Patent Publication Number: US-8126677-B2

Title: Analyzing surface structure using scanning interferometry

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to Provisional Application No. 61/013,732, entitled “ANALYZING SURFACE STRUCTURE USING SCANNING INTERFEROMETRY,” filed on Dec. 14, 2007, the entire contents of which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The disclosure relates to using scanning interferometry to analyze the surface structure of a test object, and more particularly, to analyze the surface topography and/or features of a complex surface structure of the test object. 
     BACKGROUND 
     Scanning interferometry is used to gain information about a test object. Information about, for example, the surface structure can be relevant to flat-panel display (FPD) metrology, e.g., the characterization of FPD components, semiconductor wafer metrology, and in-situ analysis of thin films and dissimilar materials. Examples of relevant information include besides the surface topography itself, features of a complex surface structure, such as thin film parameters (thickness or index of refraction), discrete structures of dissimilar materials, and discrete structures that are under-resolved by the optical resolution of an interference microscope. 
     Interferometric techniques are commonly used to measure the profile of a surface of an object. To do so, an interferometer combines measurement light reflected from the surface of interest with reference light reflected from a reference surface to produce an interferogram. Fringes in the interferogram are indicative of spatial and structural variations between the surface of interest and the reference surface. 
     A scanning interferometer scans the optical path length difference (OPD) between the reference and measurement light of the interferometer over a range comparable to or larger than the coherence length of the interfering light. For multiple scan-positions, a detector measures the intensity of the interfering light, which is the basis for a scanning interferometry signal (hereafter also interferometry signal). For surface interferometry, for example, multiple camera pixels can be used to measure a spatial interferogram at each scan position, with each camera pixel measuring an interferometry signal for a corresponding location of the test surface over the range of scan positions. An interferometry signal is typically characterized by a sinusoidal carrier modulation (the “fringes”) with bell-shaped fringe-contrast envelope. 
     A limited coherence length of the interfering light can be produced, for example, by using a white-light source, which is referred to as scanning white light interferometry (SWLI). A typical SWLI signal features a few fringes localized near the zero OPD position which is defined as an equal optical path length for the reference and measurement light. 
     The conventional idea underlying interferometric metrology is to derive features of an object from the interferometry signal. The analysis can be performed in a scan domain, i.e., using the interferometry signal depending on the scan-coordinate, or in a frequency domain, i.e., using an interferometry spectrum derived from the interferometry signal. 
     For surface profiling, the first approach includes, for example, to locate the peak or center of the envelope, assuming that this position corresponds to the zero OPD of a two-beam interferometer for which one beam reflects from the object surface. The second approach includes, for example, calculating the rate of change of the phase of the transformed interferometry signal with the wavelength, assuming that an essentially linear slope is directly proportional to a surface height of the test object. This latter approach is referred to as Frequency Domain Analysis (FDA). See, for example, U.S. Pat. No. 5,398,113, U.S. Pat. No. 7,106,454, U.S. Pat. No. 7,271,918, the contents of which are herein incorporated by reference. 
     Conventional techniques used for surface characterization (e.g., ellipsometry and reflectometry) rely on the fact that the complex reflectivity of an unknown optical interface depends both on its intrinsic characteristics (material properties and thickness of individual layers) and on three properties of the light that is used for measuring the reflectivity: wavelength, angle of incidence, and polarization state. In practice, characterization instruments record reflectivity fluctuations resulting from varying these parameters over known ranges. 
     SUMMARY 
     Scanning interferometers can be used to analyze surface structure of a test object based on an interferometry signal. The analysis of the interferometry signal can involve a comparison of the interferometry signal with a set of model signals, each model signal being indicative for a specific feature (parameter) of the object, for which it is modeled. The comparison yields a merit value on that the determination of a test object parameter is based. 
     In general, in a first aspect, the invention features a method that includes comparing a scanning interferometry signal obtained for a location of a test object to each of multiple model signals corresponding to different model parameters for modeling the test object, wherein for each model signal the comparing comprises calculating a correlation function between the scanning interferometry signal and the model signal to identify a surface-height offset between the scanning interferometry signal and the model signal and, based on the identified surface-height offset, calculating a height-offset compensated merit value indicative of a similarity between the scanning interferometry signal and the model signal for a common surface height. The method further includes, based on the respective merit values for the different model signals, determining a test object parameter at the location of the test object. 
     In another aspect, an interferometer includes an optical system configured to obtain an scanning interferometry signal from a surface location of an object and a processor. The processor includes code configured to: 
     i) receive multiple model signals corresponding to different model parameters for modeling the test object, compare the scanning interferometry signal to each of multiple model signals, wherein for each model signal the comparing comprises calculating a correlation function between the scanning interferometry signal and the model signal to identify a surface-height offset between the scanning interferometry signal and the model signal and, based on the identified surface-height offset, calculating a height-offset compensated merit value indicative of a similarity between the scanning interferometry signal and the model signal for an approximated common surface height; and 
     ii) based on the respective merit values for the different model signals, determine a test object parameter at the location of the test object. 
