fourier transform of the image in  4 A.)  FIGS. 4C and 4D  show how these formed images combine on the image sensor  240  with the heterodyne beam  250 . 
         [0047]    Referring now to  FIG. 2B  (in which certain elements of  FIG. 2A  have been removed for clarity), a heterodyne light beam  250  is made incident on image sensor  240 . The size and shape of beam heterodyne light beam  250  preferably covers the entire sensor  240 , although in theory smaller beams could be used provided that it is supported by optical sampling. Light beams  230  and  250  arrive coherently so that they properly interfere. The angle of heterodyne light beam  250  is based on a variety of factors, discussed more fully below. 
         [0048]    Referring now to  FIG. 5 , at step  510 , the system captures a snapshot of the incoming light beams  230  and  250  as incident on image sensor  240 . At step  515 , the resulting captured image is tested to determine whether a well-formed image is formed. A well-formed image is an image with a high fringe contrast due to beams  230  and  250  arriving coherently. The level of coherency and fringe contrast need only be to a degree sufficient for interferometer  200  to effectively differentiate the fringe patterns from ambient noise, which differs from interferometer to interferometer. If not, at step  517  the system parameters are adjusted (e.g., adjusting the relative intensity or optical path difference of light  225  and  250 ) and control returns to steps  510  for another snapshot. 
         [0049]    At step  520 , the captured image is subject to a two (2) dimensional Fourier transform. Each pixel (or other subpoint of data for the image) of the original image will therefore generate a corresponding real and imaginary number; the collection of all real numbers forms a real image, and the collection of all the imaginary numbers forms a imaginary image. It is not necessary to display the resulting real or imaginary images, but it may be desirable for diagnostic or other test purposes. 
         [0050]    At step  525 , a phase image is formed by processing the real and imaginary number for each pixel of the transformed image per the formula acrtan (i/r). Several available computer products such as MATLAB are already programmed with this functionality and can be used. 
         [0051]    Referring now to  FIGS. 6A-6B , within the resulting phase image  610  will be found a wave front map  620  of the object  205  under inspection, its complex conjugate  630 , and a silhouette of the optical transfer function (“OTF”)  640  of the optic under inspection  205 . For ease of discussion, the phase date is omitted, only showing the images in black and white. The wave front map  620  is the information of interest. At step  530 , the wave front map is at least partially isolated from the remainder of phase image  610  and displayed at step  535 . This can be as simple as zooming in on the outline or using known software to extract/copy the image of interest from the background, such as shown in  FIG. 6B . The wave front map  620  will be a highly accurate wave front map of the object under inspection  205  to within one wavelength of the incident light. Applicants refer to this process as “OSHIFT.” 
         [0052]    To create the above phase image  610 , the heterodyne beam preferably is incident at either a specific angle or range of angles. Specifically, the position of wave front map  620  relative to OTF  640  is dependent upon the angle of heterodyne beam, in that the two will be closer together for a smaller angle and farther apart for a larger angle. The minimum angle is the angle at which the map  620  would be adjacent to OTF  640  without overlap; if the angle were any smaller, map  620  and OTF  640  would overlap with corrupted results in the overlap region. The maximum angle is the angle at which the map  620  would be adjacent to the outer edge of phase image  610 ; if the angle were any larger, part of map  620  would be cut off and the desired data therein would be lost. 
