Abstract:
A method of measuring properties of a substrate, the method involving: illuminating a spot on the substrate with a standing wave measurement beam to generate a return measurement beam, the standing wave measurement beam characterized by a standing wave pattern; generating an electrical signal from the return measurement beam; causing the standing wave pattern to be at a succession of different positions on the surface of the substrate; and for each of the succession of different positions of the standing wave pattern, acquiring measurement data from the electrical signal.

Description:
This application claims the benefit of U.S. Provisional Application No. 60/507,675, filed Oct. 1, 2003. 

   BACKGROUND OF THE INVENTION 
   A number of different applications of catadioptric imaging systems for far-field and near-field interferometric confocal and non-confocal microscopy have been described such as in commonly owned U.S. Pat. No. 6,552,852 (ZI-38) entitled “Catoptric And Catadioptric Imaging Systems” and U.S. Pat. No. 6,717,736 (ZI-43) entitled “Catoptric And Catadioptric Imaging Systems;” U.S. Provisional Patent Applications No. 60/447,254, filed Feb. 13, 2003, entitled “Transverse Differential Interferometric Confocal Microscopy,” (ZI-40); No. 60/448,360, filed Feb. 19, 2003, entitled “Longitudinal Differential Interferometric Confocal Microscopy for Surface Profiling,” (ZI-41); No. 60/448,250, filed Feb. 19, 2003, entitled “Method and Apparatus for Dark Field Interferometric Confocal Microscopy,” (ZI-42); No. 60/442,982, filed Jan. 28, 2003, entitled “Interferometric Confocal Microscopy Incorporating Pinhole Array Beam-Splitter,” (ZI-45); No. 60/459,425, filed Apr. 1, 2003, entitled “Apparatus and Method for Joint Measurement Of Fields Of Scattered/Reflected Orthogonally Polarized Beams By An Object In Interferometry,” (ZI-50); No. 60/485,507, filed Jul. 7, 2003, entitled “Apparatus And Method For High Speed Scan For Sub-Wavelength Defects And Artifacts In Semiconductor Metrology,” (ZI-52); No. 60/485,255, filed Jul. 7, 2003, entitled “Apparatus and Method for Ellipsometric Measurements with High Spatial Resolution,” (ZI-53); No. 60/501,666, filed Sep. 10, 2003, entitled “Catoptric And Catadioptric Imaging Systems With Adaptive Catoptric Surfaces,” (ZI-54); No. 60/602,046, filed Aug. 16, 2004, entitled “Apparatus And Method For Joint And Time Delayed Measurements Of Components Of Conjugated Quadratures Of Fields Of Reflected/Scattered Beams By An Object In Interferometry,” (ZI-57); No. 60/506,715, filed Sep. 26, 2003, entitled “Catoptric and Catadioptric Imaging Systems Comprising Pellicle Beam-Splitters and Non-Adaptive and Adaptive Catoptric Surfaces,” (ZI-56); and No. 60/611,564, filed Sep. 20, 2004, entitled “Catoptric Imaging Systems Comprising Pellicle Beam-Splitters and Non-Adaptive and/or Adaptive Catoptric Surfaces,” (ZI-58); and U.S. patent application Ser. No. 10/778,371, filed Feb. 13, 2004, entitled “Transverse Differential Interferometric Confocal Microscopy,” (ZI-40); Ser. No. 10/782,057, filed Feb. 19, 2004, entitled “Longitudinal Differential Interferometric Confocal Microscopy for Surface Profiling,” (ZI-41); Ser. No. 10/782,058, filed Feb. 19, 2004, entitled “Method and Apparatus for Dark Field Interferometric Confocal Microscopy,” (ZI-42); Ser. No. 10/765,229, filed Jan. 27, 2004, entitled “Interferometric Confocal Microscopy Incorporating Pinhole Array Beam-Splitter,” (ZI-45); Ser. No. 10/816,180, filed Apr. 1, 2004, entitled “Apparatus and Method for Joint Measurement Of Fields Of Scattered/Reflected or Transmitted Orthogonally Polarized Beams By An Object In Interferometry,” (ZI-50); Ser. No. 10/886,010, filed Jul. 7, 2004, entitled “Apparatus And Method For High Speed Scan For Sub-Wavelength Defects And Artifacts In Semiconductor Metrology,” (ZI-52); Ser. No. 10/886,157, filed Jul. 7, 2004, entitled “Apparatus and Method for Ellipsometric Measurements with High Spatial Resolution,” (ZI-53); No. 10/938,408, filed Sep. 10, 2004, entitled “Catoptric And Catadioptric Imaging Systems With Adaptive Catoptric Surfaces,” (ZI-54); No. 10/948,959, filed Sep. 24, 2004, entitled “Catoptric and Catadioptric Imaging Systems with Pellicle and Aperture-Array Beam-Splitters and Non-Adaptive and Adaptive Catoptric Surfaces”. In addition, U.S. patent application (ZI-48) Ser. No. 10/218,201, entitled “Method for Constructing a Catadioptric Lens System,” filed Apr. 1, 2004 described one way to make some of these catadioptric lens systems. These patents, patent applications, and provisional patent applications are all by Henry A. Hill and the contents of each are incorporated herein in their entirety by reference. 
   SUMMARY OF THE INVENTION 
   Methods and apparatuses are described for achieving enhanced resolution of high spatial frequency components of images generated in non-interferometric microscopy using standing wave illumination of a substrate and in interferometric microscopy using a standing wave reference beam and/or standing wave measurement beam and wherein a measurement object may also be used simultaneously as a reference object. The enhanced resolution is achieved in microscopy systems operating in either a reflection or a transmission mode. The resolution for the high spatial frequency components is enhanced by approximately a factor of 2 in one dimension or two orthogonal dimensions with respect to that achievable in other imaging systems. 
   Taught herein is how to improve lateral spatial resolution for high spatial frequency components in interferometric microscopy by using a standing wave beam as a measurement beam incident on a measurement object and a reference beam that may or may not comprise a standing wave beam wherein the measurement object may also serve simultaneously as the reference object. Also taught herein is how to use standing wave beams in interference microscopy to obtain the improvement in lateral spatial resolution concurrently with a depth discrimination that is similar to or better than the depth discrimination obtained in a confocal microscopy system. 
   Taught herein, in addition in interferometric microscopy, the use of an object that is being imaged as the element that combines the measurement beam and reference beam subsequently detected to generate an electrical interference signal. 
   The standing wave measurement beam is generated by superimposing two beams at a spot on an object that have the same nominal angle of incidence on the object, coplanar planes of incidence, and the angle between the incident beams is equal to the sum of the respective angles of incidence. The state of polarization of each of the two beams may be either a s or p polarization. A standing wave reference beam when used is generated in a like manner by superimposing two beams that have the same nominal angle of incidence at a spot on a beam combining element, coplanar planes of incidence, and the angle between the incident beams is equal to the sum of the respective angles of incidence. 
   Also taught herein is in non-interferometric microscopy the practice of using a standing wave beam for illumination of an object that is being imaged. 
   The net result of using a standing wave or standing wave beams in either non-interferometric or interferometric microscopy is an improvement in lateral spatial resolution for high spatial frequency components by a factor of approximately 2. 
   In general, in one aspect, the invention features a method of measuring properties of a substrate. The method involves: illuminating a spot on the substrate with a standing wave measurement beam to generate a return measurement beam; generating an electrical signal from the return measurement beam; causing the standing wave pattern to be at a succession of different positions on the surface of the substrate; and for each of the succession of different positions of the standing wave pattern, acquiring measurement data from the electrical signal. 
   Other embodiments include one or more of the following features. The measurement beam is characterized by a wavelength, λ, and each position of among the succession of different locations after a first location is separated from the preceding position by a distance of less than one wavelength, λ. The method also includes: illuminating an object with a standing wave reference beam to generate a return reference beam; and interfering the return measurement beam and the return reference beam to generate an interference beam, wherein generating the electrical signal involves generating the electrical signal from the interference beam. Illuminating the object with a standing wave reference beam involves interfering two reference beams at the object to generate the standing wave reference beam. Illuminating the spot with a standing wave measurement beam involves interfering two measurement beams at the spot on the substrate to produce the standing wave measurement beam. The object is the substrate. The substrate has a front side and a back side and illuminating the spot on the substrate with the standing wave measurement beam involves directing a first measurement beam and a second measurement beam at the spot from the backside of the substrate to produce the standing wave measurement beam, the first and second measurement beams traveling in opposite directions relative to the surface of the substrate and wherein the return measurement beams emanates from the front side of the substrate. The two measurements beams lie in a first plane and the two reference beams lie in a second plane, and the method further also involves orienting the first and second planes to be orthogonal to each other. The method further includes: generating an input beam; and deriving the first and second measurement beams and the first and second reference beams from the input beam. Acquiring measurement data from the electrical signal involves measuring components of conjugated quadratures of fields of scattered/reflected beams by the substrate. Causing the standing wave pattern to be at the succession of different positions on the surface of the substrate involves moving the substrate. Alternatively, it involves scanning the standing wave pattern over the substrate. Generating the electrical signal from the interference beam involves directing the interference beam onto a detector. Generating the electrical signal from the interference beam involves directing the interference beam onto a fluorescent material to generate an optical signal and directing the optical signal onto a detector to generate the electrical signal. 
   In general, in another aspect, the invention features a method of interferometrically measuring properties of an object. The method involves: illuminating a spot on the object with a standing wave measurement beam to generate a return measurement beam; interfering the return measurement beam with a reference beam to generate an interference beam; generating an electrical interference signal from the interference beam; and acquiring measurement data from the electrical interference signal. 