     In another aspect, a method includes comparing a scanning interferometry signal obtained for each of multiple locations of a test object to each of multiple model signals corresponding to different model parameters for modeling the test object, wherein for each test object location and each model signal the comparing comprises calculating a correlation function between the scanning interferometry signal and the model signal based on a frequency domain representation of the scanning interferometry signal and a frequency domain representation of the model signal to identify a surface-height offset between the scanning interferometry signal and the model signal and, based on the identified surface-height offset, calculating a height-offset compensated merit value indicative of a similarity between the scanning interferometry signal and the model signal for a common surface height. The method further includes, based on the respective merit values for the different model signals at each of the different test object locations, determining one or more test object parameters at each test object location. 
     In another aspect, a method includes, for at least one model signal of a set of model signals, calculating a height-offset compensated merit value indicative of a similarity between a scanning interferometry signal and the model signal for a common surface height, wherein calculating the height-offset compensated merit value includes performing a correlation of the scanning interferometry signal or information derived thereof and the model signal or information derived thereof, and based on the correlation, determining a height-dependent phase slope between a frequency domain representation of the interferometry signal and a frequency domain representations of the model signal and compensating the phases of the coefficients of at least one of the frequency domain representations of the interferometry signal and the model signal. The method further includes, based on the height-offset compensated merit value, determining a test object parameter. 
     Implementations may include one or more of the following features. 
     In some embodiments, the calculated correlation function can be based on a frequency domain representation of the scanning interferometry signal and a frequency domain representation of the model signal. 
     In some embodiments, calculating the correlation function can include inverse transforming the product of the frequency domain representations of the scanning interferometry signal and the model signal into the scan coordinate domain. 
     In some embodiments, the identified surface-height offset can correspond to a peak in the calculated correlation function. The peak can be determined by interpolating the correlation function between scan-positions. 
     In some embodiments, identifying the surface-height offset can include determining a phase difference between the scanning interferometry signal and the model signal. 
     In some embodiments, determining the phase difference can include determining a complex phase of the correlation function at a peak positioning the correlation function. 
     In some embodiments, calculating the height-offset compensated merit value can include compensating a frequency domain representation of the scanning interferometry signal or a frequency domain representation of the model signal with a linear phase term having a slope corresponding to the identified surface-height offset and quantifying the similarity between the scanning interferometry signal and the model signal following the phase compensation. 
     The quantification of the similarity between the scanning interferometry signal and the model signal following the phase compensation can be performed in the frequency domain. 
     In some embodiments, a phase compensation can be applied to the frequency domain representation of the scanning interferometry signal to produce a frequency domain representation of the scanning interferometry signal corresponding to a surface height common to that used for modeling the model signal. 
     The phase compensation of the frequency domain representation of the interferometry signal can include multiplying a spectral component with a linear phase factor exp(−iKζ offset ), where K is the fringe frequency component and ζ offset  is the identified surface-height offset. 
     The phase compensation of the frequency domain representation of the interferometry signal can include multiplying a spectral component with a phase factor exp(−iA peak ), where A peak  is the complex phase of the correlation function at a peak of the calculated correlation function. 
     The phase compensation of the frequency domain representation of the interferometry signal can include removing a linear portion of the phase change within the spectrum. 
     The phase compensation comprises removing a phase difference between the interferometry spectrum and the model spectrum arising from the surface-height offset between the scanning interferometry signal and the model signal. 
     In some embodiments, calculating the height-offset compensated merit value can be based on a frequency domain representation of the scanning interferometry signal and a frequency domain representation of the model signal. 
     In some embodiments, calculating the height-offset compensated merit value can be restricted to a region of interest in the frequency domain. 
     In some embodiments, calculating the height-offset compensated merit value can be based on a least-square difference between the phase-compensated interferometry spectrum and the model spectrum. 
     In some embodiments, calculating the height-offset compensated merit value can be based on a complex phase of the correlation function at the peak position. 
     In some embodiments, calculating the height-offset compensated merit value can be based on the peak value of the correlation function at the peak position. 
     In some embodiments, calculating the height-offset compensated merit value can be based on normalizing the frequency domain representation of the scanning interferometry signal or the frequency domain representation of the model signal. 
     In some embodiments, the model parameters corresponding to the model signal can include one or more of thin film thickness and thin film index. The model parameters corresponding to the model signals can further include one or more parameters relating to an under-resolved surface feature. 
     In some embodiments, the under-resolved surface feature can be an array feature defining a diffractive grating. 
     In some embodiments, determining a test object parameter can include determining more than one test object parameter based on the respective merit values. 
     In some embodiments, the determined test object parameter can correspond to one or more of surface height, thin film thickness, and thin film index of refraction. The determined test object parameter can further correspond to one of the model parameters for the model signals. 
     In some embodiments, determining a test object parameter can include identifying a matching model signal based on comparing the height-offset compensated merit values. 
     Determining the test object parameter can be based on the matching model signal. 
     In some embodiments, determining the test object parameter can include corrections based on a complex phase of the correlation function at the peak. 
     In some embodiments, the method can further include outputting the test object parameter. 
     In some embodiments, comparing a scanning interferometry signal to each of multiple model signals and determining a test object parameter can be repeated for each of multiple scanning interferometry signals corresponding to different surface locations of the test object. 
