Abstract:
In one embodiment, an interferometer system comprises an unequal path interferometer assemble comprising; a first reference flat having a first length L 1  in a first dimension, a second reference flat having a second length L 2  in the first dimension, a cavity D 1  defined by a distance between the first reference flat and the second reference flat, and a receptacle to receive an object in the cavity such that an optical path remains open between the first reference flat and the second reference flat, and a radiation targeting assembly to direct a collimated radiation beam to the interferometer assembly, a radiation collecting assembly to collect radiation received from the interferometer assembly, and a controller comprising logic to; vary a wavelength of the collimated radiation beam, record interferograms formed by a plurality of surfaces, extract phases of each of the interferograms for each of the plurality of surfaces to produce multiple phase maps, and determine each phase map from its corresponding interferogram, using a weighted least-square algorithm.

Description:
RELATED APPLICATIONS 
   This application claims the benefit of U.S. Provisional Patent Application Ser. No. 60/938,337, filed May 16, 2007, entitled MEASURING THE SHAPE, THICKNESS VARIATION, AND MATERIAL INHOMOGENEITY OF A WAFER, the disclosure of which is incorporated herein by reference in its entirety. 

   BACKGROUND 
   This invention relates to radiation-based inspection techniques, and more particularly to interferometric profilometry systems and methods which may be used to measure the shape, thickness, and material inhomogeneity of a wafer. 
   Phase shifting interferometry (PSI) is a highly accurate and efficient phase measuring method applied to a variety of applications including optical testing, surface profilometry, surface roughness estimation, and surface displacement measurement. The fundamental concept of PSI is that the phase of an interferogram can be extracted accurately by acquiring a set of phase-shifted interferograms. The phase shifts between interferograms are produced by changing the optical path difference (OPD) between the measurement surface and a reference surface. The phase shifts also can be achieved by changing the wavelength of the radiation used, if the OPD between the measurement surface and the reference surface is not zero. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a schematic illustration of an interferometer assembly according to embodiments. 
       FIG. 2A  is a flowchart illustrating operations of a method which may be used to measure the shape, thickness, and material inhomogeneity of a wafer according to an embodiment. 
       FIG. 2B  is a flowchart illustrating operations of a method which may be used to measure the shape, thickness, and material inhomogeneity of a wafer according to an embodiment. 
       FIG. 3  is a schematic illustration of an integrated visible pilot beam for non-visible interferometric device according to an embodiment. 
   

