This application claims the benefit of U.S. Provisional Application No. 60/485,255, filed Jul. 7, 2003. 
    
    
     TECHNICAL FIELD 
     This invention generally relates to systems and methods for making ellipsometric measurements. 
     BACKGROUND OF THE INVENTION 
     Prior art teaches the practice of ellipsometric measurements with a relatively low spatial resolution of approximately 30 microns or larger. The spatial resolution of 30 microns or larger is generally a direct consequence of two ellipsometric measurement requirements. The two ellipsometric measurement requirements are (1) a relatively large angle of incidence of a beam on the surface of the object and (2) the angle of incidence is relatively well confined, e.g., ≲0.1 radians. 
     The relatively low spatial resolution property of the prior art sets the limits on the spatial resolution that is available to examine non-uniform surface properties of the substrate. 
     SUMMARY OF THE INVENTION 
     This limitation of prior art ellipsometric measurements is lifted in various embodiments of the present invention by using information contained in measured non-forward scattering by the non-uniform substrate surface. 
     If the substrate surface is uniform, there will be no non-forward scattering of an incident beam other than that produced by diffraction at the stop of the imaging system. If however, there is a non-uniformity in the substrate surface, each spatial frequency component of the non-uniformity of the substrate surface will generate a non-forward scattering at an associated angle of diffraction in a one-to-one mapping. The non-forward scattering amplitudes of each of the associated harmonics contain information about the respective amplitudes of the harmonic spatial structure of the surface and the spatial frequencies of the non-forward scattering contain information about the spatial frequencies of the non-uniform harmonic spatial structure of the surface through the one-to-one mapping. 
     Measurements in various embodiments of the present invention comprise measurements of forward scattered or reflected fields of orthogonally polarized beams by a substrate surface at the large angles of incidence and reflection of traditional ellipsometric measurements, measurements of non-forward scattered fields of orthogonally polarized beams by the surface of a substrate, and measurements of forward or reflected fields of orthogonally polarized beams by the substrate surface at angles of reflection corresponding to the angles of diffraction used in the measurements of non-forward scattered fields. The measured amplitudes of conjugated quadratures of forward scattered or reflected fields contain information about the zero spatial frequency component of the surface structure. The amplitudes of the conjugated quadratures of non-forward scattered fields are analyzed for high spatial frequency information about non-uniform spatial structure. The highest spatial frequency of the measured amplitudes of the conjugated quadratures of the fields corresponds to a sub-wavelength resolution. The information obtained at the zero spatial frequency with an angle of reflection of traditional ellipsometric measurements and the high spatial frequency information about non-uniform spatial structure obtained from the analysis of the amplitudes and phases of the conjugated quadratures of non-forward scattered fields and corresponding forward scattered fields are superimposed to obtain a high spatial resolution ellipsometric image of the substrate surface. The superposition of spatial frequency information can be cast as a Fourier series representation. The high spatial resolution may be one to two orders of magnitude larger than that obtained in prior art ellipsometric measurements. 
     The analysis of the amplitudes and phases of the conjugated quadratures of non-forward scattered fields and corresponding forward scattered fields for high spatial frequency information about non-uniform spatial structure is formulated as a “forward” problem, an “inverse” problem or a combination of the two approaches. In the forward problem, certain properties and structure for a non-uniform surface is assumed and the effect of the assumed properties and non-uniform surface structure is calculated through the application of Maxwell&#39;s equations. The results of the calculations are subsequently compared to the measured amplitudes of the conjugated quadratures of the non-forward scattered fields and corresponding forward scattered fields and differences are used to select a next trial for the assumed properties and non-uniform surface. A solution of the forward problem is found if the procedure converges such that the differences are less than respective measurement errors although the solution may not be unique. In the inverse problem, the measured amplitudes of the conjugated quadratures of non-forward scattered fields and corresponding forward scattered fields are used as input information to integral-differential equations derived from Maxwell&#39;s equations. 
     It is important to note that the non-zero spatial frequency components of the measured amplitudes and phases of conjugated quadratures are obtained generally as interferometric dark field measurements. 
     Bi- and quad-homodyne detection methods are used with respect to two polarization states of the measurement beam. As a consequence, ellipsometric measurements of the present invention may be obtained more accurately and also in a scanning mode. 
     Two-dimensional arrays of images are generated jointly or substantially jointly. This feature of the present invention generates information about ellipsometric properties of the substrate that has enhanced signal-to-noise ratios with respect to lateral variations of ellipsometric properties of the substrate. 
     In general, in one aspect, the invention features a method of detecting non-uniform ellipsometric properties of a substrate surface wherein the non-uniform ellipsometric properties are characterized by a characteristic dimension. The method involves: generating an input beam for illuminating a spot at a selected location on or in the substrate, the spot having a size L that is substantially larger than the characteristic dimension; deriving a measurement beam and a reference beam from the input beam; directing the measurement beam onto the substrate as an incident measurement beam that illuminates the spot at that selected location on or in the substrate to produce a scattered measurement beam; for each orientation of a plurality of different orientations of the reference beam relative to the scattered measurement beam, interfering the scattered measurement beam with the reference beam to produce a corresponding interference beam, wherein each of the different orientations of the reference beam is selected to produce a peak sensitivity for a portion of the scattered measurement beam that emanates from the object at a corresponding different diffraction angle of a plurality of diffraction angles; for each orientation of the plurality of different orientations of the reference beam relative to the scattered measurement beam, converting the interference beam for that selected location into an interference signal; and using the interference signals for each orientation of the plurality of different orientations to determine whether any non-uniform ellipsometric properties that are characterized by the characteristic dimension are present anywhere within a region on or in the substrate defined by the spot at that selected location. 
     Other embodiments include one or more of the following features. The plurality of diffraction angles is selected to represent a basis set of spatial frequency components. More specifically, the plurality of diffraction angles is selected to represent a set of harmonic spatial frequency components. The plurality of diffraction angles is selected to represent eight harmonic spatial frequency components or sixteen harmonic spatial frequency components. The incident measurement beam is at an angle of incidence θ I  with respect to a direction that is normal to the surface of the object, wherein the plurality of diffraction angles includes a diffraction angle θ D  relative to the direction that is normal to the surface of the substrate, wherein the characteristic dimension is equal to Λ, wherein the incident measurement beam is characterized by a wavelength λ, and wherein the diffraction angle θ D  satisfies the following relationship: Λ[sin(θ I )−sin(θ D )]=λ. 
     Still other embodiments include one or more of the following features. The input beam includes two orthogonally polarized beams and the measurement derived from the input beam also includes two orthogonally polarized beams. Using the interference signals for each orientation of the plurality of different orientations involves measuring one or more values for the interference signal for each orientation of the plurality of different orientations and then using the measured interference signal values to compute coefficients of a series expansion representing a measured reflectance of the substrate. The series expansion is a Fourier series expansion. The method also involves solving a forward problem to develop a model of the surface of the substrate that satisfactorily fits the series expansion representing the measured reflectance of the substrate. Or the method involves solving an inverse problem to determine properties of the substrate from the series expansion representing the measured reflectance of the substrate. The method further involves: for a plurality of different orientations of the incident measurement beam relative to the normal of the substrate, interfering the scattered measurement beam with the reference beam to produce a corresponding interference beam; for each orientation of the plurality of different orientations of the incident measurement beam, converting the resulting interference beam for that selected location into an interference signal; and using the interference signals for the plurality of different orientations of the incident measurement beam along with the interference signals for each orientation of the plurality of different orientations to determine whether any non-uniform ellipsometric properties that are characterized by the characteristic dimension are present anywhere within a region on or in the substrate defined by the spot at that selected location. 
     Still yet other include one or more of the following features. The size of the spot L is at least three times greater than the characteristic dimension or at least an order of magnitude larger than the characteristic dimension. The method further involves performing the steps of generating, deriving, directing, interfering, and converting for each of a sequence of different selected locations on or in the object, wherein the first-mentioned selected location is one of the plurality of different selected locations. Generating the input beam involves generating a first beam at a first wavelength and a second beam at a second wavelength that is different from the first wavelength, the first and second beams being coextensive and sharing the same temporal window. For each orientation of a plurality of different orientations of the reference beam relative to the scattered measurement beam, for each of a plurality of successive time intervals, introducing a corresponding different shift in a selected parameter of the first beam and introducing a different corresponding shift in the selected parameter of the second beam, wherein the selected parameters are selected from a group consisting of phase and frequency. Using the interference signals for each orientation of the plurality of different orientations to determine whether any non-uniform ellipsometric properties are present involves: for each orientation of a plurality of different orientations of the reference beam relative to the scattered measurement beam: (1) for each of the plurality of successive time intervals, measuring a value of the interference signal; and (2) from the measured values of the interference signal for the plurality of successive time internals, computing the orthogonal components of conjugated quadratures of fields of the corresponding scattered measurement beam; and using the two computed orthogonal components of conjugated quadratures of fields of the corresponding scattered measurement beam for each orientation of the plurality of different orientations of the reference beam to determine whether any non-uniform ellipsometric properties are present within the spot. 
     Still yet other embodiments include one or more fo the following features. Each of the first and second beams includes a first component and a second component that is orthogonal to the first component, wherein the selected parameter of the first beam is the phase of the second component of the first beam, and wherein the selected parameter of the second beam is the phase of the second component of the second beam. The selected parameter of the first beam is the frequency of the first beam, and wherein the selected parameter of the second beam is the frequency of the second beam. 
     An advantage of at least one embodiment of the present invention is that one-dimensional, two-dimensional, and three-dimensional high spatial resolution ellipsometric images of an object are generated with the measurement of amplitudes and phases of fields of non-forward and forward scattered beams by an object based on interferometric measurements. 
     Another advantage of at least one embodiment of the present invention is that one-dimensional, two-dimensional, and three-dimensional high spatial resolution ellipsometric images of an object are generated with joint or substantially joint measurements of conjugated quadratures of fields of non-forward scattered fields by an object based on interferometric measurements. 
     Another advantage of at least one embodiment of the present invention is that high spatial resolution ellipsometric images of an object are generated with simultaneous measurements of amplitudes of fields of non-forward and forward scattered fields by an array of a large number of spots in or on an object in interferometric measurements. 
     Another advantage of at least one embodiment of the present invention is that high spatial resolution ellipsometric images of an object are generated with simultaneous measurements of joint or substantially joint measurements of conjugated quadratures of fields of non-forward and forward scattered fields by a large array of spots in or on an object in interferometric measurements. 
     Another advantage of at least one embodiment of the present invention is that bi- and quad-homodyne detection methods can be used to obtain conjugated quadratures of fields of beams reflected/scattered by a substrate that is being imaged. 
     Another advantage of at least one embodiment of the present invention is that variants of the bi- and quad-homodyne detection methods can be used to obtain joint or substantially joint measurements of conjugated quadratures of fields of reflected/scattered orthogonally polarized beams by a substrate that is being imaged. 
     Another advantage of at least one embodiment of the present invention is that relative phase shifts between the arrays of reference and measurement beams can be introduced by changing the frequencies of components of the input beam. 
     Another advantage of at least one embodiment of the present invention is that relative phase shifts from a predetermined set of relative phase shifts can be introduced between the arrays of reference and measurement beams in certain embodiments of the present invention. 
     Another advantage of at least one embodiment of the present invention is high spatial resolution ellipsometric imaging of a substrate with a sub-wavelength lateral resolution may be obtained with a working distance of the order of a mm. 
     Another advantage of at least one embodiment of the present invention is high spatial resolution ellipsometric imaging of a substrate with a sub-wavelength lateral resolution may be obtained with a working distance of the order of microns. 
     Another advantage of at least one embodiment of the present invention is high spatial resolution ellipsometric imaging of an interior portion of a substrate with a sub-wavelength lateral resolution may be obtained with a working distance of the order of a mm. 
     Another advantage of at least one embodiment of the present invention is high spatial resolution ellipsometric imaging of an interior portion of a substrate with a sub-wavelength lateral resolution may be obtained with a working distance of the order of microns. 
     Another advantage of at least one embodiment of the present invention is that the phases of the input beam components do not affect measured conjugated quadratures of fields. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1   a  is a diagram of an interferometric system. 
         FIG. 1   b  is a schematic diagram of an embodiment of a wafer inspection tool that comprises an interferometric metrology system which in turn comprises a catadioptric imaging system. 
         FIG. 1   c  is a diagram of a catadioptric imaging system. 
         FIG. 1   d  is a diagram of a pinhole array beam-splitter. 
         FIG. 1   e  is a schematic diagram of a beam-conditioner configured as a two-frequency generator and a frequency-shifter. 
         FIG. 1   f  is schematic diagram of an embodiment of a wafer inspection tool that comprises an interferometric metrology system which in turn comprises a catadioptric imaging system. 
         FIG. 1   g  is schematic diagram of another embodiment of a wafer inspection tool that comprises an interferometric metrology system which in turn comprises a catadioptric imaging system. 
         FIG. 2   a  is a schematic diagram of an achromatic astigmatic catadioptric imaging system. 
         FIG. 2   b  is a diagram showing surfaces and corresponding radii of a catadioptric imaging system. 
         FIG. 2   c  is a schematic diagram an astigmatic catadioptric imaging system. 
         FIG. 2   d  is a schematic diagram of a section of a catadioptric imaging system located near a measurement object. 
         FIG. 2   e  is a schematic diagram of a section of a catadioptric imaging system located near a measurement object and imaging an interior section of the measurement object. 
         FIG. 2   f  is a schematic diagram an astigmatic catadioptric imaging system. 
         FIG. 3  is a diagram of the angles of incidence and diffraction of beams. 
     