   Other embodiments include one or more of the following features. The method also involves causing the standing wave pattern to be at a succession of different positions on the surface of the substrate, and acquiring measurement data involves for each of the succession of different positions of the standing wave pattern, acquiring measurement data from the electrical interference signal. The measurement beam is characterized by a wavelength, λ, and each position of among the succession of different locations after a first location is separated from the preceding position by a distance of less than one wavelength, λ. The method further includes: illuminating an object with a standing wave reference beam to generate a return reference beam; and interfering the return measurement beam and the return reference beam to generate the interference beam. Illuminating the object with a standing wave reference beam involves interfering two reference beams at the object to generate the standing wave reference beam. Illuminating the spot with a standing wave measurement beam involves interfering two measurement beams at the spot on substrate to produce the standing wave measurement beam. The object is the substrate. The substrate has a front side and a back side and illuminating the spot on the substrate with the standing wave measurement beam involves directing a first measurement beam and a second measurement beam at the spot from the backside of the substrate to produce the standing wave measurement beam, the first and second measurement beams traveling in opposite directions relative to the surface of the substrate and the return measurement beams emanating from the front side of the substrate. The two measurements beams lie in a first plane and the two reference beams lie in a second plane, and the method further involves orienting the first and second planes to be orthogonal to each other. The method also involves: generating an input beam; and deriving the first and second measurement beams and the first and second reference beams from the input beam. Causing the standing wave pattern to be at the succession of different positions on the surface of the substrate involves moving the substrate. Alternatively, causing the standing wave pattern to be at the succession of different positions on the surface of the substrate involves scanning the standing wave pattern over the substrate. Generating the electrical interference signal from the interference beam involves directing the interference beam onto a detector. Alternatively, generating the electrical signal from the interference beam involves directing the interference beam onto a fluorescent material to generate a optical signal and directing the optical signal onto a detector to generate the electrical interference signal. 
   In general, in still another aspect, the invention features an interferometric system for measuring properties of a substrate. The system includes: an optical system for illuminating a spot on the substrate with a standing wave measurement beam to thereby produce a return measurement beam from that spot; an interferometer for interfering the return measurement beam with a reference beam to generate an interference beam; and a detector system for converting the interference beam to an electrical interference signal. 
   Other embodiments include one or more of the following features. The optical system includes a beam splitter for separating a first input beam into a first measurement beam and a second measurement beam and further optics for directing the first measurement beam onto the spot from one direction and the second measurement beam onto the spot from an opposite direction so as to produce the standing wave measurement beam. The further optics of the optical system is arranged to operate in a reflection mode by directing the first measurement beam and a second measurement beam at the spot from the front side of the substrate to produce the standing wave measurement beam and the return measurement beam emanates away from the front side of the substrate. Alternatively, the further optics of the optical system is arranged to operate in a transmission mode by directing the first measurement beam and a second measurement beam at the spot from the backside of the substrate to produce the standing wave measurement beam and the return measurement beam emanates away from the front side of the substrate. The interferometer includes a second optical system for illuminating an object with a standing wave reference beam to produce the return reference beam. The object is the substrate. The second optical system includes a beam splitter for separating a second input beam into a first reference beam and a second reference beam and further optics for directing the first reference beam onto the spot from one direction and the second reference beam onto the spot from an opposite direction so as to produce the standing wave reference beam. The interferometric system further includes a beam source for generating a source beam, wherein the first and second input beams are derived from the source beam. The beam source includes a beam splitter for splitting the source beam into the first and second input beams. The interferometric system further includes a transport mechanism for holding the object and scanning the object relative to the interferometer. The interferometer includes a catadioptric imaging system. The detector system includes a detector and an imaging system which focuses the interference beam onto the detector. Alternatively, the detector system includes a fluorescent material onto which the interference beam impinges to produce an optical signal, a detector, and an imaging system which directs the optical signal onto the detector. The optical signal is at a different wavelength from the interference signal. 
   An advantage of at least one embodiment of the present invention is the generation of an enhanced lateral resolution for high spatial frequency components of an image in one axis of a coordinate system. 
   Another advantage of at least one embodiment of the present invention is the generation of an enhanced lateral resolution for high spatial frequency components of an image in two orthogonal axes of a coordinate system. 
   Another advantage of at least one embodiment of the present invention is the availability of a large working distance. 
   Another advantage of at least one embodiment of the present invention is a depth discrimination against out-of-focus images similar to or better than that associated with confocal microscopy. 
   Another advantage of at least one embodiment of the present invention is the generation of a depth discrimination with respect to out-of-focus images concomitant with an enhancement in lateral spatial resolution for high spatial frequency components of an image. 
   Another advantage of at least one embodiment of the present invention is an enhanced lateral resolution with respect to high spatial frequency components of an image for non-interferometric microscopy applications. 
   Another advantage of at least one embodiment of the present invention is an enhanced lateral resolution with respect to high spatial frequency components of an image for interferometric microscopy applications. 
   Another advantage of at least one embodiment of the present invention is the operation in a dark field mode for measurement of reflection properties of a substrate. 
   Another advantage of at least one embodiment of the present invention is that enhanced lateral resolution for high spatial frequency components of an image may be achieved in both the reflection and transmission modes of operation. 
   Another advantage of at least one embodiment of the present invention is that the enhanced lateral resolution for high spatial frequency components of an image is independent of depth of an imaged plane section embedded in a substrate. 
   Another advantage of at least one embodiment of the present invention is that a substrate being imaged may also serve as reference and measurement beam objects simultaneously in interferometric microscopy measurements of the substrate. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1   a  is a schematic diagram of an interferometric system operating in a reflection mode configured with a standing wave measurement beam at a measurement object. 
       FIG. 1   b  is a schematic diagram of an interferometric system operating in a reflection mode configured with standing wave reference and measurement beams at reference and measurement objects, respectively, wherein the reference and measurement objects are the same object. 
       FIG. 1   c  is a schematic diagram of an interferometric system operating in a reflection mode configured with non-standing wave reference beam and a standing wave measurement beams at reference and measurement objects, respectively, wherein the reference and measurement objects are the same object. 
       FIG. 1   d  is a schematic diagram of an interferometric system operating in a transmission mode configured with a standing wave measurement beam at a measurement object. 
       FIG. 1   e  is a schematic diagram of an interferometric system operating in a transmission mode configured with standing wave reference and measurement beams at reference and measurement objects, respectively, wherein the reference and measurement objects are the same object. 
       FIG. 1   f  is a schematic diagram of an interferometric system operating in a transmission mode configured with non-standing wave reference beam and a standing wave measurement beams at reference and measurement objects, respectively, wherein the reference and measurement objects are the same object. 
       FIG. 1   g  is a schematic diagram of two beams in respective s and p polarization states incident on a substrate. 
       FIG. 1   h  is a schematic diagram of two beams in respective p polarization states incident on a substrate. 
       FIG. 1   i  is a schematic diagram of two beams in respective s and p polarization states incident on a substrate. 
       FIG. 1   j  is a schematic diagram of two beams incident on a substrate with one of the beams in a p polarization state. 
       FIG. 1   k  is a schematic diagram of two beams incident on a substrate with one of the beams in a s polarization state. 
       FIG. 1   l  is a graph of properties an electrical interference signal that is generated using standing wave reference and measurement beams at reference and measurement objects, respectively, wherein the reference and measurement objects are the same object. 
       FIG. 1   m  is a graph of properties of a signal from a non-interferometric system that is generated using standing wave illumination of a substrate being imaged. 
       FIG. 2   a  is schematic diagram of a interferometer system that comprises a catadioptric imaging system. 
       FIG. 2   b  is schematic diagram of a catadioptric imaging system comprising adaptive catoptric surfaces and a pellicle beam-splitter. 
   