   DETAILED DESCRIPTION 
   Described herein are exemplary systems and methods which may be used to measure the shape, thickness, and material inhomogeneity of a wafer. In the following description, numerous specific details are set forth in order to provide a thorough understanding of various embodiments. However, it will be understood by those skilled in the art that the various embodiments may be practiced without the specific details. In other instances, well-known methods, procedures, components, and circuits have not been described in detail so as not to obscure the particular embodiments. 
   Embodiments described herein may be used in conjunction with an unequal path length interferometer (such as a Fizeau or Twyman-Green interferometer), contemporaneously extracting phases of all of the individual interferograms from a set of intensity frames that record superimposed interferograms generated with wave fronts reflected from multiple reflective surfaces. These intensity frames are acquired sequentially, by changing the wavelength in a measurement system. The wavelength can be changed mechanically, or, preferably, with a tunable laser light source. In some embodiments, contemporaneous events may be defined as events that happen within a reasonable time period of one another, given the technical circumstances. 
   The method takes advantage of the fact that the phase shift which results from the wavelength shift for a given interferogram is proportional to the OPD of that interferogram. In other words, the phases of each of the interferograms in the superimposed interferograms shift at different speeds during acquisition. The method of the preferred embodiment of this invention also takes advantage of the fact that the solutions of the least-square fitting technique (which is used and which is described in greater detail subsequently) with respect to an orthogonal basis, are completely independent of each other. As a consequence, the underlying phases of interferograms carried with the solutions are fully separable also. 
   In general, the system and method of the preferred embodiment of the invention produces a phase map or profile for each interferogram from a set of superimposed interferograms. When a particular interferogram corresponds to the OPD between a measuring surface and the reference plane, the phase map (mapping profile) of this particular interferogram represents the shape of the measuring surface. When an interferogram corresponds to the OPD between the front surface and the back surface of a plate (the object being measured), the phase map of this interferogram represents the thickness, or the distribution of the refractive index, of the plate. As a consequence, the method and system disclosed can be used to measure or profile surfaces, plate thickness, and refractive index inhomogeneity of an optical element or object from superimposed multiple interferograms by using PSI. 
   The method of the preferred embodiment is capable of separating multiple interferograms superimposed on the recording plane, as long as the phase shift speeds of these interferograms are different during acquisition, and there are enough intensity frames recorded. Consequently, the method is capable of measuring shapes of multiple reflective surfaces (greater than two). In addition, the method is capable of measuring multiple plate thicknesses. 
     FIG. 1  is a schematic illustration of an interferometer assembly according to embodiments. In some embodiments, the unequal path interferometer may be a Fizeau interferometer. In some embodiments, the unequal path interferometer may be a Twyman-Green interferometer. Referring to  FIG. 1 , an interferometer assembly includes a tunable laser  110  coupled to a computer  160 , which is in turn coupled to a detector  170 . The interferometer assembly further includes a radiation directing assembly that comprises a focusing lens  112 , a beam splitter  120 , and a collimator  124 . The interferometer assembly further includes a first reference flat  130  that comprises a front surface  134  and a back surface  132  and a second reference flat  150  that comprises a front surface  152  and a back surface  154 . The region between the front surface  134  of the first reference flat  130  and the front surface  152  of the second reference flat  150  defines an interferometer cavity  136 . In some embodiments, an interferometer with two reference flats, such as the interferometer depicted in  FIG. 1 , may be able to monitor tilt changes of reference flats ( 130 ,  150 ) with every measurement thereby significantly increasing measurement repeatability. The object under test, typically a wafer  140 , may be positioned in the cavity  136  between the first reference flat  130  and the second reference flat  150 . The wafer comprises a front surface  142  and a back surface  144 . 
   In some embodiments, an interferometer with two reference flats that are larger sizes than a test object, such as the interferometer depicted in  FIG. 1 , may be able to determine the location of a testing object in the imaging plane very precisely. In such an embodiment, the interferometer may obtain the test object&#39;s edge by using a cavity map instead of using the wafer surface or thickness variation maps, therefore the location of a test object&#39;s edge may be determined in such a way as to be free from errors resulting from the surface slope at any edge of test object. 
   In some embodiments, a wafer may be placed at a distance between the first reference flat front surface  134  and the first wafer surface  142  equaling 3 T, where T the wafer optical thickness. In some embodiments, a wafer may be placed at a distance between the second reference flat front surface  152  and the second wafer surface  144  equaling 5 T. A wafer optical thickness may be determined by multiplying the refractive index n with the thickness of the wafer t (T=nt). 
   In some embodiments, an interferometer phase shifting speed may be calibrated such that the phase shift of the optical thickness T of a wafer is equal to 22.5 degrees. In such embodiments, this may be accomplished by placing a polished opaque plate in a cavity formed between reference flats. In some embodiments, an interferometer may acquire  89  intensity frames while varying the wavelength of a light source. The set of recorded intensity frames for p superimposed interferograms may be expressed as 
   
     
       
         
           
             
               
                 
                   
                     g 
                     m 
                   
                   = 
                   
                     
                       a 
                       0 
                     
                     + 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         p 
                       
                       ⁢ 
                       
                         
                           a 
                           j 
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 Φ 
                                 j 
                               
                               + 
                               
                                 θ 
                                 jm 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 , 
                 
                   
                     for 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                   
                   = 
                   1 
                 
                 , 
                 2 
                 , 
                 … 
                 ⁢ 
                 
                     
                 
                 , 
                 8 
                 , 
                 9. 
               