         [0053]    More particularly, if a “pupil width” is defined by the diameter of a circle that encloses the object under inspection  205  in pupil space, then the source of the heterodyne beam is positioned in pupil space at a distance of 1.5 pupil widths from the center of the pupil width of the object  205  under inspection. This can be seen for example in  FIG. 8A , in which the circle  810  encircles object under inspection  205 , and the source of the heterodyne beam  250  is a distance of 1.5 pupil widths from the center of circle  810 . The reason is that the object  205  under inspection will create an OTF that is two pupil widths in diameter, while the heterodyne beam itself will generate a copy of the object under inspection  205  that is one pupil width in diameter. Placing the source of heterodyne beam  250  at the noted angle will ensure no overlap between the two. The mathematical basis for the same is discussed in more detail below. 
         [0054]    Multiple heterodyne beams  250  may be made incident on the image sensor  240  from different points of origin. The processing is the same as in  FIG. 5 , save that the resulting plane image at step  525  will include a different outline/conjugate pair  620 / 630  for each heterodyne beam  250 . At step  530 , the multiple  620  of the object under inspection  205  and the contents within are at least partially isolated from the remainder of the phase image  610 . The contents of each outline  720  is then combined through known techniques (e.g., averaging, weighted averaging, discarding the worst image, etc.) which reduces the potential effects noise and aids in phase unwrapping on any individual image. The resulting combined wave front map will be a highly accurate map of the object under inspection  205  to within one wavelength of the incident light  225 .  FIG. 6C  shows the phase image  610  for a circular mirror with two heterodyne beams (again, phase data is omitted for clarity). 
         [0055]    The incoming angle of any heterodyne beam must comply with the specific beam requirements as set forth above. When using multiple beams, the incoming beam angles must also be set such that none of the maps  620  within plane image  610  overlaps, else the contents would provide corrupted data. To avoid this overlap, any incoming heterodyne beams  250  must not overlap in autocorrelation space with any other heterodyne beam or its conjugate. 
         [0056]    As the variations of the object under inspection  205  will be on the order of nanometers, use of any color light for heterodyne beam  250  would be appropriate. However, if for any reason the variations would be large enough to challenge the accuracy range at lower wavelengths, then light with higher wavelengths could be used. Also, the range of accuracy could be increased beyond one wavelength using known techniques for applying multiple heterodyne beams of different colors. All heterodyne beams  250  and the point light source  210  preferably originate from a common light source. 
         [0057]    The processing disclosed above to capture of an image on the image sensor and produce the resulting wavefront map is performed by appropriate software stored on a computing platform, such as the hard drive of a stand alone computer, computer network and/or the Internet. The architecture of the platform and/or software is not critical, and all implantations are contemplated by the invention: 
         [0058]    The following sections address the mathematical basis for the overview of the embodiments as set forth above. The following assumptions and conventions will be used:
       It is assumed that the optic under inspection  205  is a single surface conjugate to the exit pupil which can impart a phase screen;   Pupil and autocorrelation space are coordinates (u,v);   A phase screen in the complex pupil is described by θ(u,v). This phase screen is the article of interest;   Image space (image sensor  250 ) coordinates (x,y);   Phase in image space φ(x,y);   The symbol for complex conjugation is *;   The flip that occurs in coherent transfer (pupil) space is ignored for mathematical clarity;   Going from the pupil plane to the image plane is approximated by a forward transform           or F{·};   Going from the image to the pupil is approximated by an inverse transform (         ) or F −1 {·}   The symbol for a Fourier transform pair is          ;
 