    
    
     DETAILED DESCRIPTION 
     An apparatus and method is described for ellipsometric measurements of properties of a substrate surface with a high wavelength or sub-wavelength spatial resolution. The apparatus and method for the measurement of high spatial resolution ellipsometric properties comprises interferometric measurement of fields of forward and non-forward scattered beams by the substrate surface. The ellipsometric measurements comprise measurement of harmonic spatial frequency components of conjugated quadratures of fields of a beam reflected/scattered by the substrate surface, e.g., the surface of a non-patterned or patterned wafer. The surface may comprise non-uniform ellipsometric properties wherein a characteristic scale of the non-uniform ellipsometric properties may be of the order of or less than the wavelength of the beam. The conjugated quadrature amplitudes of the harmonic spatial frequency components are measured with a low spatial resolution. The harmonic spatial frequency components of reflected/scattered fields preferentially contain information about the non-uniform ellipsometric properties of the surface, i.e. a two-dimensional and/or three dimensional spatial structure at the surface, of the non-patterned or patterned wafer. The use of the low spatial resolution in the measurement of the conjugated quadrature amplitudes permits smaller values for the angular widths of beams used in the determination of the harmonic spatial frequency components, i.e., resolution of an imaging system is inversely proportional to the numerical aperture of the imaging system. The operation comprising measurement of conjugated quadratures of the non-forward scattered beams has the important property that the measurement of the non-forward scattered components is made operating in a dark field mode. The use of a dark field mode leads to improved signal-to-noise ratio for measurement of the ellipsometric properties. Information about the non-forward scattered fields is obtained from joint or substantially joint measurements of conjugated quadratures of the corresponding amplitudes using either a bi-homodyne or quad-detection method or variants thereof for different polarization states of the fields. The information about the non-forward scattered fields may be obtained from joint or substantially joint measurements of 2-dimensional arrays of conjugated quadratures in a scanning mode. 
     A general description of the technique and procedure of various embodiments of the present invention will first be given followed by a technical description of various embodiments. The technique and procedure are based on the spatial frequency response of a surface of a substrate. The two-dimensional section may correspond to a surface of the substrate or to an internal two-dimensional section of the substrate. A substrate reflectance can be expanded into a Fourier series of spatial harmonics with each representing a spatial frequency component. The conjugated quadratures of the spatial frequency component represent properties of the surface of the substrate. The conjugated quadratures of the spatial frequency component are written as 
                     F   ⁡     (     x   ,   y   ,     Λ   x     ,     Λ   y     ,     ζ   x     ,     ζ   y       )       =       A   ⁡     (     x   ,   y   ,     Λ   x     ,     Λ   y     ,     ζ   x     ,     ζ   y       )       ×             (   1   )                       ⁢         cos   ⁡     (         2   ⁢           ⁢   π   ⁢           ⁢   x       Λ   x       +     ζ   x       )       ⁢           ⁢     cos   ⁡     (         2   ⁢           ⁢   π   ⁢           ⁢   y       Λ   y       +     ζ   y       )         ,                                 F   ~     ⁡     (     x   ,   y   ,     Λ   x     ,     Λ   y     ,     ζ   x     ,     ζ   y       )       =         A   ~     ⁡     (     x   ,   y   ,     Λ   x     ,     Λ   y     ,     ζ   x     ,     ζ   y       )       ×             (   2   )                       ⁢       sin   ⁡     (         2   ⁢           ⁢   π   ⁢           ⁢   x       Λ   x       +     ζ   x       )       ⁢           ⁢     sin   ⁡     (         2   ⁢           ⁢   π   ⁢           ⁢   y       Λ   y       +     ζ   y       )                                 
where F(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and {tilde over (F)}(x,y,Λ x ,Λ y ,ζ x ,ζ y ) are conjugated quadratures of the spatial frequency component, A(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and Ã(x,y,Λ x ,Λ y ,ζ x ,ζ y ) are the amplitudes of the conjugated quadratures of the spatial frequency components, and Λ x  and Λ y  are the wavelengths of the spatial frequency components in the x and y directions, respectively, such that 1/Λ x  and 1/Λ y  are the corresponding spatial frequencies. Phase offsets ζ x  and ζ y  in Equations (1) and (2) are determined by the location of the source of the corresponding spatial frequency components in or on the two dimensional section of the substrate relative to a Cartesian coordinate system.
 
     With respect to the present embodiment, the autocorrelation lengths of A(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and Ã(x,y,Λ x ,Λ y ,ζ x ,ζ y ) in the x and y directions and/or of the envelop function are larger, e.g., a factor of three or an order of magnitude or more, than the smaller of the spatial wavelengths Λ x  and Λ y  and generally much less than corresponding dimensions of substrate surface being imaged. The autocorrelation lengths and the associated spatial wavelengths Λ x  and Λ y  are substantially independent parameters controlled by the design of the apparatus used in embodiments of the present invention. 
     Amplitudes of conjugated quadratures of a spatial frequency component of a substrate reflectance are measured interferometrically. The interferometric measurements will have a peak in detection sensitivity for surface structures that have a characteristic dimension in the x direction of the order of Λ x /2 to Λ x /4 wherein it has been assumed that Λ x ≲Λ y  so as to illustrate important properties with a simplified description. Accordingly, the fractional spatial wavelength Λ x /4 is preferably selected to be of the order of the corresponding characteristic dimension of surface structure for which information is desired. For other spatial wavelengths, there will be a reduced sensitivity. 
     It is of particular interests to note that in addition to a peak in detection sensitivities for surface structures that have corresponding spatial frequencies, the detection sensitivities will be reduced for a non-corresponding spatial frequency components. In particular, the detection sensitivity for zero spatial frequency components will be substantially zero. The zero sensitivity for detection of the corresponding zero spatial frequency component permits a particularly important mode of operation, a dark field interferometric mode. In the dark field interferometric mode, the measurements of the amplitudes A(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and Ã(x,y,Λ x ,Λ y ,ζ x ,ζ y ) are made with a significantly increased signal-to-noise ratio (see for example the discussion of statistical noise in commonly owned U.S. Pat. No. 5,760,901 (ZI-05) entitled “Method And Apparatus For Confocal Interference Microscopy With Background Amplitude Reduction And Compensation” and U.S. Pat. No. 6,480,285 B1 (ZI-08) entitled “Multiple Layer Confocal Interference Microscopy Using Wavenumber Domain Reflectometry And Background Amplitude Reduction And Compensation” of which both are by Henry A. Hill and of which the contents are incorporated herein in their entirety by reference. 
     In the embodiments described herein, ellipsometric information about the zero spatial frequency component is obtained interferometrically in a forward scattering or reflection mode of operation at the traditional large angles of incidence and reflection of ellipsometry. 
     Because of the characteristic size of the spot being imaged is larger than the wavelength of the spatial frequency components, the spatial frequency components will be preferentially diffracted at particular angles in respective planes. For a given optical wavelength λ for a measurement beam, the particular angle at which a spatial frequency component of the substrate reflectance will diffract the beam in a corresponding plane is given by a grating equation
 
Λ[sin θ I −sin θ D ]=λ  (3)
 
where θ I  and θ D  are angles of incidence and diffraction as shown diagrammatically in  FIG. 3 . It is evident from Equation (3) that in order to obtain the smallest Λ for a given optical wavelength λ,
 
θ D ≅−θ I ,  (4)
 
θ I ≈1.  (5)
 
     Amplitudes of conjugated quadratures of spatial frequency components of the substrate reflectance are measured for a set of harmonics of a spatial frequency for each of two orthogonal polarization states of the measurement beam. For the set of harmonics of spatial frequency, there corresponds a set of θ D  with a one-to-one mapping between the two sets. In addition, corresponding zero spatial frequency components of fields reflected for the values of θ D  of the set of θ D  are measured by setting θ I =θ D . Each of the measured harmonics and corresponding measured zero spatial frequency components of fields are then used to determine a transformed amplitude corresponding to the amplitude of the spatial frequency component of the reflectance of the substrate surface. The procedure used to determine the transformed amplitude may be posed as a forward problem or as an inverse problem. 
     In the forward problem, a model is assumed for the substrate structure and the properties of reflected/scattered beams are calculated using Maxwell&#39;s equations. The computed properties are then compared to the measured properties. If there is a satisfactory fit of the measured properties with the computed properties, the assumed model is considered to be a representation of the substrate structure. If the fit is not within the measurement errors, the differences between the computed and measured properties are used to guide the selection of a modified model of the substrate structure and the procedure previous procedure repeated. The procedure is repeated until the model converges to a stable solution. If the procedure does not converge satisfactorily, it is necessary to repeat the whole procedure starting with a different initial model of the substrate structure. 
     In the inverse problem, the measured scattered fields are used as input information to integral-differential equations derived from Maxwell&#39;s equations, e.g., see “Inverse Methods in Electromagnetic Imaging,” parts 1 and 2, Ed. By Wolfgang-M. Boerner, NATO ASI Series, 1983 (Reidel), and analogous procedures such as used in seismology of the earth to determine properties of the earth&#39;s interior. 
     The set of transformed amplitudes represent a Fourier series expansion of the substrate surface reflectance. The set of transformed amplitudes of the conjugated quadratures are next added together to generate a high spatial resolution image of the ellipsometric properties of the substrate surface. 
     The physics relating to the description of the diffracted beam and to the description of the corresponding reflected beam, i.e., the angle of diffraction is the same as the angle of reflection, are closely related. It is for this reason that the measured zero spatial frequency components of fields reflected at angles the same as the angles of diffraction used to obtain the non-zero spatial frequency components are beneficially used in the forward problem or inverse problem. 
     An example set of values for θ I  and θ D  is listed in Table 1 that can be used for generation of a set of eight spatial harmonics that are harmonically related. A value of 75 degrees is assumed for θ I  and a maximum negative value of −60 degrees is assumed for θ D  which fixes the value of the highest spatial frequency or the smallest spatial wavelength that can be measured in the example. The remaining set of values of θ D  in Table 1 corresponds to those values of θ D  that generate spatial wavelengths that are harmonics, i.e., first, second, . . . , of the shortest spatial wavelength. Another example set of values for θ I  and θ D  is listed in Table 2 that may be used for generation of a set of sixteen spatial harmonics. The values of θ I  and θ D  is listed in Table 2 are determined by the same procedure as used in deriving the θ I  and θ D  values listed in Table 2 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Values Of θ D  For Eight Harmonics 
               
             
          
           
               
                 Harmonic 
                 θ I   
                 θ D   
               
               
                 Number 
                 degrees 
                 degrees 
               
               
                   
               
             
          
           
               
                 1 
                 75.000 
                 47.471 
               
               
                 2 
                   
                 30.527 
               
               
                 3 
                   
                 16.197 
               
               
                 4 
                   
                 2.863 
               
               
                 5 
                   
                 −10.314 
               
               
                 6 
                   
                 −24.082 
               
               
                 7 
                   
                 −39.571 
               
               
                 8 
                   
                 −60.000 
               
               
                   