             
             
               
                 ( 
                 1 
                 ) 
               
             
           
         
       
     
   
   where g m  is the mth acquisition, a 0  is the background, a j  is the modulation of jth interferogram, Φ j  is the phase of jth interferogram related to optical path difference of the testing surface and the reference mirror or the optical path difference between testing surfaces, and θ jm  is the mth phase shift for jth interferogram. In some embodiments, an interferometer may exact phases of all individual interferograms from the set of intensities by solving the following equation for X, 
   
     
       
         
           
             
               
                 
                   
                     A 
                     · 
                     X 
                   
                   = 
                   Y 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 where 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
           
             
               
                 
                   
                     A 
                     jk 
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         1 
                       
                       89 
                     
                     ⁢ 
                     
                       
                         w 
                         m 
                       
                       ⁢ 
                       
                         
                           ϕ 
                           m 
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ϕ 
                           k 
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                   
                 
                 , 
                 
                   
                     X 
                     j 
                   
                   = 
                   
                     x 
                     j 
                   
                 
                 , 
                 
                   
                     Y 
                     j 
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         1 
                       
                       89 
                     
                     ⁢ 
                     
                       
                         w 
                         m 
                       
                       ⁢ 
                       
                         I 
                         m 
                       
                       ⁢ 
                       
                         
                           ϕ 
                           j 
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       ϕ 
                       0 
                     
                     = 
                     1 
                   
                   , 
                   
                     
                       
                         ϕ 
                         
                           
                             2 
                             ⁢ 
                             j 
                           
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         m 
                         ) 
                       
                     
                     = 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         jm 
                       
                     
                   
                   , 
                   
                     
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ϕ 
                           
                             2 
                             ⁢ 
                             j 
                           
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     = 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         jm 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                     = 
                     1 
                   
                   , 
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                   ⁢ 
                   
                       
                   
                   , 
                   
                     
                       89 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       j 
                     
                     = 
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                 ⁢ 
                 
                   
 
                 
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                                 w 
                                 
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                                     1 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     … 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     89 
                                   
                                 
                               
                               = 
                             
                           
                           
                             
                               [ 
                               0.0001 
                             
                           
                           
                             0.0007 
                           
                           
                             0.0029 
                           
                           
                             0.0084 
                           
                           
                             0.0207 
                           
                           
                             0.0446 
                           
                           
                             0.0877 
                           
                           
                             0.1603 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             0.2762 
                           
                           
                             0.4531 
                           
                           
                             0.7130 
                           
                           
                             1.0821 
                           
                           
                             1.5914 
                           
                           
                             2.2768 
                           
                           
                             3.1787 
                           
                           
                             4.3425 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             5.8177 
                           
                           
                             7.6570 
                           
                           
                             9.9151 
                           
                           
                             12.646 
                           
                           
                             15.904 
                           
                           
                             19.738 
                           
                           
                             24.188 
                           
                           
                             29.286 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             35.052 
                           
                           
                             41.490 
                           
                           
                             48.588 
                           
                           
                             56.318 
                           
                           
                             64.630 
                           
                           
                             73.459 
                           
                           
                             82.719 
                           
                           
                             92.305 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             102.09 
                           
                           
                             111.94 
                           
                           
                             121.71 
                           
                           
                             131.22 
                           
                           
                             140.32 
                           
                           
                             148.83 
                           
                           
                             156.59 
                           
                           
                             163.44 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             169.25 
                           
                           
                             173.90 
                           
                           
                             177.29 
                           
                           
                             179.36 
                           
                           
                             180.05 
                           
                           
                             179.36 
                           
                           
                             177.29 
                           
                           
                             173.90 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             169.25 
                           
                           
                             163.44 
                           
                           
                             156.59 
                           
                           
                             148.83 
                           
                           
                             140.32 
                           
                           
                             131.22 
                           
                           
                             121.71 
                           
                           
                             111.94 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             102.09 
                           
                           
                             92.305 
                           
                           
                             82.719 
                           
                           
                             73.459 
                           
                           
                             64.630 
                           
                           
                             56.317 
                           
                           
                             48.588 
                           
                           
                             41.490 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             35.052 
                           
                           
                             29.286 
                           
                           
                             24.188 
                           
                           
                             19.738 
                           
                           
                             15.904 
                           
                           
                             12.646 
                           
                           
                             9.9151 
                           
                           
                             7.6570 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             5.8177 
                           
                           
                             4.3425 
                           
                           
                             3.1787 
                           
                           
                             2.2768 
                           
                           
                             1.5914 
                           
                           
                             1.0821 
                           
                           
                             0.7130 
                           
                           
                             0.4531 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             0.2762 
                           
                           
                             0.1603 
                           
                           
                             0.0877 
                           
                           
                             0.0446 
                           
                           
                             0.0207 
                           
                           
                             0.0084 
                           
                           
                             0.0029 
                           
                           
                             0.0007 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                           
                           ⁢ 
                           0.0001 
                           ] 
                         