Assuming unit illumination of the optic under inspection the complex pupil of interest is defined as
       
 
         [0000]      Pupil( u,v )=Mask( u,v ) e   i2πθ(u,v)    (1) 
         [0000]    where Mask(u,v) is the binary (or shaded) real amplitude mask in pupil space and θ(u,v) is the phase screen applied to the pupil. For example, continuing with the mouse-shaped object under inspection  205 , a mouse-shaped-shaped mask  910  is present in pupil space shown in  FIG. 9A . The amplitude image formed by the test beam is then: 
         [0000]        F {Pupil( u,v )}= A   2 ( x,y ) e   i2πφ(x,y) .   (2) 
         [0000]      FIG. 9B  shows as example of the diffraction limited image (in logarithmic scale) formed by the mask in  FIG. 9A .  FIG. 9C  shows the OTF mathematically stretched in order to highlight the silhouette of the OTF  920  location. 
         [0069]    In image space the heterodyning beam is a plane wave with amplitude A 1  and due to the angle of arrival a linear phase. The Fourier transform of a tilted plane wave results is a displaced delta function in pupil space: 
         [0000]        A   1 ∂( u±u   o   ,v )           A   1   e   Fi2πxu     o   .   (3) 
         [0070]    The separation u o  in pupil space must be enough to avoid interference in the data. The addition of the heterodyning beam (δ-fcn) in pupil space is shown in  FIG. 10A-C  (where the delta function is exaggerated for clarity).  FIGS. 10A-C , A 1  and A 2  were set so that both “apertures” transmit equal amounts of light.  FIG. 10A  is an example of a binary mask  1010  with a single heterodyne beam  1020  (exaggerated for clarity) representing the heterodyning beam  250  in the pupil plane.  FIG. 10B  is the logarithmic stretch of the diffraction limited PSF due to that mask and δ-fcn.  FIG. 10C  shows the OTF mathematically scaled to enhance the silhouettes. In addition to the OTF  1030 , a silhouette of the exact pupil  1040  and a silhouette of its complex conjugate  1050  now appear as a result of the heterodyning beam  250 . 
         [0071]    The PSF resulting from the mask in  FIG. 9B  is similar in shape to that in  FIG. 10B , but now displays fringing properties (although the fringes are so close together for the scale of  FIG. 10B  that they form a mostly uniform background). The difference is far more distinct in  FIGS. 9C  v.  10 C, where the presence of heterodyne beam  250  generates an exact copy of the complex pupil to appear as a side lobe in the OTF. This can be seen as the silhouette of the mouse-shaped in pupil  1040 . The side lobe associated with the silhouette of the complex conjugate  1050  of the complex pupil is also seen. 
         [0072]    The embodiment of the present invention may be understood by considering the autocorrelation of  FIG. 10A . It is illustrative to envision autocorrelation as the convolution of  FIG. 10A  with its complex conjugate. In this manner it is noted that when the δ-fcn is sufficiently separated from the pupil mask, the convolution of the complex pupil and the delta function equals exactly the complex pupil. The convolution of the complex pupil with its conjugate gives the traditional OTF. Finally the complex conjugate δ-fcn convolves with the pupil yielding an exact copy of the conjugate of the complex pupil. There is no new information in the conjugate side lobe; other than a sign change and dimension flip in phase space it is exactly equal to the other side lobe. 
         [0073]    Continuing with the mathematical derivation, the full description of the “OSHIFT” complex pupil is the addition of equations (1) &amp; (3), 
         [0000]        O SHIFT_Pupil( u,v )= A   1 ∂( u−u   o   ,v )+Mask( u,v ) e   i2πθ(u,v)    (4) 
         [0000]    Given the Fourier pairs in equations (2) &amp; (3) the intensity image is 
         [0000]        I ( x,y )=| A   1   e   i2πxu     o     +A   2 ( x,y ) e   i2πφ(x,y) | 2    (5a) 
         [0000]        I ( x,y )=| A   1 | 2   +|A   2 ( x,y )| 2   +A   1   A   2 ( x,y )( e   i2π(φ(x,y)−xu     o     )   +e   −i2π(φ(x,y)−xu     o     ) ).   (5b) 
         [0074]    The inverse Fourier transform of the intensity image is 
         [0000]        F   −1   {I ( x,y )}= F   −1   {|A   1 | 2   }+F   −1   {|A   2 ( x,y )| 2   }+F   −1   {A   1   A   2 ( x,y )( e   i2π(φ(x,y)−xu     o     )   +e   −i2π(φ(x,y)−xu     o     ) )}.   (6) 
         [0075]    The first two Fourier terms contribute, respectively, to the traditional OTF and a DC term associated with the δ-fcn intensity (the center image of  FIG. 10C ). The last Fourier term represents the OSHIFT side lobes resulting from the heterodyning beam. Given sufficient separation of the side lobes, these last terms do not interfere with the traditional OTF and are the terms of interest. Dropping the constant A 1 , the transform of the side lobes may be re-written using equations (2) &amp; (3) 
         [0000]        F   −1   {A   2 ( x,y )( e   i2π(φ(x,y)−xu     o     )   +e   −i2π(φ(x,y)−xu     o     ) )}=Pupil( u−u   o   ,v )+Pupil*( u+u   o   ,v )   (7) 
         [0000]    The salient property of the mathematics is that an exact copy of the complex pupil (and its conjugate) appears in the Fourier transform of a single OSHIFT intensity image. Furthermore the information of interest, the wave front induced by a phase screen at the pupil, may be found by masking the region of interest and taking the arctangent 
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         [0076]    Referring now again to  FIGS. 7 and 8 , assuming that the image sensor has a resolution of d p  (e.g., a pixel pitch of a CCD, or granularity of film, etc.) the test beam has a wavelength of λ and converges at an f/# then the optical sampling factor is: 
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                           f 
                           / 
                           # 
                         