     In some embodiments, the method can further include obtaining the scanning interferometry signals for the multiple surface locations. 
     In some embodiments, the scanning interferometry signals for the multiple surface locations can be obtained using a scanning interferometer that images the multiple locations onto an imaging detector. 
     In some embodiments, the interferometry signal can be obtained by imaging test light emerging from the test object to interfere with reference light on a detector, and varying an optical path length difference from a common source to the detector between interfering portions of the test and reference light, wherein the test and reference light are derived from the common source, and wherein the interferometry signal corresponds to an interference intensity measured by the detector as the optical path length difference is varied. 
     In some embodiments, the test and reference light can have a spectral bandwidth greater than 5% of a central frequency for the test and reference light. 
     The common source can have a spectral coherence length, and the optical path length difference can be varied over a range larger than the spectral coherence length to produce the scanning interferometry signal. 
     In some embodiments, optics used to direct test light onto the test object and image it to the detector can define a numerical aperture for the test light greater than 0.8. 
     In some embodiments, the method can further include accounting for systematic contributions to the scanning interferometry signal arising from a scanning interferometer system used to acquire the scanning interferometry signal. The method can further include calibrating the systematic contributions of the scanning interferometry system using a test-object having known properties. 
     In some embodiments of the interferometer, the code can be further configured to transform the scanning interferometer signal and the model signal into the frequency domain and calculate the correlation function based in the transformed signals. 
     In some embodiments, the code can be further configured to compensate a frequency domain representation of the scanning interferometry signal or a frequency domain representation of the model signal with a linear phase term having a slope corresponding to the identified surface-height offset and quantifying the similarity between the scanning interferometry signal and the model signal following the phase compensation. 
     In some embodiments, the processor can further include code configured to generate one of the model signals based on model parameters. 
     In some embodiments, the code can be also configured to determine a test object parameter map associated with a surface of the test object. The test object parameter map can be based on a height parameter, on a thin film parameter, and/or on an under-resolved surface feature parameter. 
     In some embodiments, the processor can be further configured to output information about the determined test object parameter. 
     In some embodiments, the optical system can include a multi-element detector configured to obtain an interferometry signal from each of multiple surface locations of the object, and wherein the processor is configured to determine information about a test object parameter at each of the multiple surface locations based on the obtained interferometry signals. 
     In another aspect, the invention features a process for making a display panel, including providing a component of the display panel, determining information about the component using a method or interferometer discussed with respect to the aforementioned aspects, wherein the component corresponds to the test object and the information is based on the test object parameter, and forming the display panel using the component. 
     Implementations of the process can include one or more of the following features and/or features of other aspects. For example, the component can include a pair of substrates separated by a gap and the information comprises information about the gap. Forming the display panel can include adjusting the gap based on the information. In some embodiments, forming the display panel includes filling the gap with a liquid crystal material. 
     The component can include a substrate and a layer of a resist on the substrate. The information can include information about the thickness of the layer of resist. The layer of resist can be a patterned layer, and the information can include information about a dimension or an overlay error of a feature of the patterned layer. Forming the display can include etching a layer of material under the layer of resist. 
     The component can include a substrate that includes spacers and the information can include information about the spacers. Forming the display can include modifying the spacers based on the information. 
     The details of one or more embodiments are set forth in the accompanying drawings and the description below. Other features and advantages will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic drawing of a Mirau-type scanning interferometry system. 
         FIG. 2  is a flow chart of an interferometry method for determining a surface structure. 
         FIG. 3  shows an example of an interferometry signal. 
         FIG. 4  shows an example of a model signal. 
         FIG. 5  shows examples of a model signal for different thin-film thicknesses. 
         FIG. 6  is a flow chart illustrating a library search. 
         FIG. 7  is a plot of a correlation function of an interferometry signal and a model signal. 
         FIG. 8  shows plots of the real and imaginary parts of the Fourier spectrum for an interferometry signal and a model signal. 
         FIG. 9  shows a plot comparing an interferometry signal and a matching model signal. 
         FIG. 10  is a plot of the value of a merit function for model signals for different thin-film thicknesses. 
         FIG. 11  shows a plot of a 2D-profile of an etched trench in a substrate with a thin-film. 
         FIG. 12  shows a plot of a 3D surface profile of a TFT area for a flat-panel display. 
         FIG. 13  shows Fourier magnitude and Fourier phase associated with a thin film interferometry signal. 
         FIG. 14A  is a schematic showing a device exemplary of the film structure resulting from the deposition of a dielectric over copper features deposited on a substrate. 
         FIG. 14B  is a schematic diagram of the device shown in  FIG. 14A  after undergoing chemical mechanical processing. 
         FIG. 15A  is a schematic diagram showing a top down view of an object which includes a substrate, e.g., a wafer, and an overlying layer, e.g., photoresist layer. 
         FIG. 15B  is a schematic diagram showing a side on view of the object. 
         FIG. 16A  is a schematic diagram of a structure suitable for use in solder bump processing. 
         FIG. 16B  is a schematic diagram of the structure from  FIG. 16A  after solder bump processing has occurred. 
         FIG. 17A  is a schematic diagram of an LCD panel composed of several layers. 
         FIG. 17B  is a flow chart showing various steps in LCD panel production. 
         FIG. 17C  is a diagram of an embodiment inspection station for LCD panels including an interferometric sensor. 
     