               
             
          
         
       
     
     The width Δθ of the beam diffracted by the spatial frequency component of the substrate reflectance will be determined by an autocorrelation length l of A(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and Ã(x,y,Λ x ,Λ y ,ζ x ,ζ y ) as 
                     Δ   ⁢           ⁢   θ     ≅       λ   l     ⁢   sec   ⁢           ⁢       θ   R     .               (   6   )               
The autocorrelation length l is determined by properties of a detector imaging system.
 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Values Of θ D  For Sixteen Harmonics 
               
             
          
           
               
                 Harmonic 
                 θ I   
                 θ D   
               
               
                 Number 
                 degrees 
                 degrees 
               
               
                   
               
             
          
           
               
                 1 
                 75.000 
                 58.367 
               
               
                 2 
                   
                 47.471 
               
               
                 3 
                   
                 38.494 
               
               
                 4 
                   
                 30.527 
               
               
                 5 
                   
                 23.169 
               
               
                 6 
                   
                 16.197 
               
               
                 7 
                   
                 9.465 
               
               
                 8 
                   
                 2.863 
               
               
                 9 
                   
                 −3.701 
               
               
                 10 
                   
                 −10.314 
               
               
                 11 
                   
                 −17.070 
               
               
                 12 
                   
                 −24.082 
               
               
                 13 
                   
                 −31.502 
               
               
                 14 
                   
                 −39.571 
               
               
                 15 
                   
                 −48.723 
               
               
                 16 
                   
                 −60.000 
               
               
                   
               
             
          
         
       
     
     A corollary to the large value of the autocorrelation length l relative to λ [see Equation (6)] is a lower spatial resolution such as encountered in prior art ellipsometric measurements. However, a second corollary is an opposite effect with respect to the density of measurements of A(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and Ã(x,y,Λ x ,Λ y ,ζ x ,ζ y ) required for a survey of a portion of a substrate surface. The autocorrelation length l is typically selected to be larger than Λ x  by a factor ξ that is the order of 10 or more. Thus, the time required to execute a survey of the portion of the substrate surface is reduced by a factor of ξ 2 . The effect of the factor ξ 2  offsets in part the effect of the increased number of measurements required to obtain the measurements of the non-zero spatial frequency components of the conjugated quadratures of the scattered fields. 
     The sampling frequency for the measurement of A(x,y,Λ x ,Λ y ,ζ x ,ζ y ) and Ã(x,y,Λ x ,Λ y ,ζ x ,ζ y ) is preferably approximately equal to twice the spatial frequency 1/l, i.e., 2/l. In addition, the sampling spatial frequency is preferably selected to correspond to the spatial frequency of the pitch of pixel location of a CCD detector in the x and/or y directions. 
     The embodiments described herein may be used to measure the ellipsometric effects of propagation of signals, e.g., thermal or acoustical, in a section by using a probe beam in addition to the measurement beam where the probe beam precedes the measurement beam by a time τ. The probe beam and the measurement beams may have different optical wavelengths. An advantage of certain embodiments to measure temporal response of the substrate is the high spatial frequency resolution in the plane of the section. 
     In the following description of the different embodiments, many elements of the different embodiments perform like functions and are indicated with the same numerals in different respective figures of the embodiments. 
     A general description of embodiments incorporating the present invention will first be given for interferometer systems wherein the bi- and quad-homodyne detection methods and variants thereof are used in interferometer systems for making joint or substantially joint measurements of conjugated quadratures of fields of beams backscattered by a measurement object. Referring to  FIG. 1   a , an interferometer system is shown diagrammatically comprising an interferometer  10 , a source  18 , a beam-conditioner  22 , detector  70 , an electronic processor and controller  80 , and a measurement object or substrate  60 . Source  18  is a pulsed or shuttered source that generates input beam  20  comprising one or more frequency components. Beam  20  is incident on and exits beam-conditioner  22  as input beam  24  that comprises a single polarized component or two orthogonally polarized components. Each of the orthogonally polarized components comprises two or more different frequency components. The measurement beam components of the frequency components of input beam  24  are coextensive in space and have the same temporal window function and the corresponding reference beam components are coextensive in space and have the same temporal window function although the measurement beam components and the reference beam components may not be spatially coextensive. Reference and measurement beams may be generated in either beam-conditioner  22  from a set of beams or in interferometer  10  for each of the two or more frequency components of input beam  24 . Measurement beam  30 A generated in either beam-conditioner  22  or in interferometer  10  is incident on substrate  60 . Measurement beam  30 B is a return measurement beam generated as either a portion of measurement beam  30 A reflected and/or scattered by substrate  60 . Return measurement beam  30 B is combined with the reference beam in interferometer  10  to form output beam  34 . 
     Output beam  34  is detected by detector  70  to generate either one or more electrical interference signals per source pulse for the bi-homodyne or quad-homodyne detection methods or variants thereof as signal  72 . Detector  70  may comprise an analyzer to select common polarization states of the reference and return measurement beam components of beam  34  to form a mixed beam. Alternatively, interferometer  10  may comprise an analyzer to select common polarization states of the reference and return measurement beam components such that beam  34  is a mixed beam. 
     In the practice, known phase shifts are introduced between the reference and measurement beam components of output beam  34  by two different techniques. In the first technique, phase shifts are introduced between corresponding reference and measurement beam components for each of the frequency components of output beam  34  as a consequence of a non-zero optical path difference between the reference and measurement beam paths in interferometer  10  and corresponding frequency shifts introduced to the frequency components of input beam  24  by beam-conditioner  22  and/or source  18  as controlled by signals  74  and  92  from electronic processor and controller  80 . In the second technique, phase shifts are introduced between the reference and measurement beam components for each of the frequency components of input beam  24  by beam-conditioner  22  as controlled by signals  74  and  92  from electronic processor and controller  80 . 
     There are different ways to configure source  18  and beam-conditioner  22  to meet the input beam requirements of the different embodiments of the present invention. Examples of beam-conditioners that may be used in either first or the second technique comprise combinations of a two frequency generator and phase shifting type of beam-conditioner such as described in commonly owned U.S. Provisional Patent Application No. 60/442,858 (47) entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered Beams by an Object in Interferometry” and U.S. patent application Ser. No. 10/765,369, filed Jan. 27, 2004 (ZI-47) and entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted Beams by an Object in Interferometry” of which both are to Henry A. Hill and of which the contents of provisional and non-provisional applications are herein incorporated in their entirety by reference. 
     Another example of beam-conditioner that may be used in either the first or the second technique comprise combinations of multiple frequency generators and phase shifting types of beam-conditioners such as subsequently described herein with respect to  FIG. 1   e.    
     With a continuation of the description of different ways to configure source  18  and beam-conditioner  22  to meet the input beam requirements of different embodiments of the present invention, source  18  will preferably comprise a pulsed source. There are a number of different ways for producing a pulsed source [see Chapter 11 entitled “Lasers”,  Handbook of Optics,  1, 1995 (McGraw-Hill, New York) by W. Silfvast]. Each pulse of source  18  may comprise a single pulse or a train of pulses such as generated by a mode locked Q-switched Nd:YAG laser. A single pulse train is referenced herein as a pulse and a pulse and a pulse train are used herein interchangeably. 
     Source  18  may be configured in certain embodiments of the present invention to generate two or more frequencies by techniques such as described in a review article entitled “Tunable, Coherent Sources For High-Resolution VUV and XUV Spectroscopy” by B. P. Stoicheff, J. R. Banic, P. Herman, W. Jamroz, P. E. LaRocque, and R. H. Lipson in  Laser Techniques for Extreme Ultraviolet Spectroscopy , T. J. McIlrath and R. R. Freeman, Eds., (American Institute of Physics) p 19 (1982) and references therein. The techniques include for example second and third harmonic generation and parametric generation such as described in the articles entitled “Generation of Ultraviolet and Vacuum Ultraviolet Radiation” by S. E. Harris, J. F. Young, A. H. Kung, D. M. Bloom, and G. C. Bjorklund in  Laser Spectroscopy I , R. G. Brewer and A. Mooradi, Eds. (Plenum Press, New York) p 59, (1974) and “Generation of Tunable Picosecond VUV Radiation” by A. H. Kung,  Appl. Phys. Lett.  25, p 653 (1974). The contents of the three cited articles are herein incorporated in their entirety by reference. 
     The output beams from source  18  comprising two or more frequency components may be combined in beam-conditioner  22  by beam-splitters to form coextensive measurement and reference beams that are either spatially separated or coextensive as required in certain embodiments of the present invention. The frequency shifting of the various components required in certain embodiments of the present invention may be introduced in source  18  for example by frequency modulation of input beams to parametric generators and the phase shifting of reference beams relative to measurement beams in beam-conditioner  22  may be achieved by phase shifters of the optical-mechanical type comprising for example prisms or mirrors and piezoelectric translators or of the electro-optical modulator type. 
     The general description is continued with reference to  FIG. 1   a . Input beam  24  is incident on interferometer  10  wherein reference beams and measurement beams are generated. The reference beams and measurement beams comprise one or two arrays of reference beams and one or two arrays of measurement beams, respectively, for measurements using measurement beams that comprise a single polarization state or two orthogonal polarization states, respectively, wherein the arrays may comprise arrays of one element. The arrays of measurement beams are focused on and/or in measurement object  60  and arrays of return measurement beams are generated by reflection/scattering by measurement object  60 . The arrays of reference beams and return measurement beams are combined by a beam-splitter to form one or two arrays of output beams using measurement beams that comprise a single polarization state or two orthogonal polarization states, respectively. The arrays of output beams are mixed with respect to state of polarization either in interferometer  10  or in detector  70 . The arrays of output beams are subsequently focused to spots on pixels of a multipixel detector and detected to generate the array of electrical interference signals  72 . 
     The conjugated quadratures of fields of return measurement beams are obtained by using a single-, double-, bi-, quad-homodyne detection method or variants thereof. The bi- and quad-homodyne detection methods are described for example in cited U.S. Provisional Patent Application No. 60/442,858 (47) and U.S. patent application Ser. No. 10/765,369, filed Jan. 27, 2004 (ZI-47) and entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted Beams by an Object in Interferometry”. The variants of the bi- and quad-homodyne detection methods are described for example in cited U.S. Provisional Patent Application No. 60/459,425 (ZI-50) and U.S. patent application Ser. No. 10/816,180, filed Apr. 1, 2004 (ZI-50) and entitled “Apparatus and Method for Joint Measurement of Fields of Scattered/Reflected Orthogonally Polarized Beams by an Object in Interferometry”. 
     For the single-homodyne detection method, input beam  24  comprises a single frequency component and sets of four or eight measurements of the array of electrical interference signals  72  is made in non-ellipsometric or ellipsometric measurements, respectively, wherein non-ellipsometric and ellipsometric measurements correspond to measurements made with a measurement beam  30 A comprising a single polarized component or orthogonally polarized components, respectively. For each of the measurements of the array of electrical interference signals  72  in non-ellipsometric and ellipsometric measurements, known phase shifts are introduced between each reference beam component and respective return measurement beam component of output beam  34 . The subsequent data processing procedure used to extract the conjugated quadratures of fields of beams reflected and/or scattered by a substrate is subsequently described herein with respect to the first embodiment of the present invention and also for example in commonly owned U.S. Pat. No. 6,445,453 (ZI-14) entitled “Scanning Interferometric Near-Field Confocal Microscopy” by Henry A. Hill of which the contents are incorporated herein in their entirety by reference. 
     The double-homodyne detection method which is applicable to non-ellipsometric measurements uses input beam  24  comprising four frequency components and four detectors to obtain measurements of electrical interference signals that are subsequently used to obtain conjugated quadratures in non-ellipsometric measurements. Each detector element of the four detector elements obtains a different one of the four electrical interference signal values with the four electrical interference signal values obtained simultaneously to compute the conjugated quadratures for a field. Each of the four electrical interference signal values contains only information relevant to one orthogonal component of the conjugated quadratures. The double-homodyne detection used herein is related to the detection methods such as described in Section IV of the article by G. M D&#39;ariano and M G. A. Paris entitled “Lower Bounds On Phase Sensitivity In Ideal And Feasible Measurements,”  Phys. Rev. A  49, 3022-3036 (1994). Accordingly, the double-homodyne detection method does not make joint determinations of conjugated quadratures of fields wherein each electrical interference signal value contains information simultaneously about each of two orthogonal components of the conjugated quadratures. 
     In the adaptation of the double-homodyne detection method to ellipsometric measurements, input beam  24  comprises eight frequency components and eight detectors to obtain measurements of eight electrical interference signals that are subsequently used to obtain conjugated quadratures. Each detector element of the eight detector elements obtains a different one of the eight electrical interference signal values with the eight electrical interference signal values obtained simultaneously to compute the conjugated quadratures of fields of scattered/reflected orthogonally polarized fields. Each of the eight electrical interference signal values contains only information relevant to one orthogonal component of one of the two conjugated quadratures. 
     The bi- and quad-homodyne detection methods obtain measurements of electrical interference signals wherein each measured value of an electrical interference signal contains simultaneously or substantially simultaneously information about two orthogonal components of conjugated quadratures. The two orthogonal components correspond to orthogonal components of conjugated quadratures such as described in cited U.S Provisional Patent Application No. 60/442,858 (ZI-47) and cited U.S. patent application Ser. No. 10/765,369, filed Jan. 27, 2004 (ZI-47) entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted Beams by an Object in Interferometry”. 
     The variants of the bi- and quad-homodyne detection methods obtain measurements of electrical interference signals wherein each measured value of an electrical interference signal contains simultaneously or substantially simultaneously information about two orthogonal components of each of two conjugated quadratures of fields of scattered/reflected orthogonally polarized beams. The two orthogonal components of the two conjugated quadratures correspond to orthogonal components of conjugated quadratures such as described in cited U.S Provisional Patent Application No. 60/459,425 (ZI-50) and cited U.S. patent application Ser. No. 10/816,180, filed Apr. 1, 2004 (ZI-50) and entitled “Apparatus and Method for Joint Measurement of Fields of Scattered/Reflected Orthogonally Polarized Beams by an Object in Interferometry”. 
     A first embodiment of the present invention is shown schematically in  FIG. 1   b . The schematic diagram in  FIG. 1   b  is of a wafer inspection tool used to make ellipsometric measurements of the wafer surface or subsurface. The first embodiment comprises a first imaging system generally indicated as numeral  100 , pinhole array beam-splitter  12 , detector  70 , and a second imaging system generally indicated as numeral  110 . The second imaging system  110  is low power microscope having a large working distance, e.g. Nikon Nikon ELWD and SLWD objectives and Olympus LWD, ULWD, and ELWD objectives. 
     The first imaging system  100  is shown schematically in  FIG. 1   c . Imaging system  100  is a catadioptric system such as described in commonly owned U.S. Pat. No. 6,552,852 B2 (ZI-38) and U.S. Pat. No. 6,717,736 (ZI-43) both of which are entitled “Catoptric and Catadioptric Imaging System” and U.S. Provisional Patent Applications No. 60/485,507 (ZI-52) and U.S. patent application Ser. No. [tbd] (ZI-52) filed Jul. 7, 2004 both of which are entitled “Apparatus And Method For High Speed Scan For Sub-Wavelength Defects And Artifacts In Semiconductor Metrology” of which the two patents, the patent application, and the provisional patent application are to Henry A. Hill and of which the contents are incorporated herein in their entirety by reference. 
     Catadioptric imaging system  100  comprises a section, i.e., a pie section, of catadioptric imaging system  210  shown schematically in  FIG. 2   a . Elements of catadioptric imaging system  210  shown in  FIG. 2   a  comprise two different media in order to generate an achromatic anastigmat. Catadioptric imaging system  210  comprises catadioptric elements  240  and  244 , beam-splitter  248 , concentric lenses  250  and  254 , and plano convex lenses  256  and  258 . Surfaces  242 A and  246 A are convex spherical surfaces with nominally the same radii of curvature and the respective centers of curvature of surfaces  242 A and  246 A are conjugate points with respect to beam-splitter  248 . Surfaces  242 B and  246 B are concave spherical surfaces with nominally the same radii of curvature. The centers of curvature of surfaces  242 B and  246 B are the same as the centers of curvature of surfaces  246 A and  242 A, respectively. 
     The centers of curvature of the surfaces of concentric lens  250  and plano convex lens  256  are nominally the same as the center of curvature of surfaces  242 B and  246 A. The centers of curvature of the surfaces of concentric lens  254  and piano convex lens  258  are nominally the same as the center of curvature of surfaces  242 A and  246 B. The radii of curvature of surfaces  260  and  264  are nominally the same and the radii of curvature of surfaces  262  and  266  are nominally the same. There may be a small gap between the convex surface and corresponding concave surface of lenses  256  and  250 , respectively, and there may be a corresponding small gap between the convex surface and corresponding concave surface of lenses  258  and  254 , respectively. 
     The sagittal field of catadioptric imaging system  210  is a flat field and the tangential field is also a flat field for a certain object field when the Petzval sum is zero, i.e., 
                       2   ⁢           ⁢       ∑     j   =   1       p   -   1       ⁢       (       1     n   j       -     1     n     j   +   1           )     ⁢     1     r   j             +       1     n   p       ⁢     2     r   p           =   0           (   7   )               
where r j  is the radius of curvature of surface j, r p  is the radius of curvature of the mirror surface, and n j  is the index of refraction of the media located on the beam incidence side of surface j such as shown diagrammatically in  FIG. 2   b . The condition for the generation of an achromatic anastigmat at wavelength λ c  is accordingly given by the equation
 