                          
                         λ 
                       
                       
                         d 
                         p 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0077]    It is further noted that the digital pupil width of a Q=M system is N/M pixels. 
         [0078]      FIG. 8  combines scaling features contained in  FIGS. 9A-C  and  10 A-C. In order to fit the OSHIFT side lobes within the digital Fourier transform of a captured image, the pupil is preferably ¼ of the total image size and the delta function is preferably displaced 1½ pupil widths from center. This corresponds to a digital sampling of Q= 4  and angle of the heterodyning beam of 1.5/(f/#) as shown in  FIG. 7 . 
         [0079]    If the OSHIFT intensity image is N×N pixels, its digital Fourier transform is also N×N pixels.  FIG. 8A  examines spacing issues shown in  FIGS. 9A-C  and  10 A-C. The small circle  810  circumscribes the pupil whereas the larger circle  820 —twice the diameter of the circle  810 —circumscribes the OTF. The delta function (heterodyne beam  250 ), which occurs at the center of the rightmost silhouette, is shown as circle  830 . This image may then be used to derive preferable but non-limiting “rules of thumb” to separate the OSHIFT side lobes:
       The delta function must be at least 1½ pupil widths from the center;   N pixels must contain at least 4 pupil widths.       
 
         [0082]    This second criteria above along with the statement below equation (9) says that a good rule of thumb is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Q 
                       TestBeam 
                     
                     = 
                     4 
                   
                    
                   
                     
 
                   
                    
                   
                     Thus 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     / 
                     
                       # 
                       TestBeam 
                     
                   
                   = 
                   
                     4 
                      
                     
                       
                         
                           d 
                           p 
                         
                         λ 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0083]    The full angle associated with the marginal rays of the test beam is: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     
                       marginal_rays 
                        
                       _of 
                        
                       _TestBeam 
                     
                   
                   ≈ 
                   
                     
                       1 
                       
                         f 
                         / 
                         # 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0084]    Since the angle in image space varies linearly with separation in pupil space; the heterodyne beam angle relative to the chief ray of the test beam is then: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     heterodyne 
                   