    
    
     Like reference numerals in different drawings refer to common elements. 
     DETAILED DESCRIPTION 
     Scanning interferometers can be used to analyze surface structure of an object by comparing interferometry signals with model signals. Examples of surface structure include surface heights, material composition, film thickness, and optically-under-resolved surface structure. Applications for scanning interferometry include semiconductor wafer inspection, flat panel display process control, and general laboratory use. A specific example is the measurement of the photoresist thickness in the halftone region of thin film transistors used for flat panel displays. 
     The measured interference signal is acquired with an interferometry system, such as interferometry system  100  shown in  FIG. 1 . The interferometry system  100  is based on a Mirau-type interferometer with an adjustable optical path length difference (OPD) between a measurement path and a reference path. 
     Referring to  FIG. 1 , a source module  105  provides illumination light  110  to a beam splitter  115 , which directs the illumination light  110  to a Mirau interferometric objective assembly  120 . The assembly  120  includes an objective lens  125 , a reference flat  130  having a reflective coating on a small central portion thereof defining a reference mirror  135 , and a beam splitter  140 . During operation, the objective lens  125  focuses the illumination light towards an object  145  through the reference flat  130 . The object  145  is characterized by its surface height profile h(x,y), which varies over the object surface, and its complex surface structure. 
     The beam splitter  140  transmits a first portion of the focusing light to the object  145  to define measurement light  150  and reflects a second portion of the focusing light to the reference mirror  135  to define reference light  155 . Then, the beam splitter  140  recombines the measurement light  150  reflected (or scattered) from the object  145  with the reference light  155  reflected from the reference mirror  135 . The objective  125  and an imaging lens  160  image the combined light to interfere on a detector  165  (e.g. a multi-pixel camera). As the relative position of the object  145  is being scanned, the detector  165  measures the intensity of the interfering light at one or more pixels of the detector and sends that information to a computer  167  for analysis. 
     The scanning in the Mirau-type interferometry system  100  involves a piezoelectric transducer (PZT)  170  coupled to the Mirau interferometric objective assembly  120 . The PZT  170  is configured to scan the assembly  120  as a whole relative to the object  145  along the optical axis of the objective lens  125  as denoted by the scan coordinate ζ in  FIG. 1 . The Mirau-type interferometry system  100  provides scanning interferometry data at each pixel of the detector  165 . Alternatively, a PZT may be coupled to the object  145  rather than the assembly  120  to provide the relative motion there between, as indicated by PZT actuator  175 . In yet further embodiments, the scanning may be provided by moving one or both of the reference mirror  135  and the beam splitter  140  relative to the objective lens  125  along the optical axis of the objective lens  125 . 
     Source module  105  includes a spatially extended source  180 , a telescope formed by lenses  185  and  187 , and an aperture  190  positioned in the front focal plane of the lens  185  (which coincides with the back focal plane of lens  187 ). This arrangement images the spatially extended source  180  onto the pupil plane  195  of the Mirau interferometric objective assembly  120 , which is an example of Koehler imaging. The size of the aperture  190  controls the size of the illumination field on the object  145 . 
     For simplicity,  FIG. 1  shows the measurement light  150  and the reference light  155  focusing onto particular points on the object  145  and the reference mirror  130 , respectively, and subsequently interfering on a corresponding point on the detector  165 . Such light corresponds to those portions of the illumination light  110  that propagate perpendicular to the pupil plane  195  of the Mirau interferometric objective assembly  120 . Other portions of the illumination light  110  ultimately illuminate other points on the object  145  and the reference mirror  135 , which are then imaged onto corresponding points on the detector  165 . 
     The detector  165  is, for example, a multiple element (i.e., multi-pixel) camera to independently measure the interference between the measurement light  150  and reference light  155  corresponding to different points on the object  145  (i.e., to provide spatial resolution for the interference pattern). The optical resolution of the interferometry system  100  is given by its optical characteristics and the pixel size of the detector  165 . 
     Because the scanning occurs in a region where the illumination light  110  is being focused onto the object  145 , the scan varies the OPD depending on the angle of incidence. As a result, the OPD from the source  201  to the detector  165  between interfering portions of the measurement light  150  and reference light  155  scale differently with the scan coordinate ζ depending on the angle of the measurement light  150  incident on, and emerging from, the object  145 . 