     
       
         
           
             
               
                 
                   
                     ∂ 
                     
                       
                         [ 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   1 
                                 
                                 
                                   p 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         n 
                                         j 
                                       
                                     
                                     - 
                                     
                                       1 
                                       
                                         n 
                                         
                                           j 
                                           + 
                                           1 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   1 
                                   
                                     r 
                                     j 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               1 
                               
                                 n 
                                 p 
                               
                             
                             ⁢ 
                             
                               2 
                               
                                 r 
                                 p 
                               
                             
                           
                         
                         ] 
                       
                       
                         ∂ 
                         λ 
                       
                     
                   
                   = 
                   0. 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Two considerations in the selection of the radii of curvature of surfaces  242 B and  246 B and surfaces  162  and  166  are the area of the system pupil function of the imaging system  210  and the size of the object field that can be effectively used with respect to image quality. The first two considerations place competing demands of the selection of the radii of curvature of surfaces  242 B and  246 B and surfaces  162  and  166 . Third and fourth considerations are with respect to the conditions set out in Equations (7) and (8). A fifth consideration in the selection of the media of the lenses of imaging system  210  is the transmission properties of the media for the range of wavelengths to be used in an end use application. 
     For an example of an achromatic anastigmat design for deep UV operation, the media of elements  240 ,  244 ,  256 , and  258  is selected as CaF 2  and the media of concentric lenses  252  and  254  is selected as a UV grade fused silica or fluorine-doped fused silica (F—SiO 2 ). Other parameters of the example achromatic anastigmat design such as the radii of curvature of surfaces are listed in Table 3 for λ c =250 nm. With this choice of media, the operation range is down to 170 nm for UV grad fused silica and down to 155 nm for F—SiO 2 . For the achromatic anastigmat design parameters listed in Table 3, the contribution of geometric ray tracing effects is ≦40 nm for an object field of 1.5 mm in diameter and a numerical aperture NA=0.970 in the object space just outside of the plane surface of piano convex lens  258 . 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Achromatic Anastigmat Design for λ c  = 250 nm 
               
             
          
           
               
                   
                 Media 
                 j 
                 n j   
                 r j  (mm) 
               
               
                   
                   
               
             
          
           
               
                   
                 CaF 2   
                 1 
                 1.467297 
                 3.600 
               
               
                   
                 Fused Silica 
                 2 
                 1.507446 
                 9.256 
               
               
                   
                 Vacuum 
                 3 
                 1 
                 18.000 
               
               
                   
                 CaF 2   
                 4 
                 1.467297 
                 50.000 
               
               
                   
                   
               
             
          
         
       
     
     A variant of catadioptric imaging system  210  is shown in  FIG. 2   c  wherein catadioptric imaging system  100  is an anastigmat that is not achromatic. The media of elements  140  and  144  may comprise CaF 2 , BaF 2 , or SrF 2  for work down to 140 nm and UV grade fused silica for operation to 180 nm. The respective radii of the curvature for anastigmat design at λ=250 nm using CaF 2  are listed in Table 4. For anastigmat design listed in Table 4, the contribution of geometric ray tracing effects is ≦40 nm for an object field of 1.5 mm and a numerical aperture NA=0.970 in the object space just outside of the plane surface of plano convex lens  258 . 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Anastigmat Design for λ = 250 nm 
               
             
          
           
               
                   
                 Media 
                 j 
                 n j   
                 r j  (mm) 
               
               
                   
                   
               
             
          
           
               
                   
                 CaF 2   
                 1 
                 1.467297 
                 7.950 
               
               
                   
                 Air 
                 2 
                 1 
                 12.000 
               
               
                   
                 CaF 2   
                 3 
                 1.467297 
                 50.000 
               
               
                   
                   
               
             
          
         
       
     
     The respective radii of curvature for anastigmat design at λ=250 nm using fused silica are listed in Table 5. For the anastigmat design listed in Table 4, the contribution of geometric ray tracing effects is ≦40 nm for an object field of 1.5 mm and a numerical aperture NA=0.970 in the object space just outside of the plane surface of plano convex lens  258 . 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Anastigmat Design for λ = 250 nm 
               
             
          
           
               
                   
                 Media 
                 j 
                 n j   
                 r j  (mm) 
               
               
                   
                   
               
             
          
           
               
                   
                 Fused Silica 
                 1 
                 1.467297 
                 7.950 
               
               
                   
                 Air 
                 2 
                 1 
                 12.000 
               
               
                   
                 Fused Silica 
                 3 
                 1.467297 
                 50.000 
               
               
                   
                   
               
             
          
         
       
     
     The respective radii of curvature for anastigmat design at λ=250 nm using fused silica are listed in Table 5. For the anastigmat design listed in Table 5, the contribution of geometric ray tracing effects is ≦40 nm for an object field of 1.5 mm and a numerical aperture NA=0.970 in the object space just outside of the plane surface of piano convex lens  258 . 
     Intrinsic birefringence of SrF 2  is less than the intrinsic birefringence of CaF 2  and BaF 2  at 140 nm. However, the intrinsic birefringence of any one of the three crystalline materials can be accommodated in the catadioptric imaging system  210  since only an azimuthal section of the lens elements are used and that section can be selected to significantly reduce the effects of intrinsic birefringence, e.g., with the [111] axis of the crystal aligned parallel to the optic axis of catadioptric imaging system  210  and the [110] axis of the crystal aligned parallel to the plane of  FIG. 2   a.    
     Another form of catadioptric imaging system that may be used for catadioptric and catoptric imaging system  100  is the catadioptric imaging system shown schematically in  FIG. 2   f  and such as described in commonly owned U.S. Provisional Patent Application No. 60/460,129 entitled “Apparatus and Method for Measurement of Fields of Forward Scattered/Reflected and Backscattered Beams by an Object in Interferometry” and U.S. patent application Ser. No. 10/816,172, (ZI-51) filed Apr. 1, 2004 and entitled “Apparatus And Method For Measurement Of Fields Of Forward Scattered/Reflected and Backscattered Beams By An Object In Interferometry” of which each are to Henry A. Hill and of which the contents of the provisional patent application and of the patent application are herein incorporated in their entirety by reference. The catadioptric imaging system shown in  FIG. 2   f  comprises the same elements as the astigmatic catadioptric imaging system shown in  FIG. 2   c  except for the omission of element  258 . The sagittal field of catadioptric imaging system  210  shown in  FIG. 2   f  is a flat field and the tangential field is also a flat field for a certain object field when the Petzval sum is zero, i.e., 
                         (       1     n   1       -     1     n   2         )     ⁢     1     r   1         +     2   ⁢           ⁢       ∑     j   =   2       p   -   1       ⁢       (       1     n   j       -     1     n     j   +   1           )     ⁢     1     r   j             +       1     n   p       ⁢     2     r   p           =   0           (   9   )               
where r 1  corresponds to the radius of curvature of surface  260 . An example of a set of respective radii of curvature for an anastigmat design for at λ=250 nm using fused silica comprises the same radii of curvature listed in Table 5 for r 2  and r 3  and ½ of the value listed for r 1 .
 