   DETAILED DESCRIPTION 
   A general description of interferometric microscopy embodiments of the present invention will first be given. The interferometric microscopy embodiments are separated into groups according to properties of a measurement beam incident on an object or substrate; to whether the substrate is used as both the measurement and reference beam objects simultaneously or only as a measurement beam object; and to properties of a reference beam incident on a beam combining element, e.g., a beam-splitter. 
   The referenced properties of the measurement beam are with respect to a state of polarization of a standing wave beam, spatial properties of the standing wave beam, and the spatial properties relative to the state of polarization. For a reference beam comprising a standing wave beam at a beam combining element and/or at a reference object, the referenced properties of the reference beam are with respect to a state of polarization of a standing wave beam, spatial properties of the standing wave beam, and the spatial properties relative to the state of polarization or the state of polarization relative to the direction of propagation of the reference beam. For a reference beam comprising a non-standing wave beam at a beam combining element and/or reference object, the referenced properties of the reference beam are with respect to a state of polarization of the reference beam relative to the plane of incidence at a beam combining element or the state of polarization relative to a coordinate system at the beam combining element. The embodiments for use in non-interferometric microscopy are separated into groups according to the state of polarization and spatial properties of the beam illuminating the substrate. 
   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. 
   Referring to  FIG. 1   a , an interferometer system is shown diagrammatically as an interferometer system operating in a reflection mode to measure components of conjugated quadratures of fields reflected/scattered by substrate  60  with a measurement beam comprising a standing wave beam at substrate  60 . The components of the conjugated quadratures may be measured jointly. The interferometer system shown diagrammatically in  FIG. 1   a  comprises an interferometer  10 , a source  18 , a beam-conditioner  22 , a detector  70 , an electronic processor and controller  80 , and a measurement object shown as substrate  60 . Source  18  generates input beam  20 . 
   Reference and measurement beams are generated in beam-conditioner  22  and the measurement beam generated in beam-conditioner  22  is incident on substrate  60  as a standing wave beam. Beam  30  comprises a return reflected/scattered measurement beam that is generated by the reflection/scattering of the measurement beam by substrate  60 . The return reflected/scattered measurement beam  30  is combined with the reference beam by a beam combiner (not shown) in interferometer  10  to form a mixed beam. In certain embodiments, the mixed beam is incident on a thin fluorescent layer and output beam  32  comprises an optical interference beam generated by fluorescence. In certain other embodiments, output beam  32  comprises the mixed optical beam. 
   Output beam  32  is detected by detector  70  to generate an electrical interference signal  72  from either the optical interference beam generated by fluorescence from the mixed output beam in the certain embodiments or from the mixed optical beam in the certain other embodiments. The composition of the thin fluorescent layer is selected such that the decay time of the fluorescence is much shorter than the read out time of detector  70 . 
   Detector  70  may comprise in the certain other embodiments an analyzer to select common polarization states of the reference and return measurement beam components of beam  32  to form a mixed beam in lieu of beam  32  being formed as a mixed beam. 
   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 by metrology system  88  and position information acquired by metrology system  88  is transmitted as signal  76  to electronic processor and controller  80  to generate an error signal for use in the position control of stage  90 . Metrology system  88  may comprise for example linear displacement and angular displacement interferometers and cap gauges. The elevation and angular orientation of substrate  60  is controlled by transducers  86 A and  86 B according to servo control signal  78 . 
   In the practice wherein heterodyne and bi- or quad-homodyne detection methods are used, known phase shifts are introduced between the reference and measurement beam components of mixed beam  32  generated by interferometer system  10 . For heterodyne detection methods, the phase shifts are introduced as a consequence of the reference and measurement beams having different frequencies. With respect to the bi- or quad-homodyne detection methods, the phase shifts are introduced by either of two techniques. In one technique, phase shifts are introduced between the reference and measurement beam components for each of the frequency components by beam-conditioner  22  as controlled by signal  74  from electronic processor and controller  80 . In the second technique, phase shifts are introduced between the reference and measurement beam components of mixed beam  32  for each of the frequency components as a consequence of frequency shifts introduced to the frequency components of input beam  24  by beam-conditioner  22  as controlled by signal  74  from electronic processor and controller  80 . 
   In the practice wherein N-dimensional bi- and quad-homodyne detection methods are used in certain embodiments, additional phase shifts are introduced between each corresponding reference and measurement beam portion of N portions of the mixed beam generated by interferometer  10 . The additional phase shifts are in addition to those introduced in the implementation of bi- or quad-homodyne detection methods. The additional phase shifts are generated in interferometer  10  by causing for example changes in the locations of elements of adaptive catoptric surfaces. 
   The description of source  18  including a pulse mode of operation and beam-conditioner  22  is the same as the corresponding portions of the description given to the source and beam-conditioner in embodiments described in commonly owned U.S. Provisional Patent 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. 10/765,368 (ZI-47) entitled “Apparatus and Method for Joint Measurements of Conjugated Quadratures of Fields of Reflected/Scattered or Transmitted Beams by an Object in Interferometry” wherein the provisional and the non-provisional patent applications are by Henry A. Hill and the contents of which are herein incorporated in their entirety by reference and in cited U.S. Provisional Patent Application No. 60/485,255 (ZI-53), in cited U.S. Provisional Patent Ser. No. 60/602,046 filed Aug. 16, 2004 (ZI-57) entitled “Apparatus and Method for Joint And Time Delayed Measurements of Components of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted/Scattered Beams by an Object in Interferometry,” and in cited U.S. patent application Ser. No. 60/485,255 filed Jul. 7, 2003 (ZI-53) entitled “Apparatus and Method for Ellipsometric Measurements with High Spatial Resolution.” 
   The descriptions the of bi-homodyne and quad-homodyne detection methods used in certain of the embodiments of the present invention are the same as corresponding portions of the descriptions given for the descriptions of bi-homodyne and quad-homodyne detection methods in the cited U.S. Provisional Patent Application Nos. 60/442,858 (ZI-47) and 60/485,255 (ZI-53) and in cited U.S. patent application Ser. No. 10/765,368 (ZI-47) and U.S. patent application Ser. No. 60/485,255 filed Jul. 7, 2003 (ZI-53) wherein the homodyne detection methods are based on frequency encoding. The extension of the bi- and quad-homodyne detection methods to N-dimensional bi- and quad-homodyne detection methods based on a combination of frequency encoding and either amplitude or phase modulations or permutations is implemented in the embodiments by the use for example of conjugate pairs of adaptive reflective surfaces of subsequently described catadioptric imaging system  410 A (see  FIGS. 2   a  and  2   b ) as optical switches or as π phase shifters, respectively. 
   The extension of the bi- and quad-homodyne detection methods to N-dimensional bi- and quad-homodyne detection methods may also be based on a combination of frequency encoding, polarization encoding, and either amplitude or phase modulations or permutations. The description of bi- and quad-homodyne detection methods based on a combination of frequency and polarization encoding is the same as the corresponding description given in cited U.S. Provisional Patent Application No. 60/459,425 (ZI-50) and in cited U.S. patent application Ser. No. 60/459,425 filed Apr. 1, 2003 (ZI-50) entitled “Apparatus and Method for Joint Measurement Of Fields Of Orthogonally Polarized Beams Scattered/Reflected By An Object In Interferometry.” 
   Interferometer  10  of the first embodiment of the present invention is shown schematically in  FIG. 2   a . Interferometer  10  of the first embodiment comprises a first imaging system generally indicated as numeral  410 A, and a second imaging system generally indicated as numeral  10 B. The second imaging system  10 B may comprise a low power microscope having a large working distance, e.g. Nikon ELWD and SLWD and Olympus LWD, ULWD, and ELWD objectives or a high resolution catadioptric imaging system such as described in cited U.S. Pat. No. 6,552,852 (ZI-38) and No. 6,717,736 (ZI-43). 