     This difference in how the OPD varies with the scan coordinate ζ introduces a limited coherence length of the light measured at each pixel of the detector  165 . Thus, the interference signal (as a function of scan coordinate ζ) is typically modulated by an envelope having a spatial coherence length on the order of λ/2(NA) 2 , where λ is the nominal wavelength of the illumination light and NA is the numerical aperture of the assembly  120 . To increase the limited spatial coherence, the assembly  120  in the scanning interferometry system  100  can define a large numerical aperture NA, e.g., greater than about 0.7 (or more preferably, greater than about 0.8, or greater than about 0.9). The interference signal can also be modulated by a limited temporal coherence length associated with the spectral bandwidth of the illumination source  180 . Depending on the configuration of the interferometry system  100 , one or the other of these limited coherence length effects may dominate, or they may both contribute substantially to the overall coherence length. 
       FIG. 2  shows an exemplary flow chart of the analysis of interferometry signal based on a surface-height offset compensation. To acquire interferometry signals for the object  145 , the interferometry system  100  scans mechanically or electro-optically the optical path difference between the reference and measurement path. The measurement light  150  is directed along the measurement path to the object  145  and after reflection interferes with the reference light  155 . The OPD at the beginning of the scan depends on the local surface height of the object  145 . The intensity of the interfering light is detected with the detector  165 . During the scan, the computer  167  records experimental intensity data I ex (x, y, ζ) for each image point or camera pixel x,y in successive camera frames (step  200 ). Neglecting any influence of the interferometry system  100  (e.g. detector sensitivity), the experimental intensity data I ex (x, y, ζ) represent the interferometry signal. For each of multiple camera pixels corresponding to different surface locations of the object  145 , the computer  167  can record such an interferometry signal during the OPD scan. 
     In  FIG. 3 , an exemplary SWLI-signal is plotted for a single pixel. The plot shows the measured intensity as a function of the scan position ζ. The SWLI-signal is detected for a Si-substrate having a SiO 2  thin-film. Note the two SWLI-signal comprises two overlapping signals, the one on the left for the Si-substrate and the one on the right for the top surface of the SiO 2  thin-film. 
     Next, after storing the interferometry signals as a function of OPD scan position ζ, the computer performs a transformation (e.g., a Fourier Transformation) to generate a frequency-domain spectrum of the interferometry signal (step  210 ). This interferometry spectrum contains both magnitude and phase information as a function of the spatial frequency of the interferometry signal in the scanning dimension. An example for analyzing the interferometry signal in the frequency domain is disclosed in the commonly owned U.S. Pat. No. 5,398,113 by Peter de Groot and entitled “Method and Apparatus for Surface Topography Measurements by Spatial-Frequency Analysis of Interferograms,” the contents of which are incorporated herein by reference. 
     The analysis of the measured interferometry signal is based on signal modeling. Specifically, one calculates and stores model signals as entries of a model library or one calculates the library entries when needed. The signal modeling can be performed with the same computer  167  or another computer (step  220 ). 
     The signal modeling is based on some user input about the object surface structure, e.g., about a film stack (step  230 ) and on a characterization of the interferometry system  100 , e.g., by using pupil plane imaging (step  240 ). With that information, one calculates the entries of the library, e.g., model signals for a parameter skew of the object  145 . For example, one generates a library of theoretical predictions for frequency-domain spectra for a variety of surface parameters and a system model for the interferometer. These model spectra can cover a range of possible thin film thicknesses, surface materials, and surface textures. In some embodiments, the model spectra are calculated for a constant surface height, e.g., for zero OPD. Thus, in such embodiments, the library does not contain information regarding the position of the object along the scan coordinate but contains information about the type of complex surface structure and the interaction of the object  145 , the optical system, the illumination system, and detection system. 
     Turning now to an exemplary generation of a library of SWLI model signals, a SWLI signal is the sum of the interference signals over all the rays passing through the pupil and over all the wavelengths of the light source. Incoherent superposition allows calculating a model signal I(L,ζ) for a specific film thickness L as an inverse Fourier Transform: 
                     I   ⁡     (     L   ,   ζ     )       =       ∫     -   ∞     ∞     ⁢       ρ   ⁡     (     L   ,   K     )       ⁢     exp   ⁡     (       -   ⅈ     ⁢           ⁢   K   ⁢           ⁢   ζ     )       ⁢     ⅆ   k                 (   1   )               
where ρ(L,K) are the Fourier components at a fringe frequencies K. A fringe frequency of K=4 cycle/micron (cycle=2π radians) means that the intensity oscillates through four full periods for every micron of scan motion. The fringe frequencies K correspond to the angle of incidence Ψ of a ray passing through the illumination pupil according to
 