     The location of the object plane of catadioptric imaging system  210  is outside of plano convex lens  158  and on the surface of substrate  60  which is shown diagrammatically in  FIG. 2   d . The separation of the plane surface of plano convex lens  158  and the surface of substrate  60  is h. The object plane of catadioptric imaging system  210  may also be located in the interior of substrate  60  which is shown diagrammatically in  FIG. 2   e . When used in an end use application such as the embodiments in which the resolution of catadioptric imaging system  210  is restricted to a Δθ such as described with respect to Equation (6), the spherical aberration introduced by transmission through plane surfaces shown in  FIGS. 2   d  and  2   e  generally will not impact on the performance of the imaging system  210 . 
     An advantage of the catadioptric imaging system  210  is that as a consequence of the spherical aberration introduced by transmission through plane surfaces, the effective angle of incidence θ I  can be scanned by introducing a scan in h. 
     For those end use applications where compensation is required for the spherical aberration introduced by transmission through plane surfaces, procedures may be use such as described in commonly owned U.S. Provisional Patent Application No. 60/444,707 (ZI-44) and U.S. patent application Ser. No. 10/771,785 (ZI-44) of which both are entitled “Compensation for Effects of Mismatch in Indices of Refraction at a Substrate-Medium Interface in Confocal and Interferometric Confocal Microscopy” of which both are to Henry A. Hill and the contents of which are herein incorporated in their entirety by reference. 
     The description of imaging system  100  is continued with reference to  FIG. 1   c . Lens sections  40  and  44  are sections, i.e., pie sections, of lens  240  and  244  shown in  FIG. 2   a . Lens elements  250 ,  256 ,  254 , and  258  in  FIG. 1   c  are the same elements lens elements  250 ,  256 ,  254 , and  258  in  FIG. 2   a . Convex lens  52  has a center of curvature the same as the center of curvature of convex lens  250 . Convex lenses  250  and  52  are bonded together with pinhole array beam-splitter  12  in between. Pinhole array beam-splitter  12  is shown in  FIG. 1   d . The pattern of pinholes in pinhole array beam-splitter is chosen so that the image of pinhole array beam-splitter  12  on detector  70  to match the pixel pattern of detector  70 . An example of a pattern is a two dimensional array of equally spaced pinholes in two orthogonal directions. The pinholes may comprise circular apertures, rectangular apertures, or combinations thereof such as described in commonly owned U.S. patent application Ser. No. 09/917,402 (ZI-15) entitled “Multiple-Source Arrays for Confocal and Near-field Microscopy” by Henry A. Hill and Kyle Ferrio of which the contents thereof are incorporated herein in their entirety by reference. The pinholes may also comprise microgratings such as described in commonly owned U.S. Provisional Patent Application No. 60/459,425 (ZI-50) entitled “Apparatus and Method for Joint Measurement of Fields of Scattered/Reflected Orthogonally Polarized Beams by an Object in Interferometry” and U.S. patent application Ser. No. 10/816,180, (ZI-50) filed Apr. 1, 2004 and entitled “Apparatus and Method for Joint Measurement Of Fields Of Scattered/Reflected Or Transmitted Orthogonally Polarized Beams By An Object In Interferometry” both of which each are to Henry A. Hill and of which the contents of the provisional patent applications and of the patent application are herein incorporated in their entirety by reference. A nonlimiting example of a pinhole array for pinhole array beam-splitter  12  is shown in  FIG. 1   d  having a spacing between pinholes of b with aperture size a. 
     With reference to  FIGS. 1   b  and  1   c , a first portion of input beam  24  is reflected by non-polarizing beam-splitter  54 A and incident on pinhole beam-splitter  12  after reflection by mirrors  54 B and  54 C. The angle of incidence of the reference beam at pinhole array beam-splitter  12  is selected to correspond to θ D  (see Equation (3) and related discussion) in the plane of  FIG. 1   c  for which information is to be obtained. The reference beam may alternatively be incident on pinhole array beam-splitter  12  from the opposite side with an angle of incidence equal to θ D  in the plane of  FIG. 1   c . The direction of propagation of the reference beam incident from the opposite side is the mirror image about the surface of pinhole array beam-splitter  12  of the direction of propagation of the reference beam shown in  FIG. 1   c . A portion of the beam reflected by beam-splitter  54 A and incident on pinhole array beam-splitter  12  is reflected/scattered as reference beam components of output beam  32 A and  32 B (see  FIG. 1   b ). 
     A general property of various embodiments of the present invention is that the integrated interference cross-term between the reference beam and the respective return measurement beam across the spot by the detector  70  will have a peak in detection sensitivity for the reflected/scattered return measurement beam components of beams  28 C that have an angle of diffraction corresponding to θ D . By the selection of the angle of incidence of the reference beam at pinhole array beam-splitter  12  to correspond to θ D , the reflected/scattered return measurement beam components of beams  28 C across the spot on beam-splitter  12  remains in phase with the reference beam. As a consequence, the interference cross-term between the reference beam and the respective return measurement beam across the spot comprises a constant term and an oscillatory term. In this case, the interference cross-term integrates across the spot to a meaningful non-zero value proportional to the constant term. Whereas the projection of the spatial wavelength onto beam splitter  12  for the other diffraction angles of the return measurement beam is not equal to the projection of the wavelength of the reference beam onto beam-splitter  12  so that the return measurement beam components of beams  28 C across the spot on beam-splitter  12  do not remain in phase with the reference beam. Accordingly, the interference cross-term for those diffraction directions oscillates across the spot comprising two oscillatory terms and therefore their contributions integrate to zero or reduced values. 
     Another general property of various embodiments of the present invention is that the since the reference beam is generated with the angle of incidence at pinhole array beam-splitter  12  selected to correspond to θ D , the phases of conjugated quadratures corresponding to the interference cross-term between the reference beam and the reflected/scattered return measurement beam components of beams  28 C in the electrical interference signal  72  generated by detection of mixed output beams by detector  70  is ζ x  and ζ y  (see Equations (1) and (2) and associated description) with no dependences on either x or y. This is an important feature since the phase represented in conjugated quadratures is a function only of the reflecting properties and location of the defect and/or artifact in addition to a fixed offset error in the interferometric metrology system. A corollary statement is that the accuracy to which the location of a defect and/or artifact on or in substrate being imaged can be measured is not affected by displacements of a pinhole corresponding to a detector or of a detector pixel used in measuring the respective conjugated quadratures other than contributing to a phase redundancy mod 2π. 
     A second portion of beam  24  is transmitted by beam-splitter  54 A as measurement beam  24 A after reflection by mirror  54 D. Measurement beam  24 A is incident on substrate  60  and a portion thereof is reflected and/or scattered as return measurement beam components of beams  28 C. The angle of incidence of the measurement beam at substrate  60  is selected to correspond to θ I  (see Equation (3) and related discussion). Return measurement beam components of beam  28 C are imaged by catadioptric imaging system  100  to pinhole array beam-splitter  12  and a portion thereof is transmitted as return measurement beam components of output beams  32 A and  32 B. 
     The next step is the imaging of output beams  32 A and  32 B by imaging system  110  to an array of spots that coincide with the pixels of a multi-pixel detector such as a CCD to generate an array of electrical interference signals  72 . The array of electrical interference signals is transmitted to signal processor and controller  80  for subsequent processing. 
     Conjugated quadratures of fields of the return measurement beam are obtained by single-homodyne detection in the first embodiment of the present invention. For the single-homodyne detection, a set of four measurements of electrical interference signals  72  is made. For each of the four measurements of the electrical interference signals  72 , a known phase shift is introduced between the reference beam component and respective return measurement beam component of output beams  32 A and  32 B. A nonlimiting example of a known set of phase shifts comprise 0, π/4, π/2, and 3π/2 radians. 
     Input beam  24  comprises in the first embodiment one frequency component. The phase shifts are generated in the first embodiment by shifting the frequency of the input beam  24  between known frequency values. There is a difference in optical path length between the reference beam components and the respective return beam components of output beams  32 A and  32 B and as a consequence, a change in frequency of input beam  24  will generate a corresponding phase shift between the reference beam components and the respective return beam components of output beams  32 A and  32 B. 
     For an optical path difference L between the reference beam components and the respective return measurement beam components of output beams  32 A and  32 B, there will be for a frequency shift Δf a corresponding phase shift φ where 
                   φ   =     2   ⁢           ⁢   π   ⁢           ⁢     L   ⁡     (       Δ   ⁢           ⁢   f     c     )                 (   10   )               
and c is the free space speed of light. Note that L is not a physical path length difference and depends for example on the average index of refraction of the measurement beam and the return measurement beam paths. For an example of a phase shift φ=π, 3π, 5π, . . . and a value of L=0.25 m, the corresponding frequency shift Δf=600 MHz, 1.8 GHz, 3.0 GHz, . . .
 
     The frequency of input beam  24  is determined by beam-conditioner  22  according to control signal  74  generated by electronic processor and controller  80 . Source  18  of input beam  20 , such as a laser, can be any of a variety of single frequency lasers. 
     Two different modes of operation are described for the acquisition of the four electrical interference signal values. The first mode to be described is a step and stare mode wherein substrate  60  is stepped between fixed locations for which image information is desired. The second mode is a scanning mode. In the step and stare mode for generating a one-, a two-, or a three-dimensional image of substrate  60 , substrate  60  is translated by stage  90  wherein substrate  60  is mounted on wafer chuck  84  with wafer chuck  84  mounted on stage  90 . The position of stage  90  is controlled by transducer  82  according to servo control signal  78  from electronic processor and controller  80 . The position of stage  90  is measured with respect to a reference system by metrology system  88  and position information acquired by metrology system  88  is transmitted to electronic processor and controller  80  to generate an error signal for use in the position control of stage  90 . The reference system may correspond to the reference frame of an inspection tool used for wafer inspection tool. Metrology system  88  may comprise for example linear displacement and angular displacement interferometers and cap gauges. 
     Electronic processor and controller  80  translates stage  90  to a desired position and then acquires the set of four electrical interference signal values corresponding to the set of four phase shifts 0, π/4, π/2, and 3π/2. After the acquisition of the sequence of four electrical interference signal values, electronic processor and controller  80  repeats the procedure for the next desired position of stage  90 . The elevation and angular orientation of substrate  60  is controlled by transducers  86 A and  86 B. 
     The second of the two modes for the acquisition of the electrical interference signal values is next described wherein the electrical interference signal values are obtained with the position of stage  90  scanned in one or more directions. In the scanning mode, source  18  is pulsed at times controlled by signal  92  from signal processor and controller  80 . Source  18  is pulsed at times corresponding to the registration of the conjugate image of pinholes of pinhole array beam-splitter  12  with positions on and/or in substrate  60  for which image information is desired. 
     There will be a restriction on the duration or “pulse width” of a beam pulse τ p1  produced by source  18  as a result of the continuous scanning used in the scanning mode of the first embodiment. Pulse width τ p1  will be a parameter that in part controls the limiting value for spatial resolution in the direction of a scan to a lower bound of
 
τ p1 v,  (11)
 
where v is the scan speed. For example, with a value of τ p1 =50 nsec and a scan speed of v=0.20 m/sec, the limiting value of the spatial resolution τ p1 v in the direction of scan will be
 
τ p1   v= 10 nm.  (12)
 
     Source  18  and beam-conditioner  22  may be configured for certain end use applications wherein there is a time delay Δτ between the temporal profiles of two or more of the frequency components of input beam  24 . There will be a restriction on the magnitude of the time delay Δτ according to properties of quantities being measured. For example, Δτ will be a parameter that in part controls the limiting value for spatial resolution in the direction of a scan to a lower bound of
 
vΔτ.  (13)
 
For the example of Δτ=50 nsec and a scan speed of v=0.20 m/sec, the limiting value of the spatial resolution vΔτ in the direction of scan will be the same value as given in Equation (12).
 