   The description of interferometer  10 , source  18 , beam-conditioner  22 , detector  70 , and electronic processor and controller  80  is the same as corresponding portions of the descriptions of catoptric and catadioptric imaging systems given in commonly owned U.S. Patent Provisional Patent Application No. 60/506,715 (ZI-56) entitled “Catoptric and Catadioptric Imaging Systems Comprising Pellicle Beam-Splitters And Non-Adaptive And Adaptive Catoptric Surfaces” by Henry A. Hill, David Fischer, and Steven Hamann; in cited U.S. Provisional Patent Application Ser. No. 60/602,046 filed Aug. 16, 2004 (ZI-57) entitled “Apparatus and Method for Joint And Time Delayed Measurements of Components of Conjugated Quadratures of Fields of Reflected/Scattered and Transmitted/Scattered Beams by an Object in Interferometry”; and in U.S. patent application Ser. No. 60/506,715 filed Sep. 26, 2003 (ZI-56) entitled “Catoptric and Catadioptric Imaging Systems Comprising Pellicle Beam-Splitters And Non-Adaptive And Adaptive Catoptric Surfaces” by Henry A. Hill, David Fischer, and Steven Hamann for which the contents of the first of the two provisional patent applications and the utility patent application are incorporated herein in their entirety by reference. 
   A number of different catadioptric imaging systems for far-field and near-field interferometric confocal microscopy have been described such as in commonly owned U.S. Pat. No. 6,552,852 (ZI-38) entitled “Catoptric And Catadioptric Imaging Systems;” U.S. Pat. No. 6,717,736 (ZI-43) entitled “Catoptric And Catadioptric Imaging Systems;” U.S. Provisional Patent Applications No. 60/447,254 (ZI-40) entitled “Transverse Differential Interferometric Confocal Microscopy,” No. 60/448,360 (ZI-41) entitled “Longitudinal Differential Interferometric Confocal Microscopy,” No. 60/448,250 (ZI-42) entitled “Thin Film Metrology Using Interferometric Confocal Microscopy,” No. 60/442,982 (ZI-45) entitled “Interferometric Confocal Microscopy Incorporating Pinhole Array Beam-Splitter,” No. 60/459,493 (ZI-48) entitled “Method For Manufacture Of Catadioptric Lens System,” No. 60/459,425 (ZI-50) entitled “Apparatus and Method for Joint Measurement Of Fields Of Orthogonally Polarized Beams Scattered/Reflected By An Object In Interferometry,” in cited No. 60/485,255 (ZI-53), No. 60/501,666 (ZI-54) entitled “Catoptric and Catadioptric Imaging Systems With Adaptive Catoptric Surfaces,” and filed Sep. 18, 2004 (ZI-58) entitled “Catoptric Imaging Systems Comprising Pellicle and/or Aperture-Array Beam-Splitters and Non-Adaptive and/or Adaptive Catoptric Surfaces;” and U.S. patent application Ser. No. 10/778,371 (ZI-40) entitled “Transverse Differential Interferometric Confocal Microscopy,” Ser. No. 10/782,057 (ZI-41) entitled “Longitudinal Differential Interferometric Confocal Microscopy,” Ser. No. 10/782,058 (ZI-42) entitled “Thin Film Metrology Using Interferometric Confocal Microscopy,” Ser. No. 10/765,229 (ZI-45) entitled “Interferometric Confocal Microscopy Incorporating Pinhole Array Beam-Splitter,” Ser. No. 10/816,201 (ZI-48) entitled “Method For Manufacture Of Catadioptric Lens System,” Ser. No. 10/816,180 (ZI-50) entitled “Apparatus and Method for Joint Measurement Of Fields Of Orthogonally Polarized Beams Scattered/Reflected By An Object In Interferometry,” in cited patent application Ser. No. 60/485,255 filed Jul. 7, 2003 (ZI-53) entitled “Apparatus and Method for Ellipsometric Measurements with High Spatial Resolution,” and aplication Ser. No. 60/501,666 filed Sep. 10, 2003 (ZI-54) entitled “Catoptric and Catadioptric Imaging Systems With Adaptive Catoptric Surfaces.” The two cited patents, the seven not previously cited patent applications, and the eight not previously cited provisional patent applications are all by Henry A. Hill and the contents of each of which are incorporated herein in their entirety by reference. Other forms of non-catoptric or non-catadioptric microscopy imaging systems may be used for interferometer  10  without departing from the spirit or scope of the present invention. 
   With reference to  FIG. 1   a , source  18  is preferably a pulsed source that generates beam  20 . Beam  20  is incident on and exits beam-conditioner  22  as input beam  24  that has two different frequency components for each polarization state or polarization component represented. Portions of the different frequency components of input beam  24  are spatially separated into two input beam components wherein each of the two spatially separated input beam components comprises a portion of each of the two different frequency components of each of the polarization states or polarization components represented. One of the two input beam components is split off by mirror  54 A as measurement beam  24 A and the other of the two input beam components is not incident on mirror  54 A as reference beam  24 B. The first and second portions corresponding to the two input beam components of input beam  24  have the same temporal window function. 
   A standing wave measurement beam is generated at substrate  60  from measurement beam  24 A by a combination of a beam-splitter and two mirrors as shown diagrammatically in  FIG. 1   a . The reference beam is incident on a beam combining element in interferometer  10  (e.g. see beam combiner  12  in  FIG. 2   a ) and may be a standing wave beam such as described for the measurement beam or a non-standing wave beam at the beam combining element. 
   The interferometer system shown diagrammatically in  FIG. 1   b  represents an interferometric imaging system operating in a reflecting mode to measure components of conjugated quadratures of fields reflected/scattered by substrate  60  for certain embodiments of the present invention wherein substrate  60  serves as both the reference and measurement beam objects simultaneously and the measurement and reference beams comprise standing wave beams at the respective reference and measurement objects, i.e. substrate  60 . Referring to  FIG. 1   b , the interferometer system is shown diagrammatically comprising an imaging system  100 , source  18 , beam-conditioner  22 , detector  70 , electronic processor and controller  80 , and measurement object shown as substrate  60 . 
   In the interferometer system shown in  FIG. 1   b , both the measurement beam and the reference beam comprise standing wave beams at substrate  60 . The return beam  31  is the interference beam representing the combined return measurement and reference beams from substrate  60 . The plane of the two measurement beams forming the standing wave measurement beam at substrate  60  and the plane of the two reference beams forming the standing wave reference beam at substrate  60  are both shown as lying in the plane of  FIG. 1   b  in order to simplify the diagram for illustration purposes only. In practice, the two planes will generally be at some angle with respect to each other, e.g., orthogonal. 
   The interferometer system shown diagrammatically in  FIG. 1   c  represents an interferometric imaging system operating in a reflecting mode to measure amplitudes of conjugated quadratures of fields reflected/scattered by substrate  60  for certain embodiments of the present invention wherein substrate  60  serves as both the reference and measurement beam objects simultaneously and the measurement and reference beams comprise a standing wave measurement beam and a non-standing wave reference beam, respectively, at the measurement substrate  60 . Referring to  FIG. 1   c , the interferometer system is shown diagrammatically comprising an imaging system  100 , source  18 , beam-conditioner  22 , detector  70 , electronic processor and controller  80 , and measurement object shown as substrate  60 . Imaging system  100  may comprise the imaging system  10  with respect to an imaging function. 
   In the interferometer system shown in  FIG. 1   c , the measurement beam comprises a standing wave beam at substrate  60 . The reference beam is shown as lying in the plane of  FIG. 1   c  with a non-zero angle of incidence in order to simplify the diagram for illustration purposes only. In practice, the direction of propagation of the reference beam may lie in different plane and the angle of incidence may be zero. 
   The interferometer system shown diagrammatically in  FIG. 1   d  represents an interferometer imaging system operating in the transmission mode to measure the components of conjugated components of fields transmitted/scattered by substrate  60  and further represents embodiments of the present invention wherein the measurement beam comprises a standing wave beam at substrate  60 . 
   The interferometer system shown diagrammatically in  FIG. 1   e  represents an interferometric imaging system operating in a transmission mode to measure amplitudes of conjugated quadratures of fields transmitted/scattered by substrate  60  for certain embodiments of the present invention wherein substrate  60  serves as both the reference and measurement beam objects simultaneously and the measurement and reference beams comprise standing wave beams at substrate  60 . Referring to  FIG. 1   e , the interferometer system is shown diagrammatically comprising an imaging system  100 , source  18 , beam-conditioner  22 , detector  70 , electronic processor and controller  80 , and measurement object shown as substrate  60 . 
   In the interferometer system shown in  FIG. 1   e , both the measurement beam and the reference beam comprise standing wave beams at substrate  60 . The plane of the two measurement beams forming the standing wave measurement beam at substrate  60  and the plane of the two reference beams forming the standing wave reference beam at substrate  60  are both shown as lying in the plane of  FIG. 1   e  in order to simplify the diagram for illustration purposes only. In practice, the two planes will generally be at some angle with respect to each other, e.g., orthogonal. 