 K= 4πβ/λ  (2)
 
where β=cos(Ψ) is the directional cosine of the incident angle Ψ and λ is one of the wavelengths within the optical spectrum of the light source. The Fourier components ρ(L,K) are weighting coefficients that indicate how much of the interference effect comes from the particular combinations of incident angle Ψ and wavelength λ and give rise to a fringe frequency K according to Eq. (2). The Fourier components ρ(L,K) values also include complex phase information characteristic of the object surface and of the system-level dispersion. SWLI tools have a broad range of non-zero Fourier components ρ(L,K) and corresponding oscillations in the intensity data I(L,ζ). For a film-free surface, constructive interference in Eq. (1) happens only near the zero-ζ scan position.
 
     The coefficient ρ(L,K) for each fringe frequency K is proportional to a single integral over the wavenumbers k=2π/λ in the source spectrum: 
                       ρ   ⁡     (     L   ,     K   &gt;   0       )       =       ∫     k   =     K   /   2       ∞     ⁢       Sys   ⁡     (     β   ,   k     )       ⁢       m   *     ⁡     (     L   ,   β   ,   k     )       ⁢       ⅆ   k       k   2             ,           (   3   )               
where m(L, β, k) is the object reflectivity for a thin film structure of thickness L, and the system characteristics independent of the object together are collected into a variable Sys(β,k). The system characteristics, here assumed circularly symmetric, include the transmissivity t(β,k) of the measurement path, the reflectivity r(β,k) of the reference path, the assumed axially-symmetric distribution U(β) of light in the pupil plane, and the effective optical spectrum V(k) of the light source and of the detector taken together:
 
Sys(β, k )= U (β) r (β, k ) t *(β, k ) V ( k )  (4)
 
The directional cosine β appearing in Eq. (3) is a function of both the fringe frequency K and wavenumber k according to Eq. (2), and is linked therefore to the variable k of integration.
 