     The frequency of input beam  24  is controlled by signal  74  from electronic processor and controller  80  to correspond to a frequency of a set of four frequencies that will yield the desired phase shift of the set of four phase shifts between the reference and return measurement beam components of output beams  32 A and  32 B. In the first mode for the acquisition of the electrical interference signal values, each of the sets of four electrical interference signal values corresponding to the set of four phase shift values are generated by a single pixel of detector  70 . In the second mode for the acquisition of the electrical interference signal values, the set of four electrical interference signal values corresponding to the set of four phase shift values are generated by a conjugate set of four different pixels of detector  70 . Thus in the second mode of acquisition, the differences in pixel efficiency and the differences in sizes of pinholes in pinhole array beam-splitter  12  need to be compensated in the signal processing by electronic processor and controller  80  to obtain conjugated quadratures of fields of return measurement beam components. 
     The advantage of the second mode is that the electrical interference signal values are acquired in a scanning mode which increases through put of the interferometric confocal microscopy system of the first embodiment. 
     The description beam-conditioner  22  is the same as the description given for the two frequency generator and frequency-shifter described in commonly owned U.S. Provisional Application No. 60/442,858 (ZI-47) entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered Beams by an Object in Interferometry” and U.S. patent application Ser. No. 10/765,369, filed Jan. 27, 2004 (ZI-47) and entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted Beams by an Object in Interferometry” of which both are to Henry A. Hill and of which the contents of provisional and non-provisional applications are herein incorporated in their entirety by reference. 
     Reference is made to  FIG. 1   e  where beam-conditioner  22  is first described generally as a two-frequency generator and a frequency-shifter. Beam-conditioner  22  may be operated to generate a beam  24  that has either a single frequency-shifted component or two frequency-shifted components. Beam-conditioner  22  is operated in the single frequency-shifted mode for the first embodiment. 
     Beam-conditioner  22  comprises acousto-optic modulators  1120 ,  1126 ,  1130 ,  1132 ,  1142 ,  1146 ,  1150 ,  1154 ,  1058 , and  1062 ; beam-splitter  1168 ; and mirror  1166 . Input beam  20  is incident on acousto-optic modulator  1120  with a plane of polarization parallel to the plane of  FIG. 1   e . A first portion of beam  20  is diffracted by acousto-optic modulator  1120  as beam  1122  and then by acousto-optic modulator  1126  as beam  1128  having a polarization parallel to the plane of  FIG. 1   e.  A second portion of beam  20  is transmitted as a non-diffracted beam  1124  having a plane of polarization parallel to the plane of  FIG. 1   e . For beam-conditioner  22  operated to generate the single frequency-shifted component for beam  24 , the acoustic power to acousto-optic modulator  1120  is switched between two states. One state is the off state where the amplitude of diffracted beam  1122  in zero and in the on state, the amplitude of non-diffracted beam  1124  is nominally zero. The on or off states of acousto-optic modulator  1120  is controlled by signal  74  generated by electronic processor and controller  80   
     Acousto-optic modulators  1120  and  1126  may be of either the non-isotropic Bragg diffraction type or of the isotropic Bragg diffraction type. The frequency shifts introduced by acousto-optic modulators  1120  and  1126  are of the same sign and equal to ½ of a frequency shift Δf that will generate in interferometer  10  a π/2 phase difference between a reference beam and a measurement beam that have a difference in frequency equal to frequency shift Δf. The direction of propagation of beam  1128  is parallel to the direction of propagation of beam  1124 . 
     Continuing with  FIG. 1   e , beam  1128  is incident on acousto-optic modulator  1132  and is either diffracted by acousto-optic modulator  1132  as beam  1134  or transmitted by acousto-optic modulator  1132  as beam  1136  according to control signal  74  from electronic processor and controller  80 . When beam  1134  is generated, beam  1134  is diffracted by acousto-optic modulators  1142 ,  1146 , and  1150  as a frequency shifted beam component of beam  1152 . The frequency shifts introduced by acousto-optic modulators  1132 ,  1142 ,  1146 , and  1150  are all in the same direction and equal in magnitude to Δf/2. Thus the net frequency shift introduced by acousto-optic modulators  1132 ,  1142 ,  1146 , and  1150  is ±2Δf. The net frequency shifts introduced by acousto-optic modulators  1120 ,  1126 ,  1132 ,  1142 ,  1146 , and  1150  are Δf and Δf±2Δf that in turn will generate a respective relative phase shifts of π/2 and π/2±π between the respective reference and measurement beams in interferometer  10 . 
     When beam  1136  is generated, beam  1136  is transmitted as a non-frequency shifted beam component of beam  1152 . The frequency shift introduced by acousto-optic modulators  1120 ,  1126 ,  1132 , and  1150  is Δf and will generate a respective relative phase shift of π/2 between the respective reference and measurement beams in interferometer  10 . 
     Beam  1124  is incident on acousto-optic modulator  1130  and is either diffracted by acousto-optic modulator  1130  as beam  1140  or transmitted by acousto-optic modulator  1130  as beam  1138  according to control signal  74  from electronic processor and controller  80 . When beam  1140  is generated, beam  1140  is diffracted by acousto-optic modulators  1154 ,  1158 , and  1162  as a frequency-shifted beam component of beam  1164 . The frequency shifts introduced by acousto-optic modulators  1130 ,  1154 ,  1158 , and  1162  are all in the same direction and equal to ±Δf/2. Thus the net frequency shift introduced by acousto-optic modulators  1130 ,  1154 ,  1158 , and  1162  is ±2Δf and will generate a relative phase shift of π between the respective reference and measurement beams in interferometer  10 . The net frequency shift introduced by acousto-optic modulators  1120 ,  1130 ,  1154 ,  1158 , and  1162  is ±2Δf and will generate a respective relative phase shift of ±π between the respective reference and measurement beams in interferometer  10 . 
     When beam  1138  is generated, beam  1138  is transmitted as a non-frequency shifted beam component of beam  1164 . The corresponding frequency shift introduced by acousto-optic modulators  1120 ,  1130 , and  1162  is 0 and will generate a respective relative phase shift of 0 between the respective reference and measurement beams in interferometer  10 . 
     Beams  1152  and  1164  may be used directly as input beam  24  when an embodiment requires spatially separated reference and measurement beams for an input beam. When an embodiment of the present invention requires coextensive reference and measurement beams as an input beam, beam  1152  and  1164  are next combined by beam-splitter  1168  to form beam  24 . Acousto-optic modulators  1120 ,  1126 ,  1130 ,  1132 ,  1142 ,  1146 ,  1150 ,  1154 ,  1158 , and  1162  may be either of the non-isotropic Bragg diffraction type or of the isotropic Bragg diffraction type. Beams  1152  and  1164  are both polarized in the plane of  FIG. 1   e  for either non-isotropic Bragg diffraction type or of the isotropic Bragg diffraction type and beam-splitter  1168  is of the non-polarizing type. 
     In the first embodiment of the present invention, the processing of the measured arrays of sets of four measured electrical interference signal values for the determination of conjugated quadratures of fields of return measurement beams is described herein as a special case of the bi-homodyne detection method and is also described for example in cited U.S. Pat. No. 6,445,453 (ZI-14). 
     Referring to the bi-homodyne detection method such as described in cited U.S Provisional Patent Application No. 60/442,858 (ZI-47) and cited U.S. patent application Ser. No. 10/765,369, filed Jan. 27, 2004 (ZI-47) entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted Beams by an Object in Interferometry” and U.S. Provisional Patent Application No. 60/485,507 (ZI-52) and U.S. patent application Ser. No. [t.b.d.] (ZI-52) filed Jul. 7, 2004 both of which are entitled “Apparatus And Method For High Speed Scan For Sub-Wavelength Defects And Artifacts In Semiconductor Metrology” wherein conjugated quadratures are obtained jointly or substantially jointly, a set of four electrical interference signal values is obtained for each spot on and/or in substrate  60  being imaged. The set of four electrical interference signal values S j , j=1, 2, 3, 4 used for obtaining conjugated quadratures of fields for a single a spot on and/or in a substrate being imaged is represented for the bi-homodyne detection method within a scale factor by the formula 
                     S   j     =       P   j     ⁢           ⁢       ∑     m   =   1     2     ⁢     {               ξ   j   2     ⁢            A   m          2       +       ζ   j   2     ⁢            B   m          2       +       η   j   2     ⁢            C   m          2       +                   ζ   j     ⁢     η   j     ⁢           ⁢   2   ⁢          B   m          ⁢          C   m          ⁢           ⁢   cos   ⁢           ⁢     φ       B   m     ⁢     C   m     ⁢     ɛ     m   ,   j             +                   ξ   j     ⁢     ζ   j     ⁢           ⁢   2   ⁢          A   m          ⁢          B   m          ⁢           ⁢   cos   ⁢           ⁢     φ       A   m     ⁢     B   m     ⁢     ɛ     m   ,   j             +                   ɛ     m   ,   j       ⁢     ξ   j     ⁢       η   j     ⁡     [     1   -       (     -   1     )     m       ]       ⁢          A   m          ⁢          C   m          ⁢           ⁢   cos   ⁢           ⁢     φ       A   m     ⁢     C   m           +                 ɛ     m   ,   j       ⁢     ξ   j     ⁢       η   j     ⁡     [     1   +       (     -   1     )     m       ]       ⁢          A   m          ⁢          C   m          ⁢           ⁢   sin   ⁢           ⁢     φ       A   m     ⁢     C   m                 }                 (   14   )               
where coefficient A m  represents the amplitude of the reference beam corresponding to the frequency component of the input beam  24  that has index m; coefficient B m  represents the amplitude of the background beam corresponding to reference beams A m ; coefficient C m  represents the amplitude of the return measurement beam corresponding to reference beam A m ; P j  represents the integrated intensity of the first frequency component of the input beam  24  pulse j of a sequence of 4 pulses; and the values for ε m,j  are listed in Table 6. There are other set of values for ε m,j  that may be used in embodiments of the present invention wherein the other set of values for ε m,j  satisfy the conditions set out in subsequent Equations (15) and (16) herein.
 
     
       
         
               
             
               
               
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 6 
               
             
             
               
                   
               
               
                 ε m,j   
               
             
          
           
               
                   
                 m 
                   
               
             
          
           
               
                 j 
                 1 
                 2 
               
               
                   
               
             
          
           
               
                 1 
                 1 
                 1 
               
               
                 2 
                 1 
                 −1 
               
               
                 3 
                 −1 
                 −1 
               
               
                 4 
                 −1 
                 1 
               
               
                   
               
             
          
         
       
     
     The change in the values of ε m,j  from 1 to −1 or from −1 to 1 corresponds to changes in relative phases of respective reference and measurement beams. The coefficients ξ j , ζ j , and η j  represent effects of variations in properties of a conjugate set of four pinholes such as size and shape used in the generation of the spot on and/or in substrate  60  and the sensitivities of a conjugate set of four detector pixels corresponding to the spot on and/or in substrate  60  for the reference, background, and the return measurement beam, respectively. In the single-frequency single-homodyne detection operating in a non-scanning mode, the conjugate set of pinholes corresponds to a single pinhole and the conjugate set of four pixels corresponds to a single pixel. The conjugate set of four pinholes comprise pinholes of pinhole array beam-splitter  12  that are conjugate to a spot in or on the substrate being imaged at different times during the scan. 
     The relationships cos φ A     2     C     2   =sin φ A     1     C     1    and cos φ A     4     C     4   =sin φ A     3     C     3    have been used in deriving Equation (14) without departing from either the scope or spirit of the present invention since cos φ A     2     C     2   =±sin φ A     1     C     1    and cos φ A     4     C     4   =±sin φ A     3     C     3    by control of the relative phase shifts between corresponding reference and return measurement beam components in beam  32 . 
     It has also been assumed in Equation (14) that the ratios |A 2 |/|A 1 | and |A 4 |/|A 3 | are not dependent on j or on the value of P j . In order to simplify the representation of S j  so as to project the important features without departing from either the scope or spirit of the present invention, it is also assumed in Equation (14) that the corresponding ratios of the amplitudes of the return measurement beams are not dependent on j or on the value of P j . However, the ratios |C 2 |/|C 1 | and |C 4 |/|C 3 | will be different from the ratio |A 2 |/|A 1 | and |A 4 |/|A 3 |, respectively, when the ratios of the amplitudes of the measurement beam components corresponding to A 2  and A 1  are different from the ratio |A 2 |/|A 1 | and corresponding to A 4  and A 3  are different from the ratio |A 4 |/|A 3 |. 
     The change in phases φ A     m     B     m     ε     m,j    for a change in ε m,j  may be different from π for embodiments where phase shifts are introduced between the arrays of reference and measurement beams by changing the frequency of an input beam component. It may be of value in evaluating the effects of the background beams to note that the factor cos φ B     m     C     m     ε     m,j    may be written as cos [φ A     m     C     m   +(φ B     m     C     m     ε     m,j   −φ A     m     C     m   )] where the phase difference (φ B     m     C     m     ε     m,j   −φ A     m     C     m   ) is the same as the measured phase φ A     m     B     m     ε     m,j   . 
     It is evident from inspection of Equation (14) that the components of conjugated quadratures ε m,j |C m |cos φ A     m     C     m    and ε m,j |C m |sin φ A     m     C     m    in Equation (14) are functions that have mean values of zero since 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               4 
                             