   The interferometer system shown diagrammatically in  FIG. 1   f  represents an interferometric imaging system operating in a transmission mode to measure components of conjugated quadratures of fields transmitted/scattered by substrate  60  for certain embodiments of the present invention wherein substrate  60  serves as both the reference and measurement beam objects simultaneously and the measurement and reference beams comprise a standing wave measurement beam and a non-standing wave reference beam, respectively, at measurement object  60 . Referring to  FIG. 1   f , the interferometer system is shown diagrammatically comprising an imaging system  100 , source  18 , beam-conditioner  22 , detector  70 , electronic processor and controller  80 , and measurement object shown as substrate  60 . 
   Polarization States and Spatial Properties of Measurement and Reference Beams 
   Reference is made to  FIGS. 1   g - 1   k  for the discussion of properties of measurement and reference beams at an interface corresponding to the interferometric and non-interferometric microscopy systems operating in a reflection mode. The interface may be either a surface of substrate  60  or a surface associated with an element of an interferometer serving the function of a beam combining element. 
   In  FIG. 1   g , beams  1024 A- 1  and  1024 A- 2  in a p polarization states are measurement beams derived from measurement beam  24 A as shown in  FIG. 1   b  and superimposed at a spot on substrate  60  to form a standing wave measurement beam. The angles of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are nominally the same and indicated as θ 1  in  FIG. 1   g . The planes of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are coplanar with the x-z plane. Beams  1024 B- 1  and  1024 B- 2  in s polarization states are reference beams derived from reference beam  24 B as shown in  FIG. 1   b  and superimposed at the spot on substrate  60  to form a standing wave reference beam. The angles of incidence for reference beams  1024 B- 1  and  1024 B- 2  are nominally the same and indicated as θ 2  in  FIG. 1   g . The planes of incidence for reference beams  1024 B- 1  and  1024 B- 2  are coplanar and are shown as coplanar with the y-z plane in  FIG. 1   g . However, the coplanar planes of incidence of reference beams  1024 B- 1  and  1024 B- 2  may be at some other orientation with respect to the y-z plane. 
   When the electrical interference signal  72  is subsequently generated by detector  70  (see  FIG. 1   b ) by the detection of a mixed beam  32  comprising the reflected/scattered standing wave measurement beam and the reflected/scattered standing wave reference beam, the standing wave measurement beam will have polarization components in each of the x and z directions and the standing wave reference beam will have a polarization parallel to the x-y plane and orthogonal to the coplanar planes of incidence of reference beams  1024 B- 1  and  1024 B- 2  with respect to the y-z plane. 
   In  FIG. 1   h , beams  1024 A- 1  and  1024 A- 2  in p polarization states are measurement beams derived from measurement beam  24 A as shown in  FIG. 1   b  and superimposed at a spot on substrate  60  to form a standing wave measurement beam. The angles of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are nominally the same and indicated as θ 1  in  FIG. 1   h . The planes of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are coplanar with the x-z plane. Beams  1024 B- 1  and  1024 B- 2  in a p polarization state are reference beams derived from reference beam  24 B and superimposed at the spot on substrate  60  to form a standing wave reference beam. The angles of incidence for reference beams  1024 B- 1  and  1024 B- 2  are nominally the same and indicated as θ 2  in  FIG. 1   h . The planes of incidence for reference beams  1024 B- 1  and  1024 B- 2  are coplanar and are shown as coplanar in the y-z plane in  FIG. 1   h . However, the coplanar planes of incidence of reference beams  1024 B- 1  and  1024 B- 2  may be at some other orientation with respect to the y-z plane. 
   When the electrical interference signal  72  is subsequently generated by detector  70  (see  FIG. 1   b ) by the detection of a mixed beam comprising the reflected/scattered standing wave measurement beam and the reflected/scattered standing wave reference beam, the standing wave measurement beam will have polarization components in the x and z directions and the standing wave reference beam will have a polarization component in the z direction and a polarization component parallel to x-y plane and parallel to the coplanar planes of incidence of reference beams  1024 B- 1  and  1024 B- 2  with respect to the y-z plane. 
   In  FIG. 1   i , beams  1024 A- 1  and  1024 A- 2  in s polarization states are measurement beams derived from measurement beam  24 A as shown in  FIG. 1   b  and superimposed at a spot on substrate  60  to form a standing wave measurement beam. The angles of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are nominally the same and indicated as θ 1  in  FIG. 1   i . The planes of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are coplanar with the x-z plane. Beams  1024 B- 1  and  1024 B- 2  in p polarization states are reference beams derived from reference beam  24 B and superimposed at the spot on substrate  60  to form a standing wave reference beam. The angles of incidence for reference beams  1024 B- 1  and  1024 B- 2  are nominally the same and indicated as θ 2  in  FIG. 1   i . The planes of incidence for reference beams  1024 B- 1  and  1024 B- 2  are coplanar and are shown as coplanar with the y-z plane in  FIG. 1   i . However, the coplanar planes of incidence of reference beams  1024 B- 1  and  1024 B- 2  may be at some other orientation with respect to the y-z plane. 
   When the electrical interference signal  72  is subsequently generated by detector  70  (see  FIG. 1   b ) by the detection of a mixed beam  32  comprising the reflected/scattered standing wave measurement beam and the reflected/scattered standing wave reference beam, the standing wave measurement beam will have a polarization component in the y direction and the standing wave reference beam will have a polarization component in z direction and a polarization component parallel to x-y plane and parallel to the coplanar planes of incidence of reference beams  1024 B- 1  and  1024 B- 2 . 
   In  FIG. 1   j , beams  1024 A- 1  and  1024 A- 2  in a p polarization state are measurement beams derived from measurement beam  24 A and superimposed at a spot on substrate  60  to form a standing wave measurement beam. The angles of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are nominally the same and indicated as θ 1  in  FIG. 1   j . The planes of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are coplanar with the x-z plane. The reference beam  1024 B- 3  is a non-standing wave beam derived from reference beam  24 B (see  FIG. 1   c ) and is incident on substrate  60  at the spot at normal incidence. The polarization of reference beam  1024 B- 3  is shown in the x-z plane in  FIG. 1   j . However, the plane of polarization of reference beam  1024 B- 3  may be at some other orientation with respect to the x-z plane and the angle of incidence may be non-zero. 
   When the electrical interference signal  72  is subsequently generated by detector  70  (see  FIG. 1   c ) by the detection of a mixed beam comprising the reflected/scattered standing wave measurement beam and the reflected/scattered non-standing wave reference beam, the standing wave measurement beam will have polarization components in each of the x and z directions and the non-standing wave reference beam will have a polarization in the x-y plane and parallel to x-z plane as shown in  FIG. 1   j . However, the polarization of reference beam  1024 B- 3  may be at some other orientation with respect to the x-z plane. 
   In  FIG. 1   k , beams  1024 A- 1  and  1024 A- 2  in s polarization states are measurement beams derived from measurement beam  24 A and superimposed at a spot on substrate  60  to form a standing wave measurement beam. The angles of incidence for the two measurement beams are nominally the same and indicated as θ 1  in  FIG. 1   k . The planes of incidence for measurement beams  1024 A- 1  and  1024 A- 2  are coplanar with the x-z plane. The beam  1024 B- 3  is a non-standing wave beam derived from reference beam  24 B and incident on substrate  60  at the spot at normal incidence. The polarization of reference beam  1024 B- 3  is in the y-z plane. However, plane of polarization of reference beam  1024 B- 3  may be at some other orientation with respect to the y-z plane and the angle of incidence may be non-zero. 
   When the electrical interference signal  72  is subsequently generated by detector  70  (see  FIG. 1   c ) by the detection of a mixed beam  32  comprising the reflected/scattered standing wave measurement beam and the reflected/scattered non-standing wave reference beam, the standing wave measurement beam will have only a y polarization component and the non-standing wave reference beam will have a polarization in the x-y plane, where y is a unit vector in the y-direction and in the y-z such as shown in  FIG. 1   k . However, the polarization of reference beam  1024 B- 3  may be at some other orientation with respect to the y-z plane. 
   The description of properties of measurement and reference beams at an interface corresponding to the interferometric and non-interferometric microscopy systems operating in a transmission mode is the same as corresponding portions of the description given for  FIGS. 1   g - 1   k  with the transformation:
 