     A system characterization or calibration determines Sys (β,k) and perhaps can be calculated as an object-independent “base” library that may be applied to object surfaces m(L, β, k) as a final step in the model signal generation.  FIG. 4  illustrates the quality of the signal prediction for a solid (film-free) surface. 
     A method for generating model interference signals is disclosed in U.S. patent application Ser. No. 11/780,360 filed on Jul. 19, 2007 and entitled “GENERATING MODEL SIGNALS FOR INTERFEROMETRY,” the contents of which are herein incorporated by reference. 
       FIG. 5  shows exemplary model signals that could be used when analyzing the experimental data of  FIG. 3 . For a thin-film measurement, which is an example of a common application of model-based SWLI analysis, one looks for a film thickness L assuming that the film materials are known. Thus, the film thickness L is the variable model parameter, and one approach to comparing experiment to theory is to calculate in advance a library of possible signals for comparison over a range (or skew) of film thicknesses. The model signals are then stored as their Fourier or frequency-domain equivalents ρ(L,K) calculated e.g. from Eq. (3). Of course, if the software is quick enough, one could calculate the model signals on the fly, rather than storing them. But given that potentially a large number of image pixels all with the same model parameter skew will be analyzed, it might be of advantage to use a pre-determined library. Looking at the model signals of  FIG. 5 , which are modeled for film-thicknesses of 0 nm, 500 nm, and 1000 nm), one could guess that the SiO 2  thickness contributing to the interference signal of  FIG. 3  is close to 1000 nm thick. 
     In a matching operation (step  250 ), the experimental interferometry signal is compared to the library by means of a library search that identifies a matching model signal.  FIG. 6  illustrates an example flowchart of a library search that is used to analyze the object  145  for surface structure information. One acquires an interferometry signal (step  600 ) and generates a library of model signals (step  610 ). Then, one compares the interferometry signal and the model signal (step  630 ). Based on the comparison, one identifies the matching model signal (step  640 ) that is used for the determination of test object parameters characterizing the surface structure. 
     In the case of a thin film of unknown thickness ( FIG. 3 ), the library for a single surface type, e.g. SiO 2  on Si, can range over many possible film thicknesses with, for example, the top surface height always equal to zero. Other examples of a surface structure are a surface roughness, for which the adjustable parameter may be roughness depth and/or spatial frequency, and an under-resolved grating structure. 
     Referring to the matching operation (step  250 ) shown in  FIG. 2 , the object  145  is analyzed in 2D on a pixel by pixel basis. Thus, one selects Fourier data for a data point (pixel) of the object (step  260 ). Then, one selects an entry of the library, e.g. a model signal or spectrum (step  270 ). Using a correlation function of the interferometry and model signal, one determines the relative position of the interferometry signal and the model signal, i.e., the surface-height offset (step  280 ). The surface-height offset corresponds to a phase term, based on which one compensates the interferometry signal, the model spectrum or both, e.g. in the frequency domain (step  290 ). Then, one calculates a height-offset compensated merit value by calculating a frequency-domain difference between the interferometry signal and the model signal (step  300 ). 
     The calculation of the merit value is repeated for the complete library or a subset of entries of the library (loop  310 ). Then, one identifies the “best” merit value, i.e., the library entry (or an interpolation of library entries) that best fulfills a criteria associated with the merit function. Based on that merit value and/or the associated model signal, one determines one or more test object parameters, e.g., thin film thickness and surface height (step  320 ). 
     This procedure is repeated for all pixels of interest (loop  330 ), and the test object parameters are presented, for example, as 3D images of the film thickness and height (step  340 ). 
     In what follows a mathematical description of the analysis is provided. 
     In some embodiments, one compares the model and interferometry signals in a frequency domain (e.g., Fourier-Transform domain). Because Eq. (1) is an inverse Fourier Transform, one can generate the comparable experimental Fourier coefficients q ex (x, y, K) from the forward transform of the experimental intensity data I ex (x, y, ζ) 
                       q   ex     ⁡     (     x   ,   y   ,   K     )       =       ∫     -   ∞     ∞     ⁢         I   ex     ⁡     (     x   ,   y   ,   ζ     )       ⁢     exp   ⁡     (     ⅈ   ⁢           ⁢   K   ⁢           ⁢   ζ     )       ⁢       ⅆ           ⁢   ζ     .                 (   5   )               
The experimental coefficients q ex (x, y, K) contain a phase term that is a linear function of the surface height h(x,y):
 
 q   ex ( x,y,K )=ρ ex ( x,y,K )exp [ iKh ( x,y )].  (6)
 
The term Kh(x,y) is the height-dependent phase slope that can complicate a direct comparison of the Fourier coefficients q ex (x, y, K) with theoretically predicted Fourier coefficients ρ(L,K) based on surface structure alone, independent of surface height. Thus, at first one estimates h(x,y) well enough to remove its phase contribution from q ex (x, y, K), leaving only the height-independent portion ρ ex (x, y, K).
 
     Besides compensating for the height dependent phase on the experimental side, one can consider the phase on the model side or on both sides. In these cases, the phase compensation can correspond to propagating the experimental interferometry signal and the model signal to a scan position that optimizes the height independent overlap when comparing the two signals. 
     To determine the height dependent phase slope, one uses a correlation technique for estimating h(x,y). Suppose one has a model signal spectrum ρ(L,K). The correlation of the experimental and model signals is given by 
                     J   ⁡     (     x   ,   y   ,   L   ,   ζ     )       =       ∫     -   ∞     ∞     ⁢         q   ex     ⁡     (     x   ,   y   ,   K     )       ⁢       ρ   *     ⁡     (     L   ,   K     )       ⁢       ⅆ   K     .                 (   7   )               
For the case where an exact match of experiment to theory has been identified, the correlation is
 
                     J   ⁡     (     x   ,   y   ,   L   ,   ζ     )       =       ∫     -   ∞     ∞     ⁢              ρ   ⁡     (     L   ,   K     )            2     ⁢   exp   ⁢     {     ⅈ   ⁢           ⁢     K   ⁡     [       h   ⁡     (     x   ,   y     )       -   ζ     ]         }     ⁢       ⅆ   K     .                 (   8   )               
The correlation should have a peak magnitude when [h(x,y)−ζ]=0. The peak can be found by searching through the scan positions ζ to find the discrete position best ζ best (x, y, L) (corresponding to a specific camera frame) that gives the peak value for |J(x, y, L, ζ)|. The position best ζ best (x, y, L) can be refined to a value ζ fine (x, y, L) by, e.g., 2 nd -order interpolation between camera frames.
 