                   ≈ 
                   
                     
                       1.5 
                       
                         f 
                         / 
                         # 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0085]    Applicants note that the side lobes can be “pushed” into the corners of the image to allow for some latitude in the design or a smaller pupil size may be used. 
         [0086]    Referring now to  FIG. 11 , an embodiment of an interferometer  1100  is shown. In this embodiment, interferometer  1100  utilizes two heterodyning beams. As discussed above, this will create two wavefront map/conjugate pairs in the phase image, which provides for data averaging and reduces phase unwrapping. The object under inspection  1105  is in this case a deformable mirror (DM). 
         [0087]    Since the interferometer  1100  may be far from the optic under inspection  1105 , a long coherence length laser  1102  (˜10 m) is utilized. Laser light from laser  1102  passes through a microscope objective and pin hole  1104 , thus forming a point of light source  1110 . Spherical waves of light pass through a culminating lens  1120  and are converted into planar (flat) waves  1125  with an appropriate beam size to interrogate object under inspection  1105 . Fold mirrors  1126  and  1128  are provided for space considerations, as they permit the embodiment to be mounted entirely inside a box that can be conveniently placed on an optical table. 
         [0088]    Light beam  1125  then passes through a λ/2 plate  1130  before reaching a polarizing beam splitter (PBS)  1132 . Both of these provide variation in the intensity ratio between the test and heterodyning beams (see step  520  above). After PBS  1132  the beam paths separate, independently interacting with an ND filter  1134 , focusing lens  1135 , beams splitters  1136  and  1138 , image sensor  1140  (shown as a CCD), and first and second heterodyne beam tilt mirrors  1142  and  1144 , and fold mirror  1146 . The function of each is best described in the individual light pathways as set forth in  FIGS. 12A-C . 
         [0089]      FIG. 12A  illustrates the transformation of part of light beam  1125  into a test beam for image sensor  1140 . PBS  1132  partially reflects beam  1125  toward the object under inspection  1105 , (in this case a reflective surface) passing through ND filter  1134  before passing through PBS  1132 . Fold mirror  1146  reflects the beam in the direction of focusing lens  1135 , which is designed to form a cone according to equation  11  above to create the image plane on image sensor  1140 . The beam passes through 50/50 beam splitter  1138 , where it is joined by two incoming heterodyne beams and incident on image sensor  1140 . 
         [0090]      FIGS. 12B and 12C  illustrate the creation of the two heterodyne beams. PBS  1132  partially transmits beam  1125  toward 50/50 beam splitter  1136 . Beam splitter  1136  reflects 50% of light toward heterodyne beam  1  tilt mirror  1142 , while the remaining light transmits toward heterodyne beam  2  tilt mirror  1144 . (While mirrors  1142  and  1144  are shown in perpendicular alignment, they are actually at angles appropriate to properly position the heterodyne beams relative to image sensor  1140 .) The heterodyne beams are created by the mirrors, and are recombined and directed to image sensor  1140  by beam splitters  1136  and  1138 . 
         [0091]      FIGS. 14A-14E  show how the above interferometer  110  scans an object under inspection. In this non-limiting example, the object under inspection is a circular mirror  1105  as shown in  FIG. 14A . For reference,  FIG. 14B  shows object  1105  and two heterodyne beams in pupil space, (although as discussed above the embodiment does not utilize that image).  FIG. 14C  shows the resulting image plane formed on image sensor  1140  (zoomed in for clarity in  FIG. 15 ) for which interferometer  1110  takes a snapshot as discussed in step  510  above. 
         [0092]    After image processing, the resulting phase image is shown in  FIG. 14D . As discussed above, within the phase image are two (2) wave front maps  1420  of the object  1405  under inspection, two (2) complex conjugates  1430 , and an phase transfer function (“FTF”)  1440  (the phase of the OTF discussed previously). The two wave front maps can be combined to produce a final wave front map  1450  in  FIG. 14E . 
         [0093]      FIG. 13  shows another embodiment of an interferometer, in which individual component parameters are provided.  FIG. 16  shows yet another embodiment of an interferometer an in situ fiber. In this version one fiber interrogates an entire optical system. Near the image plane other fiber(s) are collimated and utilized as the heterodyning beam(s). This embodiment is useful for interrogating entire optical systems. 
         [0094]    It is noted that the foregoing examples have been provided merely for the purpose of explanation and are in no way to be construed as limiting of the present invention. While the present invention has been described with reference to certain embodiments, it is understood that the words which have been used herein are words of description and illustration, rather than words of limitation. Changes may be made, within the purview of the appended claims, as presently stated and as amended, without departing from the scope and spirit of the present invention in its aspects. Although the present invention has been described herein with reference to particular means, materials and embodiments, the present invention is not intended to be limited to the particulars disclosed herein; rather, the present invention extends to all functionally equivalent structures, methods and uses, such as are within the scope of the appended claims.