θ 1             π−θ 1 .  (1)

   The description of the reference beam and measurement beam properties at a beam combining element in interferometer  10  when comprising standing wave beams is the same as corresponding portions of the description given for  FIGS. 1   g - 1   k.    
   Properties of Electric Fields Associated with Incident Beams 
   A general description of the electric fields of the measurement and reference beams in  FIGS. 1   g  and  1   h  is next given. Referring to  FIGS. 1   g  and  1   h , the measurement beam electric field E p   (1)  of the p standing wave beam at substrate  60  is formed by the superposition of the two measurement beams  1024 A- 1  and  1024 A- 2  with p polarizations propagating in the x-z plane. The electric field E p   (1)  may be written as 
   
     
       
         
           
             
               
                 
                   E 
                   p 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 E 
                                 
                                   p 
                                   , 
                                   0 
                                 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           ix 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             θ 
                                             1 
                                           
                                           ⁢ 
                                           
                                             sin 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 k 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 sin 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   θ 
                                                   1 
                                                 
                                                 ⁢ 
                                                 x 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         z 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           1 
                                         
                                         ⁢ 
                                         
                                           cos 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               k 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               sin 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 θ 
                                                 1 
                                               
                                               ⁢ 
                                               x 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             z 
                             ≥ 
                             0 
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 E 
                                 
                                   p 
                                   , 
                                   0 
                                 
                                 
                                   ′ 
                                   ⁡ 
                                   
                                     ( 
                                     1 
                                     ) 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           ix 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             θ 
                                             1 
                                             ′ 
                                           
                                           ⁢ 
                                           
                                             sin 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   kn 
                                                   ′ 
                                                 
                                                 ⁢ 
                                                 sin 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   θ 
                                                   1 
                                                   ′ 
                                                 
                                                 ⁢ 
                                                 x 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         z 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           1 
                                           ′ 
                                         
                                         ⁢ 
                                         
                                           cos 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 kn 
                                                 ′ 
                                               
                                               ⁢ 
                                               sin 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 θ 
                                                 1 
                                                 ′ 
                                               
                                               ⁢ 
                                               x 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   kn 
                                   ′ 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                   ′ 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   κ 
                                   ′ 
                                 
                                 ⁢ 
                                 z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sec 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                   ′ 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             z 
                             &lt; 
                             0 
                           
                           , 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   where E p,0   (1)  and E′ p,0   r(1)  are the amplitudes of the electric field component of the incident and refracted beams, respectively, n 1 ′ and κ′ are the real and imaginary components of the complex refractive index for z&lt;0, θ′ is the angle of refraction of the beam, i=√{square root over ((−1))}, λ is the wavelength for the two beams and wavenumber k=2π/λ, and x and z are a unit vectors in the x- and z-directions, respectively. The relative amplitudes E′ p,0   r(1) /E p,0   (1)  can be found for example in Section 7.3 of the book by J. D. Jackson entitled  Classical Electrodynamics  (Wiley, Second Edition). The time dependence exp[iωt] of E p   (1)  has been suppressed in Equation (2) and in subsequent equations where ω is the angular frequency of the beams. 
   The electric field E s   (2)  of the s standing wave reference beam at an interface in  FIG. 1   g  is formed by the superposition of two beams propagating in the y-z plane. The electric field E s   (2)  may be written as: 
   
     
       
         
           
             
               
                 
                   E 
                   s 
                   
                     ( 
                     2 
                     ) 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 E 
                                 
                                   s 
                                   , 
                                   0 
                                 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       k 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         θ 
                                         2 
                                       
                                       ⁢ 
                                       y 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             z 
                             ≥ 
                             0 
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 E 
                                 
                                   s 
                                   , 
                                   0 
                                 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         kn 
                                         ′ 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         θ 
                                         2 
                                         ′ 
                                       
                                       ⁢ 
                                       y 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   kn 
                                   ′ 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                   ′ 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   κ 
                                   ′ 
                                 
                                 ⁢ 
                                 z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sec 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                   ′ 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           z 
                           &lt; 
                           0. 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
     
   
   where E s,0   (2)  and E′ s,0   r(2)  are the amplitudes of the electric field component of the incident and refracted beams, respectively, and θ 2  is the angle of incidence of the two beams. The relative amplitudes E′ s,0   r(2) /E s,0   (2)  can be found for example in cited Section 7.3 of the book by Jackson. 
   The electric field E p   (2)  of the p standing wave reference beam at an interface in  FIG. 1   h  is formed by the superposition of two beams propagating in the y-z plane. The electric field E p   (2)  may be written as: 
                   E   p     (   2   )       =     {             2   ⁢       E     p   ,   0       (   2   )       ⁡     [             iy   ⁢           ⁢   cos   ⁢           ⁢     θ   2     ⁢     sin   ⁡     (     k   ⁢           ⁢   sin   ⁢           ⁢     θ   2     ⁢   y     )         +               z   ⁢           ⁢   sin   ⁢           ⁢     θ   2     ⁢     cos   ⁡     (     k   ⁢           ⁢   sin   ⁢           ⁢     θ   2     ⁢   y     )               ]       ⁢     ⅇ       -   ⅈ     ⁢           ⁢   k   ⁢           ⁢   cos   ⁢           ⁢     θ   2     ⁢   z         ,             z   ≥   0     ,                 2   ⁢       E     p   ,   0       ′   ⁡     (   2   )         ⁡     [             iy   ⁢           ⁢   cos   ⁢           ⁢     θ   2   ′     ⁢     sin   ⁡     (       kn   ′     ⁢   sin   ⁢           ⁢     θ   2   ′     ⁢   y     )         +               z   ⁢           ⁢   sin   ⁢           ⁢     θ   2   ′     ⁢     cos   ⁡     (       kn   ′     ⁢   sin   ⁢           ⁢     θ   2   ′     ⁢   y     )               ]       ⁢     ⅇ       -   ⅈ     ⁢           ⁢     kn   ′     ⁢   cos   ⁢           ⁢     θ   2   ′     ⁢   z       ⁢     ⅇ     k   ⁢           ⁢     κ   ′     ⁢   z   ⁢           ⁢   sec   ⁢           ⁢     θ   2   ′           ,             z   &lt;   0     ,                     (   4   )               
where E p,0   (2)  and E′ p,0   r(2)  are the amplitudes of the electric field component of the incident and refracted beams, respectively,
 
   The measurement beam electric field E s   (1)  in  FIG. 1   i  of a s standing wave measurement beam at substrate  60  is formed by the superposition of two measurement beams propagating in the x-z plane. The electric field E s   (1)  may be written as 
   
     
       
         
           
             
               
                 
                   E 
                   s 
                   
                     ( 
                     1 
                     ) 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 E 
                                 
                                   s 
                                   , 
                                   0 
                                 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   y 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       k 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         θ 
                                         1 
                                       
                                       ⁢ 
                                       x 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             z 
                             ≥ 
                             0 
                           
                           , 
                         
                       
                     
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 E 
                                 
                                   s 
                                   , 
                                   0 
                                 
                                 
                                   ′ 
                                   ⁡ 
                                   
                                     ( 
                                     1 
                                     ) 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   y 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         kn 
                                         ′ 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         θ 
                                         1 
                                         ′ 
                                       
                                       ⁢ 
                                       x 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   ⅈ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   kn 
                                   ′ 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                   ′ 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   κ 
                                   ′ 
                                 
                                 ⁢ 
                                 z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sec 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                   ′ 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           z 
                           &lt; 
                           0. 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   The description of the measurement beam electric fields in  FIGS. 1   j  and  1   k  are the same as the description of the measurement beam electric fields in  FIGS. 1   h  and  1   i , respectively. 
   Lateral Resolution 
   The discussion of the lateral resolution is in terms of properties of the electrical interference signal  72  that results from two laterally separated infinitesimal elements of substrate  60 . The discussion of the lateral resolution is also presented for an optical system that has a square aperture at the respective imaging system pupil so as to simplify the discussion although other apertures, e.g. round, or apodized apertures may be used. 
   The amplitude of a beam reflected/scattered by an infinitesimal element of substrate  60  imaged by interferometer  10  or  100  will be proportional to the convolution of the point transfer function for the imaging system of interferometer  10  or  100  with the product of the reflection/scattering coefficient of the infinitesimal element and the corresponding amplitude of beam incident on the infinitesimal element. The primary contribution to the infinitesimal contribution δS 1  to the electrical interference signal for the case of reference and measurement beams both comprising standing wave beams at substrate  60  such as shown in  FIG. 1   h  will be due to the z components of the measurement and reference fields given by Equations (2) and (4), respectively. The contribution of the x and y components will contribute as second order terms because of the orthogonality of the x, y, and z components. The primary contribution δS 1  is accordingly written as 
   