                             ⁢ 
                             
                               ɛ 
                               
                                 m 
                                 , 
                                 j 
                               
                             
                           
                           = 
                           0 
                         
                         , 
                       
                     
                     
                       
                         
                           m 
                           = 
                           1 
                         
                         , 
                         2. 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Another important property is that the conjugated quadratures ε m,j |C m |cos φ A     m     C     m    and ε m′,j |C m′ |sin φ A     m′     C     m′    are orthogonal over the range of m=1, 2 for m≠m′ since ε m,j  and ε m′,j  are orthogonal over the range of j=1, 2, 3, 4, i.e., 
                       ∑     j   =   1     4     ⁢       ɛ     m   ,   j       ⁢     ɛ       m   ′     ,   j           =     4   ⁢           ⁢     δ     m   ,     m   ′                   (   16   )               
where δ m,m′  is the Kronecker delta defined by
 δ m,m′ =1 for m=m′, δ m,m′ =0 for m≠m′.  (17) 
     Information about conjugated quadratures |C m |cos φ A     m     C     m    and |C m |sin φ A     m     C     m    is obtained using a digital filter F m (S j ) on signals S j  that are based on the orthogonality properties of the ε m,j  as represented by Equation (16). The definition of F m (S j ) and the output of digital filter F(S j ) are 
                             F   m     ⁡     (     S   j     )       =       ⁢       ∑     j   =   1     4     ⁢       ɛ     m   ,   j       ⁢           ⁢       S   j         P   j   ′     ⁢     ξ   j     ′   ⁢           ⁢   2                           =       ⁢         ∑       m   ′     =   1     2     ⁢              A     m   ′            2     ⁢           ⁢       ∑     j   =   1     4     ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (       ξ   j   2       ξ   j     ′   ⁢           ⁢   2         )             +                     ⁢         ∑       m   ′     =   1     2     ⁢              B     m   ′            2     ⁢       ∑     j   =   1     4     ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (       ζ   j   2       ξ   j     ′   ⁢           ⁢   2         )             +                     ⁢         ∑       m   ′     =   1     2     ⁢              C     m   ′            2     ⁢           ⁢       ∑     j   =   1     4     ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (       η   j   2       ξ   j     ′   ⁢           ⁢   2         )             +                     ⁢       [     1   -       (     -   1     )     m       ]     ⁢          A   m          ⁢          C   m          ⁢           ⁢   cos   ⁢           ⁢     φ       A   m     ⁢     C   m                           ⁢         ∑     j   =   1     4     ⁢       (       P   j       P   j   ′       )     ⁢     (         ξ   j     ⁢     η   j         ξ   j     ′   ⁢           ⁢   2         )     ⁢           ⁢       ∑       m   ′     =   1     2     ⁢       ɛ     m   ,   j       ⁢           ⁢     ɛ       m   ′     ,   j               +                     ⁢       [     1   +       (     -   1     )     m       ]     ⁢          A   m          ⁢          C   m          ⁢           ⁢   sin   ⁢           ⁢     φ       A   m     ⁢     C   m                           ⁢         ∑     j   =   1     4     ⁢       (       P   j       P   j   ′       )     ⁢     (         ξ   j     ⁢     η   j         ξ   j     ′   ⁢           ⁢   2         )     ⁢           ⁢       ∑       m   ′     =   1     2     ⁢       ɛ     m   ,   j       ⁢           ⁢     ɛ       m   ′     ,   j               +                     ⁢       2   ⁢           ⁢       ∑       m   ′     =   1     2     ⁢            A     m   ′            ⁢          B     m   ′            ⁢           ⁢       ∑     j   =   1     4     ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (         ξ   j     ⁢     ζ   j         ξ   j     ′   ⁢           ⁢   2         )     ⁢           ⁢   cos   ⁢           ⁢     φ       A     m   ′       ⁢     B     m   ′       ⁢     ɛ       m   ′     ,   j                     +                     ⁢     2   ⁢           ⁢       ∑       m   ′     =   1     2     ⁢            B     m   ′            ⁢          C     m   ′            ⁢           ⁢       ∑     j   =   1     4     ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (         ζ   j     ⁢     η   j         ξ   j     ′   ⁢           ⁢   2         )     ⁢           ⁢   cos   ⁢           ⁢     φ       B     m   ′       ⁢     C     m   ′       ⁢     ɛ       m   ′     ,   j                                   (   18   )               
where ξ′ j  and P j ′ are values used in the digital filters to represent ξ j  and P j , respectively.
 
     The conjugated quadratures |C m |cos φ A     m     C     m    and |C m |sin φ A     m     C     m    correspond to the interference cross-term between the reference beam and the reflected/scattered return measurement beam components of beams  28 C in the electrical interference signal  72  generated by detection of mixed output beams by detector  70 . The phase of the conjugated quadratures |C m |cos φ A     m     C     m    and |C m |sin φ A     m     C     m   , i.e., φ A     m     C     m   , relates to ζ x  and ζ y  (see Equations (1) and (2) and associated description). 
     The parameters 
                     [       (            A   2                 A   1            )     ⁢     (            C   2                 C   1            )       ]     ,           (   19   )               [       (            A   4                 A   3            )     ⁢     (            C   4                 C   3            )       ]           (   20   )               
need to be determined in order complete the determination of a conjugated quadratures. The parameters given in Equations (19) and (20) can be measured for example by introducing π/2 phase shifts into the relative phase of the reference beam and the measurement beam and repeating the measurement for the conjugated quadratures. The ratios of the amplitudes of the conjugated quadratures corresponding to (sin φ A     1     C     1   /cos φ A     1     C     1   ) and (sin φ A     3     C     3   /cos φ A     3     C     3   ) from the first measurement divided by the ratios of the amplitudes of the conjugated quadratures corresponding to (sin φ A     1     C     1   /cos φ A     1     C     1   ) and (sin φ A     3     C     3   /cos φ A     3     C     3   ), respectively, from the second measurement are equal to
 
                       [       (            A   2                 A   1            )     ⁢     (            C   2                 C   1            )       ]     2     ,           (   21   )                   [       (            A   4                 A   3            )     ⁢     (            C   4                 C   3            )       ]     2     ,           (   22   )               
respectively.
 
     Note that certain of the factors in Equation (18) have nominal values of 4 within a scale factor, e.g., have nominal values of either 0 or 4, e.g., 
                       ∑     j   =   1     4     ⁢           ⁢       (       P   j       P   j   ′       )     ⁢     (         ξ   j     ⁢     η   j         ξ   j   ′2       )     ⁢       ∑       m   ′     =   1     2     ⁢           ⁢       ɛ     m   ,   j       ⁢     ɛ       m   ′     ,   j               ≅       ∑     j   =   1     4     ⁢           ⁢       (       P   j       P   j   ′       )     ⁢     (         ξ   j     ⁢     η   j         ξ   j   ′2       )     ⁢     δ     m   ,     m   ′             ≅     4   ⁢     δ     m   ,     m   ′                   (   23   )               
where δ m,m′  is the Kronecker delta defined by Equation (17). The scale factors corresponds to the average value for the ratio of (ξ′ j ) 2 /(ξ j η j ) assuming that the average values of P j /P j ′≅1.
 
     Certain other of the factors in Equation (18) have nominal values of zero, e.g., 
                         ∑     j   =   1     4     ⁢           ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (       ξ   j   2       ξ   j   ′2       )         ≅   0     ,         ∑     j   =   1     4     ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (       ζ   j   2       ξ   j   ′2       )         ≅   0     ,     
     ⁢         ∑     j   =   1     4     ⁢           ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (       η   j   2       ξ   j   ′2       )         ≅   0.             (   24   )               
The remaining factors,
 
                       ∑     j   =   1     8     ⁢           ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (         ξ   j     ⁢     ζ   j         ξ   j   ′2       )     ⁢   cos   ⁢           ⁢   φ   ⁢           ⁢     A     m   ′       ⁢     B     m   ′       ⁢     ɛ       m   ′     ,   j           ,     
     ⁢       ∑     j   =   1     8     ⁢           ⁢         ɛ     m   ,   j       ⁡     (       P   j       P   j   ′       )       ⁢     (         ζ   j     ⁢     η   j         ξ   j   ′2       )     ⁢   cos   ⁢           ⁢   φ   ⁢           ⁢     B     m   ′       ⁢     C     m   ′       ⁢     ɛ       m   ′     ,   j                   (   25   )               
will have nominal magnitudes ranging from of approximately zero to approximately 4 times a cosine factor and either the average value of factor (P j /P J ′) (ξ j ζ j /ξ′ j   2 ) or (P j /P J ′)(ζ j η j /ξ′ j   2 ) depending on the properties respective phases. For portion of the background with phases that do not track to a first approximation the phases of the respective measurement beams, the magnitudes of all of the terms listed in the Equation (25) will be approximately zero. For the portion of the background with phases that do track to a first approximation the phases of the respective measurement beams, the magnitudes of the terms listed in Equation (25) will be approximately 4 times a cosine factor and either the average value of factor (P j /P J ′)(ξ j ζ j /ξ′ j   2 ) or factor (P j /P J ′)(ζ j η j /ξ′ j   2 ).
 
     The two potentially largest terms in Equations (18) are generally the terms that have the factors 
               ∑       m   ′     =   1     4     ⁢           ⁢              A     m   ′              2   ⁢               ⁢   and   ⁢           ⁢       ∑       m   ′     =   1     4     ⁢           ⁢              B     m   ′              2   ⁢               .               
However, the corresponding terms are substantially eliminated in embodiments using the bi-homodyne detection method as result of the properties of the factors listed in Equation (24).
 
     The largest contribution from effects of background is represented by the contribution to the interference term between the reference beam and the portion of the background beam generated by the measurement beam  30 A. This portion of the effect of the background can be measured in embodiments of the bi-homodyne detection method by measuring the corresponding conjugated quadratures of the portion of the background with the return measurement beam component of beam  32  set equal to zero, i.e., measuring the respective electrical interference signals S j  with substrate  60  removed and with either |A 2 |=0 or |A 1 |=0 and visa versa and with either |A 4 |=0 or |A 3 |=0 and visa versa. The measured conjugated quadratures of the portion of the effect of the background can than be used to compensate for the respective background effects beneficially in an end use application if required. 
     Information about the largest contribution from effects of background amplitude ξ j ζ j 2A m B m  and phase φ A     m     B     m     ε     m,j   , i.e., the interference term between the reference beam and the portion of background beam generated by the measurement beam  30 A, may be obtained by measuring S j  for j=1, 2, . . . , 4 as a function of relative phase shift between reference beam and the measurement beam  30 A with substrate  60  removed and A p =0, p≠m, and Fourier analyzing the measured values of S j . Such information can be used to help identify the origin of the respective background. 
     Other techniques may be incorporated into embodiments of the present invention to reduce and/or compensate for the effects of background beams without departing from either the scope or spirit of the present invention such as described in commonly owned U.S. Pat. No. 5,760,901 entitled “Method And Apparatus For Confocal Interference Microscopy With Background Amplitude Reduction and Compensation,” U.S. Pat. No. 5,915,048 entitled “Method and Apparatus for Discrimination In-Focus Images from Out-of-Focus Light Signals from Background and Foreground Light Sources,” and U.S. Pat. No. 6,480,285 B1 wherein each of the three patents are by Henry A. Hill. The contents of each of the three cited patents are herein incorporated in their entirety by reference. 
     The selection of values for ξ′ j  is based on information about coefficients ξ j  for j=1, 2, . . . , 4 that may be obtained by measuring the S j  for j=1, 2, . . . , 4 with only the reference beam present in the interferometer system. In certain embodiments of the present invention, this may correspond simply blocking the measurement beam components of input beam  24  and in certain other embodiments, this may correspond to simply measuring the S j  for j=1, 2, . . . , 4 with substrate  60  removed. A test of the correctness of a set of values for ξ′ j  is the degree to which the 
               ∑       m   ′     =   1     2     ⁢           ⁢            A     m   ′              2   ⁢                     
term in Equation (18) is zero.
 