                         * 
                         
                           0.0002 
                           . 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   In such embodiments, phases may then be calculated through the equation: 
   
     
       
         
           
             
               
                 
                   
                     Φ 
                     j 
                   
                   = 
                   
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           
                             2 
                             ⁢ 
                             j 
                           
                         
                         
                           x 
                           
                             
                               2 
                               ⁢ 
                               j 
                             
                             - 
                             1 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       j 
                     
                     = 
                     1 
                   
                 
                 , 
                 2 
                 , 
                 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     p 
                     . 
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   In some embodiments, a weighted least square algorithm functions to provide an orthogonal least-square fitting to produce a phase map for each interferogram. In some embodiments, a weighted least-square algorithm may be used to produce a phase map for each interferogram with N-frame acquisition for p superimposed interferograms and may use an algorithm corresponding to: 
   
     
       
         
           
             
               
                 
                   
                     Φ 
                     j 
                   
                   = 
                   
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           
                             2 
                             ⁢ 
                             j 
                           
                         
                         
                           x 
                           
                             
                               2 
                               ⁢ 
                               j 
                             
                             - 
                             1 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       j 
                     
                     = 
                     1 
                   
                 
                 , 
                 2 
                 , 
                 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     p 
                     . 
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
       
       
         
           where x 2j  and x 2j-1  are two elements of the solution 
         
       
     
  
   
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       
                         2 
                         ⁢ 
                         p 
                       
                     
                     ⁢ 
                     
                       
                         x 
                         k 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             1 
                           
                           n 
                         
                         ⁢ 
                         
                           
                             w 
                             m 
                           
                           ⁢ 
                           
                             
                               ϕ 
                               k 
                             
                             ⁡ 
                             
                               ( 
                               m 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ϕ 
                               j 
                             
                             ⁡ 
                             
                               ( 
                               m 
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         1 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         w 
                         m 
                       
                       ⁢ 
                       
                         I 
                         m 
                       
                       ⁢ 
                       
                         
                           ϕ 
                           j 
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       j 
                     
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     2 
                     ⁢ 
                     
                       p 
                       . 
                     
                   
                 
               
             
             
               
                 ( 
                 7 
                 ) 
               
             
           
         
       
     
   