     
       
         
           
             
               
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       S 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         
                           x 
                           ′ 
                         
                         , 
                         y 
                         , 
                         
                           y 
                           ′ 
                         
                         , 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           z 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   8 
                   ⁢ 
                   
                     CR 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     E 
                     
                       p 
                       , 
                       0 
                     
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     E 
                     
                       p 
                       , 
                       0 
                     
                     
                       ( 
                       2 
                       ) 
                     
                   
                   × 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     θ 
                     1 
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           1 
                         
                         ⁢ 
                         x 
                       
                       ) 
                     
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     c 
                     ⁡ 
                     
                       [ 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             θ 
                             
                               x 
                               , 
                               0 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               - 
                               
                                 x 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   × 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     θ 
                     2 
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           2 
                         
                         ⁢ 
                         y 
                       
                       ) 
                     
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   c 
                   ⁢ 
                   
                       
                     
                         
                     
                     ⁢ 
                     
                         
                       
                           
                       
                       ⁢ 
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 θ 
                                 
                                   y 
                                   , 
                                   0 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   y 
                                   - 
                                   
                                     y 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         × 
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         V 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               
                                 φ 
                                 
                                   1 
                                   , 
                                   2 
                                 
                               
                               + 
                               φ 
                               + 
                               
                                 
                                   k 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           1 
                                         
                                       
                                       - 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   where the infinitesimal element of substrate  60  is located at (x, y, z=0), x′ and y′ are coordinates in the image space of the imaging system, δV is the infinitesimal volume of the infinitesimal element, R(x, y) is the reflection/scattering coefficient for the infinitesimal element, sin θ x,0  and sin θ y,0  represent the numerical aperture of the imaging system in the x and y directions, respectively, Δz is the displacement of the surface of substrate  60  from z=0, phase φ is the phase between measurement and reference beams determined by beam conditioner  22  or interferometer  10  as controlled by electronic processor and controller  80 , phase φ 1,2  is the phase between the reflected/scattered reference and measurement beam components of beam  32  for φ=0, and C is a proportionality constant. 
   If the properties of δS 1  given by Equation (6) are examined as a function of x′ or y′, the inferred resolution R x′  and R y′  of the imaging system in the x′ and y′ directions, respectively, are 
   
     
       
         
           
             
               
                 
                   
                     R 
                     
                       x 
                       ′ 
                     
                   
                   ≅ 
                   
                     λ 
                     
                       2 
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         
                           x 
                           , 
                           0 
                         
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 7 
                 ) 
               
             
           
           
             
               
                 
                   R 
                   
                     y 
                     ′ 
                   
                 
                 ≅ 
                 
                   
                     λ 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         
                           y 
                           , 
                           0 
                         
                       
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   However, if the properties of δS 1  given by Equation (6) are examined as a function of x or y, the inferred resolution R x  and R y  of the imaging system in the x′ and y′ directions, respectively, are different from R x′  and R y′ , respectively, given by Equations (7) and (8). For the discussion of the resolutions R x  and R y , the contributions to the electrical interference signal δS 2  of two equal infinitesimal volume elements located in the surface of substrate  60  are examined. For the two infinitesimal volume elements located at x±Δx, the corresponding δS 2  is 
   
     
       
         
           
             
               
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       S 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         
                           x 
                           ′ 
                         
                         , 
                         y 
                         , 
                         
                           y 
                           ′ 
                         
                         , 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           z 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   8 
                   ⁢ 
                   
                     CR 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     E 
                     
                       p 
                       , 
                       0 
                     
                     
                       ( 
                       1 
                       ) 
                     
                   
                   ⁢ 
                   
                     E 
                     
                       p 
                       , 
                       0 
                     
                     
                       ( 
                       2 
                       ) 
                     
                   
                   × 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     θ 
                     1 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           x 
                                           + 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             x 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 c 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         
                                           x 
                                           , 
                                           0 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           x 
                                           - 
                                           
                                             x 
                                             ′ 
                                           
                                           + 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             x 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               cos 
                               ⁡ 
                               
                                 [ 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           x 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               c 
                               ⁡ 
                               
                                 [ 
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       θ 
                                       
                                         x 
                                         , 
                                         0 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           ′ 
                                         
                                         - 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           x 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     } 
                   
                   × 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     θ 
                     2 
                   
                   ⁢ 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           2 
                         
                         ⁢ 
                         y 
                       
                       ) 
                     
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     c 
                     ⁡ 
                     
                       [ 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             θ 
                             
                               y 
                               , 
                               0 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               y 
                               - 
                               
                                 y 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   × 
                   
                       
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       V 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           [ 
                           
                             
                               φ 
                               
                                 1 
                                 , 
                                 2 
                               
                             
                             + 
                             φ 
                             + 
                             
                               
                                 k 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         θ 
                                         1 
                                       
                                     
                                     - 
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         θ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               z 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 9 
                 ) 
               
             
           
         
       
     
   
   where it has been assumed that R(x,y) and φ 1,2  are the same for the two infinitesimal volume elements for the purposes of simplifying the discussion without loss of important properties. The dependence of δS 2  on x is determined by the factor ƒ(x,x′,Δx) where 
   
     
       
         
           
             
               
                 
                   f 
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       
                         x 
                         ′ 
                       
                       , 
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   + 
                   
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       k 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           θ 
                                           1 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             x 
                                             + 
                                             
                                               Δ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               x 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       k 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           θ 
                                           
                                             x 
                                             , 
                                             0 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             x 
                                             - 
                                             
                                               x 
                                               ′ 
                                             
                                             + 
                                             
                                               Δ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               x 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           x 
                                           - 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             x 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 c 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         θ 
                                         
                                           x 
                                           , 
                                           0 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           x 
                                           - 
                                           
                                             x 
                                             ′ 
                                           
                                           - 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             x 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 10 
                 ) 
               
             
           
         
       
     
   
   Function ƒ(x,x′,Δx) is shown in  FIG. 1   l  as a function of x for different values of Δx, i.e. (Δx sin θ 1 /λ)=0, π/4, π/2, for the case of sin θ 1 =sin θ x,0 . 
   It is evident on examination of  FIG. 1   l  that lateral resolution R x  is expressed by the equation 
                   R   x     ≅       λ     4   ⁢   sin   ⁢           ⁢     θ     x   ,   0           .             (   11   )               
The lateral resolution R y  is determined by a similar analysis to be accordingly
 
   
     
       
         
           
             
               
                 
                   R 
                   y 
                 
                 ≅ 
                 
                   
                     λ 
                     
                       4 
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         
                           y 
                           , 
                           0 
                         
                       
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
         
       
     