     Information about coefficients ξ j η j  for j=1, 2, . . . , 4 may be obtained for example by scanning an artifact past the respective eight conjugate spots corresponding to the respective eight conjugate detector pixels with one of the A p ≠0 and the remaining A p =0 for p=1, 2 and measuring the conjugated quadratures component 2|A p ∥C p |cos φ A     p     C     p    or 2|A p ∥C p |sin φ A     p     C     p   , respectively. A change in the amplitude of the 2|A p ∥C p |cos φ A     p     C     p    or 2|A p  ∥C p |sin φ A     p     C     p    term corresponds to a variation in ξ j η j  as a function of j. Information about the coefficients ξ j η j  for j=1, 2, . . . , 4 may be used for example to monitor the stability of one or more elements of interferometer system  10 . 
     The bi-homodyne detection method is a robust technique for the determination of conjugated quadratures of fields. First, the conjugated quadratures amplitudes |C 1 |cos φ A     1     C     1    and |C 1 |sin φ A     1     C     1    are the primary terms in the digitally filtered values F 1 (S) and F 2 (S) as expressed by Equations (18) since as noted in the discussion with respect to Equation (24), the terms with the factors 
               ∑       m   ′     =   1     2     ⁢           ⁢              A     m   ′              2   ⁢               ⁢   and   ⁢           ⁢       ∑       m   ′     =   1     2     ⁢           ⁢            B     m   ′              2   ⁢                         
are substantially zero.
 
     Secondly, the coefficients of |C m |cos φ A     m     C     m    and |C m |sin φ A     m     C     m    terms in Equations (18) are identical. Thus highly accurate measurements of the interference terms between the return measurement beam and the reference beam with respect to amplitudes and phases, i.e., highly accurate measurements of conjugated quadratures of fields can be measured wherein first order variations in ξ j  and first order errors in normalizations such as (P j /P j ′) and (ξ j   2 /ξ′ j   2 ) enter in only second or higher order. This property translates in a significant advantage. Also, the contributions to each component of the conjugated quadratures |C m |cos φ A     m     C     m    and |C m |sin φ A     m     C     m    from a respective set of four electrical interference signal values have the same window function and thus are obtained as jointly or substantially jointly determined values. 
     Another distinguishing feature of the bi-homodyne technique is evident in Equation (18): the coefficients of reference intensity terms |A m | 2  can be made to be substantially zero by the selection of values for ξ j ′. 
     It is also evident that since the conjugated quadratures of fields are obtained jointly or substantially jointly when using the bi-homodyne detection method, there is a significant reduction in the potential for an error in tracking phase as a result of a phase redundancy unlike the situation possible in single-homodyne detection of conjugated quadratures of fields. 
     The description of processing used in the single-homodyne detection of the first embodiment is the same as the description given for the bi-homodyne detection with either of the amplitudes A 2  or A 1  set equal to zero. 
     Continuing with the description of the first embodiment of the present invention, amplitudes of the conjugated quadratures of fields of non-forward scattered fields are determined for each of the set of θ D  (see for example Tables 1 and 2) corresponding to a set of spatial harmonic frequencies for each of two orthogonal polarization states of the measurement beam. This is achieved by scanning θ D  through for example changes in h and/or the placement of an aperture of a mask at appropriate positions in the path of beam  28 A and the rotation of catadioptric imaging system  100  by 180 degrees about a respective optic axis. The scanning in θ D  may also be implemented through the rotation of elements  40  and  44  of catadioptric imaging system  100  such as shown in  FIG. 1   f  as catadioptric imaging system  100 A. 
     The scanning in θ D  may also be implemented by placing a mask selected from a set of masks placed at position  46 A in the path of beam  28 A such as shown in  FIG. 1   g  and the generation of the reference beam incident on pinhole array beam-splitter  12  with a corresponding mask from a set of masks placed in an imaging system generating the reference beam. Each mask of the set of masks and the corresponding mask of the set of masks for the reference beam comprises a different array of apertures wherein the location of an aperture corresponds to a θ D  of a single spatial harmonic. The selection of the different aperture arrays comprising the set of masks is based on optimizing the signal-to-noise ratio for measured amplitudes of the conjugated quadratures of fields of non-forward scattered fields. The properties of the subsequent detected signals may be described in terms of a Hadamard transform and properties of the Hadamard transform may be used in designing the set of masks which optimize the signal-to-noise ratio. 
     The scanning in θ D  may also be implemented by placing a phase mask selected from a set of phase masks placed at position  46 A in the path of beam  28 A such as shown in  FIG. 1   g  and the generation of the reference beam incident on pinhole array beam-splitter  12  with a corresponding mask from a set of masks placed in an imaging system generating the reference beam. Each phase mask of the set of phase masks and the corresponding phase mask of the set of phase masks for the reference beam comprises a different array of phase shifting apertures wherein the location of a phase shifting aperture corresponds to a θ D  of a single spatial harmonic and the phase shift of the phase shifting aperture comprises for example a phase shift value such as 0 or π. The design of the different phase shifting aperture arrays is based on optimizing the signal-to-noise ratio for measured amplitudes of the conjugated quadratures of fields of non-forward scattered fields. The properties of the subsequent detected signals may be described in terms of a Hadamard transform and properties of the Hadamard transform may be used in designing the set of phase masks which optimize the signal-to-noise ratio. 
     The advantage of using a set of aperture masks is an improved signal-to-noise ratio as compared to the signal-to-noise ratio obtained when using a single aperture. The advantage of using a set of phase masks is an improved signal-to-noise ratio as compared to the signal-to-noise ratio obtained when using a set of aperture masks. 
     Amplitudes of the conjugated quadratures of fields of reflected/forward scattered fields corresponding to a zero spatial frequency component are determined for each of a set of angles of reflection where each angle of reflection of the set of angles is corresponds to a θ D  of the set of θ D  (see the corresponding discussion in the paragraph following Equation (5) herein). 
     Each of the measured harmonics and corresponding measured zero spatial frequency components of fields reflected for the values of θ D  of the set of θ D  are then used to determine a transformed amplitude corresponding to the amplitude of the spatial frequency component of the reflectance of the substrate surface. The procedure used to determine the transformed amplitude may be posed as a forward problem or as an inverse problem. The set of transformed amplitudes represent a Fourier series expansion of the substrate surface reflectance. The set of transformed amplitudes of the conjugated quadratures are next added together to generate a high spatial resolution image of the ellipsometric properties of the substrate surface. 
     The physics relating to the description of the diffracted beam and to the description of the corresponding reflected beam, i.e., the angle of diffraction is the same as the angle of reflection, are closely related. It is for this reason that the measured zero spatial frequency components of fields reflected at angles the same as the angles of diffraction used to obtain the non-zero spatial frequency components are beneficially used in the forward problem or inverse problem. 
     For the first embodiment of the present invention, the resolution represented by the high spatial frequency of the measured conjugated quadratures is ≳35 nm in one direction for λ≳140 nm. 
     A variant of the first embodiment comprises the same apparatus of the first embodiment except that pinhole beam-splitter  12  is replaced with interference type beam-splitter and a pinhole array is placed at multi-pixel detector  70 . In other variants of embodiments of the present invention, pinhole array  12  is replaced by an array of microgratings such as described in cited U.S. Provisional Patent Application No. 60/459,425. 
     In the first embodiment, multi-pixel detector  70  may comprise a frame transfer CCD that is configured such that one set of CCD pixel signal values may be generated and subsequently stored on the CCD wafer while a frame of a second set of CCD pixel signal values may be generated before a readout of both the first and second set of the CCD signal values is made. The time required to store the first set of CCD signal values is generally much less than the time required to readout a set of CCD signal values for a frame transfer CCD. Thus, the advantage of the use of a frame transfer CCD is that the time between two consecutive pulses of input beam  20  and the corresponding time between measurements of electrical interference signal values can be much less than when using a non-frame transfer CCD. 
     A second embodiment of the present invention is described that comprises the interferometric confocal microscopy system of the first embodiment operated for joint or substantially joint measurement of conjugated quadratures using the bi-homodyne detection. In the second embodiment, beam-conditioner  22  is operated to generate beam  24  comprising two frequency-shifted components. 
     For generation of two frequency-shifted components of beam  24 , the acoustic power to acousto-optic modulator  1120  (see  FIG. 1   e ) is adjusted so that the intensity of diffracted beam  1122  and the intensity of non-diffracted beam  1124  are the same. The level of acoustic power in acousto-optic modulator  1120  is controlled by signal  74  generated by electronic processor and controller  80 . 
     The remaining description of the second embodiment is the same as corresponding portions of the description given of the first embodiment of the present invention. 
     In other embodiments of the present invention, conjugated quadratures of a high spatial frequency component are measured with Λ y ≅λ x  [see Equations (1) and (2)]. In the other embodiments of the present invention, catadioptric imaging system  100  comprises two sections, e.g., pie sections, of catadioptric imaging system  210  oriented 90 degrees apart about the optic axis of catadioptric imaging system  100 . In addition, beam  24  is further split by a beam-splitter (not shown in a figure) to generate reference and measurement beams for the plane defined by the orientation of the second section of catadioptric imaging system  210 . The remaining description of the other embodiments is the same as the description given for corresponding portions of the descriptions of the first and second embodiments. The advantage of the other embodiments is the acquisition of information about defects and/or artifacts that have characteristic dimensions of the order of λ/4 in the x direction, the y direction, or in the x and y directions simultaneously or substantially simultaneously. 
     In another embodiment of the present invention, the height h may be adjusted to be ≈λ/4. In that case, evanescent fields will be used to measure properties analogous to those obtained in the high spatial resolution ellipsometric measurements of the first and second embodiments of the present invention. The remaining description of the apparatus and method of another embodiment is the same as the corresponding portions of the description given for the first and second embodiments of the present invention. 
     In yet other embodiments of the present invention, information is obtained about the properties of generation and propagation of signals, e.g., electrical, thermal, or acoustical, in a section of substrate  60  by using a probe beam in addition to the measurement beam. Beam  24  comprises a probe beam that precedes the measurement beam by a time τ. The probe beam and the measurement beam may have different optical wavelengths. An advantage of the use of the yet other embodiments with respect to the measurement of temporal response of the substrate is the high spatial frequency resolution in the plane of the section. The remaining description of the yet other embodiments is the same as the description given for corresponding portions of the descriptions of the first and second embodiments of the present invention. 
     In other embodiments of the present invention, quad-homodyne detection method and variant thereof and a variant of the bi-homodyne detection method are used such as described in cited U.S. Provisional Patent Application No. 60/459,425 (ZI-50) entitled “Apparatus and Method for Joint Measurement Of Fields Of Scattered/Reflected Orthogonally Polarized Beams By An Object In Interferometry” and U.S. patent application Ser. No. 10/816,180, (ZI-50) filed Apr. 1, 2004 and entitled “Apparatus and Method for Joint Measurement Of Fields Of Scattered/Reflected Or Transmitted Orthogonally Polarized Beams By An Object In Interferometry.” 
     Other embodiments are within the following claims.
A method of detecting non-uniform ellipsometric properties of a substrate surface involving: directing a measurement beam onto a spot at a selected location on or in the substrate; for each orientation of a plurality of different orientations of the reference beam relative to the scattered measurement beam, interfering the scattered measurement beam with the reference beam to produce a corresponding interference beam, wherein each of the different orientations of the reference beam is selected to produce a peak sensitivity for a portion of the scattered measurement beam that emanates from the object at a corresponding different diffraction angle of a plurality of diffraction angles; for each orientation of the reference beam, converting the interference beam into an interference signal; and using the interference signals to determine whether any non-uniform ellipsometric properties are present anywhere within a region on or in the substrate.