   In some embodiments, the weighted least-square algorithm generates a set of optimal weights dynamically to provide the ideal basis for separating each of the individual interferograms from one another. In some embodiments, a least-square fitting algorithm may be used to exact all phases of all superimposed interferograms contemporaneously to produce multiple phase maps, each map for its corresponding interferogram. In some embodiments, contemporaneous events may be defined as events that occur within a reasonable time period of one another, given the technical circumstances. In some embodiments, extracting phases of each of the interferograms may be effected by a computer. In some embodiments, recording the multiple optical interferograms may be effected by means of a CCD camera. 
     FIG. 2A  is a flowchart illustrating operations of a method which may be used to measure the shape, thickness, and material inhomogeneity of a wafer according to an embodiment. At operation  201 , an interferometer may be initiated. In some embodiments, a test object such as, but not limited to, a wafer may be placed in the interferometer. By way of example and not limitation, the interferometer may be a Fizeau interferometer, a Twyman-Green interferometer, or the like. At operation  206 , coherent light may be supplied to a test object. In some embodiments, the coherent light may be supplied by a tunable laser or the like. At operation  211 , an interferometer may record interference patterns. In some embodiments, recording the multiple optical interferograms may be effected by means of a CCD camera. At operation  216 , interferograms may be extracted from data recorded by the interferometer. In some embodiments, extracting phases of each of the interferograms may be effected by a computer. 
   In some embodiments, analysis of these interferograms allows for the determination of various information, such as but not limited to; a test object&#39;s first surface height (operation  221 ), a test object&#39;s second surface height (operation  226 ), a test object&#39;s thickness variation (operation  231 ), a test object&#39;s material inhomogeneity (operation  236 ), and reference surface tilt ( 241 ). By way of example and not limitation, arbitrarily identifying the phase of the interferogram formed by the front reference plate  130  and the front  142  of the wafer  140  surface as A, and the phase of the interferogram formed by the front reference flat  150  and the back surface  144  of the wafer  140  as B, and the phase of the interferogram formed by both sides  142 ,  144  of the wafer  140  as T, and the phase of the interferogram formed by the cavity of the front reference flat  130  and the back reference flat  150  as C, then the surface parameters may be determined as follows: A corresponds to the front surface height of the wafer  140 , B corresponds to the back surface height of the wafer, C−(A+B) corresponds to the thickness variation in the wafer  140 , and T+A+B−C corresponds to the material inhomogeneity. 
     FIG. 2B  is a flowchart illustrating operations of a method which may be used to measure the shape, thickness, and material inhomogeneity of a wafer according to an embodiment. At operation  205  the laser is activated at a first wavelength. In operation, the computer  160  generates a signal to the tunable laser to activate the laser  110  at a first wavelength. In some embodiments, the laser  110  generates electromagnetic radiation in a range of wavelengths for which the wafer  140  is transmissive. The radiation from the laser  110  is supplied through focusing lens  112  to beam splitter  120 , from which the light passes through a collimating lens  124  to supply coherent light to a first reference flat  130 , the wafer  140 , and the second reference flat  150 . 
   A portion of the radiation incident on the first reference flat  130  is reflected. A remaining portion of the radiation incident on the first reference flat is transmitted through the first reference flat  130 . A portion of the transmitted radiation is incident on wafer  140 , a portion of which is reflected by the front surface  142  of the wafer  140 , and a portion of which is reflected by the back surface  144  of the wafer  140 . A portion of the radiation incident on the wafer  140  is transmitted through the wafer  140  onto the second reference  150 , and is reflected from the surface  152 . Further, a portion of the radiation transmitted through the first reference flat  130  is transmitted directly to the second reference flat  150 , and is reflected from the surface  152 . 
   At operation  210  radiation reflected is captured. In some embodiments, radiation reflected is captured as the wavelength of radiation is changing. The reflected radiation is directed by the beam splitter  120  to an imaging lens  172  which supplies, contemporaneously, multiple interferograms to a detector  170  (e.g., a CCD camera or other suitable recording planes). In some embodiments, contemporaneous events may be defined as events that occur within a reasonable time period of one another, given the technical circumstances. The detector  170  may include a frame grabber for storing images; alternatively, the computer  160  may be configured to provide this function. In any event, the images obtained by the detector  170  are supplied to the computer  160  for processing to produce the desired profiles in a suitable form for immediate display, or storage for subsequent utilization. At operation  215  interference patterns in the reflected radiation are captured. 