   
   The lateral resolutions R x  and R y  expressed by Equations (11) and (12) are smaller by a factor of 2 than the lateral resolutions R x′  and R y′ , respectively, expressed by Equations (7) and (8), respectively. Thus the lateral spatial resolution is enhanced by a factor of approximately 2 when using standing wave reference and measurement beams at the measurement object and the reference and measurement objects are the same object simultaneously and by using a data acquisition procedures and analysis of the measured conjugated quadratures based on object space coordinates, i.e. x instead of x′ and y instead of y′. 
   One data acquisition procedure based on object space coordinates is to keep standing wave reference and/or measurement beam patterns fixed relative to interferometer  10  or  100  and scan substrate  60  in the x and/or y directions. The scans in x and/or y directions can be implemented by translating the stage by transducer  82 . For an example, consider detector  70  comprising a 1024×1024 pixel array CCD detector with a pitch of the pixels equal to 10 microns; secondary imaging system  120  shown in  FIG. 2   a  that has a magnification of 10; magnification of catadioptric imaging system  410 A shown in  FIGS. 2   a  and  2   b  is equal to 1; the resolutions of the catadioptric imaging system R x′ =100 nm and R y′ =100 nm; the pitch of the apertures in interface  12  is equal to 1 micron; and the spatial resolutions R x =50 and R y =50. For the example, the electrical interference signal  72  is measured comprising components for each of the 1024×1024 pixels of detector  70  for an array of displacements of substrate  60  in the x and/or y directions of increments of 25 nm for an over sampling by a factor of two in both of the x and y directions. The number of increments in each of the x and/or y directions is accordingly  40 , i.e. equal to the ratio of 1 micron and 25 nm. The resulting arrays of measured electrical interference signals are subsequently processed to obtain arrays of measured conjugated quadratures. 
   Alternatively, a data acquisition procedure can be based in part on translating the respective standing wave patterns on substrate  60  by scanning the relative phases of beams generating the respective standing wave patterns, e.g. by shifting the phase of beam  1024 A 1  relative to the phase of beam  1024 B 1  in  FIG. 1   h.    
   Note that the phase of conjugated quadratures of the infinitesimal contributions δS 1  and δS 2  is not dependent on Δz when cos θ 1 =cos θ 2 . Thus the phase of the conjugated quadratures of the infinitesimal contributions δS 1  and δS 2  relates directly to information about the reflecting/scattering properties of the infinitesimal volumes without complications introduced by vertical shifts of substrate  60 . 
   The description of the electric fields of the measurement and reference beams given with respect to  FIG. 1   g  is the same as corresponding portions of the description given of the electric fields of the measurement and reference beams given with respect to  FIG. 1   h  except that the primary term in δS 1  and δS 2  arise from electric field components in the x direction. 
   The description of the electric fields of the measurement and reference beams given with respect to  FIG. 1   i  is the same as corresponding portions of the description given of the electric fields of the measurement and reference beams given with respect to  FIG. 1   g  with the roles of the measurement and reference beams exchanged and except that the primary term in δS 1  and δS 2  arise from electric field components in the y direction. 
   The description of the electric fields of the measurement and reference beams given with respect to  FIG. 1   j  is the same as corresponding portions of the description given of the electric fields of the measurement and reference beams given with respect to  FIG. 1   h  except that the primary term in δS 1  and δS 2  arise from electric field components in the x direction. 
   The description of the electric fields of the measurement and reference beams given with respect to  FIG. 1   k  is the same as corresponding portions of the description given of the electric fields of the measurement and reference beams given with respect to  FIG. 1   i  except that the primary term in δS 1  and δS 2  arise from electric field components in the y direction. 
   The properties of reflection/scattering coefficient R(x, y) for the infinitesimal volume elements for a z component of an electric field corresponds to the measurement of δS 1  or δS 2  as expressed by Equations (6) and (9). The combination of s standing wave and p standing wave beams can be used to measurement the properties of the reflection/scattering R (x, y) for x and y components of an electric field. 
   Height Profiling 
   When the reference object is not the same object as the measurement object, the phase of the corresponding conjugated quadratures is [φ 1,2 +φ+k (cos θ 1 +cos θ x,0 )Δz] and the lateral resolution R x  in the x direction is the same as expressed by Equation (11). The background discrimination in the y direction can also be enhanced when the reference beam at beam combining interface  12  is a standing wave beam with angles of incidence in the y-z plane. Thus the phase of the conjugated quadratures of the infinitesimal contributions δS 1  and δS 2  relates directly to the height profile of substrate  60  and accordingly can be used in profiling the surface of a substrate with respect to height. 
   Non-Interferometric Imaging 
   Enhanced spatial resolutions are also obtained in non-interferometric imaging. The infinitesimal contributions δI 1  and δI 2  to the corresponding intensity signal  72  are 
   
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         I 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           
                             x 
                             ′ 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     4 
                     ⁢ 
                     
                       CR 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         ⁡ 
                         
                           [ 
                           
                             E 
                             
                               s 
                               , 
                               0 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                           ] 
                         
                       
                       2 
                     
                     × 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                           ⁢ 
                           x 
                         
                         ) 
                       
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         c 
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                 ( 
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                         } 
                       
                       2 
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 14 
                 ) 
               
             
           
         
       
     
   
   The dependence of δI 2  on x is determined by the factor [ƒ(x,x′,Δx)] 2  where ƒ(x,x′,Δx) is given by Equation (10). Function [ƒ(x,x′,Δx)] 2  is shown in  FIG. 1   m  as a function of x for different values of Δx, i.e. (Δx sin θ 1 /λ)=0,θ/4,π/2, for the case of sin θ 1 =sin θ x,0 . 
   It is evident on examination of  FIG. 1   m  that lateral resolution R x  is substantially the same enhanced resolution as expressed by the Equation (11). 
   Depth Discrimination 
   It is evident on inspection of Equations (2) and (5) for the z&lt;0 solutions that the spatial properties of E p   (1)  and E S   (1)  for z&lt;0 are the same as those of standing evanescent fields propagating in the same region, i.e. exponentially decaying solutions. It is important to note that the respective spatial properties are generated advantageously with a relatively large working distance, e.g., of the order of mm&#39;s, as compared to the disadvantage of having a working distance of ˜λ/4 generally required in the generation and coupling of the evanescent fields into a refractive medium (see cited book by Jackson). This is a particularly important advantage when profiling a surface of a substrate with a large spot, e.g. ˜mm 2 , simultaneously, where the surface under examination is not flat to {tilde under (&lt;)}λ/8. Also, the magnitude of the damping term in the refractive medium may be selected in part by the selection of the wavelength λ of the measurement beam. 
   The amplitude of background beams from out-of-focus reflective image planes internal to substrate  60  will as a consequence of the exponential attenuation of beams propagating in an absorbing medium generally be lower as compared to amplitudes of background beams generated in confocal microscopy systems for a non-absorbing medium. 
   The amplitudes of a standing wave reference beam and/or a standing wave measurement beam at the spots may by increased by the incorporation of a build up cavity or resonant cavity at substrate  60  and/or the beam combining element such as described in commonly owned U.S. Provisional Patent Application No. 60/221,091 (ZI-18) entitled “Multiple-Source Arrays with Optical Transmission Enhanced by Resonant Cavities” and U.S. patent application Ser. No. 09/917,400 (ZI-18) entitled “Multiple-Source Arrays with Optical Transmission Enhanced by Resonant Cavities” for which both are by Henry A. Hill and the contents of which are herein incorporated in their entirety by reference. An example of a resonant cavity is shown in  FIG. 8   a  of the cited U.S. Provisional Patent Application No. 60/221,091 and U.S. patent application Ser. No. 09/917,400. 
   An example of an interferometer  10  is shown diagrammatically in  FIG. 2   a  which comprises a catadioptric imaging system  410 A shown schematically in  FIG. 2   b  with catoptric elements comprising adaptive reflective surfaces and with a pellicle beam-splitter. Except with respect to the description of the respective measurement beams, the description of interferometer  10  and catadioptric imaging system  410 A of  FIGS. 2   a  and  2   b  is the same as corresponding portions of the description given for interferometer  10  and catadioptric imaging system  410 A in  FIGS. 4   a  and  4   b  of cited U.S. Provisional Patent Application No. 60/506,715 (ZI-56) and U.S. patent application Ser. No. 60/506,715 filed Sep. 24, 2004 entitled “Catoptric and Catadioptric Imaging Systems Comprising Pellicle Beam-Splitters And Non-Adaptive And Adaptive Catoptric Surfaces.” 
   In the cited description, a non-standing wave measurement beam is incident on substrate  60  whereas in the present invention, a standing wave measurement beam is incident on substrate  60 . Referring to  FIG. 2   a  of the present invention, first and second portions of input beam  24  are reflected and transmitted, respectively, by beam-splitter mirror system  454 A as a measurement beam and as reference beam  424 B. When input beam  24  comprises non-coextensive reference and measurement beams, element  454 A functions as a set of mirrors to reflect the measurement beam component of beam  24  and to transmit the reference beam component beam as reference beam  424 B. Propagation of reference beam  424 B is displaced out of the plane of  FIG. 2   a  and directed toward mirror system  454 D. Reference beam  424 B exits mirror system  454 D as reference beam  424 D (see  FIG. 2   b ). Propagation of reference beam  424 D is in the plane of  FIG. 2   b  and is incident on interface  12  comprising an array of apertures and thin fluorescent spots. The measurement beam is incident on non-polarizing beam-splitter  454 C after reflection by mirror  454 B. First and second portions of the measurement beam incident on non-polarizing beam-splitter  454 C are reflected and transmitted, respectively, to form measurement beams  424 C and  424 D, respectively, after translations by optical elements  454 D and  454 E, respectively, and reflections by mirrors  454 F and  454 G, respectively. Optical elements  454 F and  454 G each comprise a pair of mirrors arranged to introduce a parallel translation of a beam. Measurement beams  424 C and  424 D in  FIGS. 2   a  and  2   b  herein are incident on substrate  60  at a spot to form a standing wave measurement beam at the spot.