   If, at operation  220 , the amount of data acquired is not sufficient, then control passes to operation  225  and the wavelength of the radiation generated by laser  110  is changed. In some embodiments, if the amount of data acquired in not sufficient, the control passed to operation  225  to keep changing its wavelength. For example, the wavelength may be increased or decreased by a predetermined amount. Control then passes back to operation  210  and the reflected radiation is captured. Operations  210 - 225  are repeated until an adequate number of data samples are acquired, whereupon control passes to operation  230  and one or more phases of interferograms are extracted from the data collected. In some embodiments, a control passed to operation  225  to stop its wavelength changing while another control passed to operation  230 . In some embodiments, the phases of interferograms may be extracted using techniques set forth in U.S. Pat. No. 6,885,461, the disclosure of which is incorporated herein by reference in its entirety. 
   At operation  235  one or more parameters are determined from the phases of interferograms extracted in operation  230 . By way of example and not limitation, arbitrarily identifying the phase of the interferogram formed by the front reference plate  130  and the front  142  of the wafer  140  surface as A, and the phase of the interferogram formed by the back reference flat  150  and the back surface  144  of the wafer  140  as B, and the phase of the interferogram formed by both sides  142 ,  144  of the wafer  140  as T, and the phase of the interferogram formed by the cavity of the front reference flat  130  and the back reference flat  150  as C, then the surface parameters may be determined as follows: A corresponds to the front surface height of the wafer  140 , B corresponds to the back surface height of the wafer, C−(A+B) corresponds to the thickness variation in the wafer  140 , and T+A+B−C corresponds to the material inhomogeneity. 
     FIG. 3  is a schematic illustration of one embodiment of a computing system which may be used to implement the computer  160  of  FIG. 1 . The computer system  300  includes a computer  308  and one or more accompanying input/output devices  306  including a display  302  having a screen  304 , a keyboard  310 , other I/O device(s)  312 , and a mouse  314 . The other device(s)  312  can include a touch screen, a voice-activated input device, a track ball, and any other device that allows the system  300  to receive input from a developer and/or a user. The computer  308  includes system hardware  320  and random access memory and/or read-only memory  330 . A file store  380  is communicatively connected to computer  308 . File store  380  may be internal such as, e.g., one or more hard drives, or external such as, e.g., one or more external hard drives, network attached storage, or a separate storage network. 
   Memory  330  includes an operating system  340  for managing operations of computer  308 . In one embodiment, operating system  340  includes a hardware interface module  354  that provides an interface to system hardware  320 . In addition, operating system  340  includes one or more file systems  350  that manage files used in the operation of computer  308  and a process control subsystem  352  that manages processes executing on computer  308 . Operating system  340  further includes a system call interface module  342  that provides an interface between the operating system  340  and one or more application modules  362 . 
   In operation, one or more application modules and/or libraries executing on computer  308  make calls to the system call interface module  342  to execute one or more commands on the computer&#39;s processor. The system call interface module  342  invokes the services of the file system(s)  350  to manage the files required by the command(s) and the process control subsystem  352  to manage the process required by the command(s). The file system(s)  350  and the process control subsystem  352 , in turn, invoke the services of the hardware interface module  354  to interface with the system hardware  320 . 
   The particular embodiment of operating system  340  is not critical to the subject matter described herein. Operating system  340  may be embodied as a UNIX operating system or any derivative thereof (e.g., Linux, Solaris, etc.) or as a Windows™ brand operating system. 
   In some embodiments, computer system  300  includes one or more modules to implement hybrid database query caching. In the embodiment depicted in  FIG. 3 , computer system  300  includes a surface analysis module  362  which implements the operations described with reference to  FIG. 2 . 
   In some embodiments, the optical thickness of a wafer  140  may be assumed to be T=nt, where n represents the refractive index of the wafer and t represents the wafer thickness. The wafer may be positioned in the cavity defined by the reference flats such that the difference between the front reference flat  130  and the front wafer surface equals 3 T, and the distance between the back reference flat  150  and the back surface is 5 T. Since the interferograms are a function of the distances or T, 3 T, 4 T, 5 T, and 9 T, respectively, the superimposed interferogram recorded by the detector  170  can be successfully separated and the phase of each interferogram can be extracted using techniques described in U.S. Pat. No. 6,885,461 or U.S. Pat. No. 6,359,692, the disclosures of which are incorporated herein by reference. 
   While the invention has been particularly shown and described with reference to a preferred embodiment and various alternate embodiments, it will be understood by persons skilled in the relevant art that various changes in form and details can be made therein without departing from the spirit and scope of the invention. While the invention has been particularly shown and described with reference to a preferred embodiment and various alternate embodiments, it will be understood by persons skilled in the relevant art that various changes in form and details can be made therein without departing from the spirit and scope of the invention. 
   Thus, although embodiments have been described in language specific to structural features and/or methodological acts, it is to be understood that claimed subject matter may not be limited to the specific features or acts described. Rather, the specific features and acts are disclosed as sample forms of implementing the claimed subject matter.