     Also in the case that the model signal is a not exactly the same as the interferometry signal, the correlation still allows identifying the position of a “best” overlap of model signal and interferometry signal. 
       FIG. 7  shows an example correlation magnitude |J(x, y, L, ζ)| of an experimental interferometry signal and a model signal. The peak represents the position of the “best” overlap. The peak corresponds also to the local surface height when the model signal is correctly matched to the interferometry signal. 
     A further refinement can be based on the complex phase A of the correlation:
 
 A ( x,y,L )=arg { J[x,y,L,ζ   fine ( x,y,L )]}.  (9)
 
The complex phase A is associated with an overall K-independent phase gap between the model signal and interferometry signal for the cases that the signals are lined up as best as possible based on the correlation magnitude, i.e., based on the signal shape. In the ideal case, if the model signal includes any expected phase shifts related to the instrument or the surface materials, the complex phase A(x, y, L) measured in this way should be zero once the correct thickness L best  has been identified. The complex phase A(x, y, L) can be preserved as a free variable to optimize the fit; but one can use the complex phase A(x, y, L) also in the merit function to evaluate the quality of that fit.
 
     Based on the refined scan position ζ fine (x, y, L) giving the height offset, one can compensate the linear phase term. For example, one can calculate an experimental signal coefficients q shift  corrected for the position within the scan and for any phase offsets with respect to the model signal:
 
 q   shift ( x,y,L,K )= q   ex ( x,y,K )exp [− iKζ   fine ( x,y,L )− iA ( x,y,L )],  (10)
 
where ζ fine (x, y, L) is the interpolated “best” match scan position for the correlation |J(x, y, L, ζ)|, and the phase gap A(x, y, L) follows from Eq. (9). If one has identified the correct thickness L best , the phase-shifted Fourier coefficients of the interferometry signal should be
 
 q   shift ( x,y,L   best   ,K )=ρ ex ( x,y,K ),  (11)
 
but for all other test values of L, one can only expect that this is approximately the case.
 
     Based on the phase compensation, one calculates a phase (height offset)-compensated merit value indicative for the quality of the fit of the model signal and the interferometry signal. A suitable measure of the quality of the match between the model signal and the interferometry signal is the least-squares difference 
                       χ   2     =       ∑   K     ⁢       [         q   shift   ′     ⁡     (     x   ,   y   ,   L   ,   K     )       -       ρ   ′     ⁡     (     L   ,   K     )         ]     2         ,           (   12   )               
where the sum is over all of the K values for which ρ(L,K)≠0; i.e., within a frequency-domain region of interest K max ≧K≧K min  defined by the expected signal bandwidth and max mm excluding noise and drift.
 
     To perform this comparison directly as in Eq. (12), the model and experimental signals have been normalized for signal strength, as indicated by the primes: 
                         q   shift   ′     ⁡     (     x   ,   y   ,   L   ,   K     )       =         q   shift     ⁡     (     x   ,   y   ,   L   ,   K     )           ∫     K   =     K   max         K   =     K   min         ⁢              q   ex     ⁡     (     x   ,   y   ,   K     )            ⁢     ⅆ   K             ,           (   13   )                   ρ   ′     ⁡     (     L   ,   K     )       =         ρ   ⁡     (     L   ,   K     )           ∫     K   =     K   max         K   =     K   min         ⁢            ρ   ⁡     (     L   ,   K     )            ⁢     ⅆ   K           .             (   14   )                 FIG. 8  shows a graphical comparison for the real and imaginary parts of the Fourier coefficients in the left and right plots, respectively. The oscillations of the coefficients are related to the film thickness—the thicker the film, the more rapid these oscillations are as a function of fringe frequency K. The smooth lines indicate the model spectrum ρ′(L,K) and the lines (showing the underlying data) indicate the phase compensated experimental coefficients q′ shift (x, y, L,K).
 
       FIG. 9  shows the experimental signal in the scan domain with the model signal (dotted) corresponding to the best match, as found by a Frequency-domain search. The experimental signal is much cleaner in  FIG. 9  than in the original data of  FIG. 3  because it is reconstructed from the region-of-interest in the frequency domain corresponding to the signal only, thus filtering out noise and low-frequency drift. 
     Although one can very nicely identify the best match by the minimum of the χ 2 -function, one may construct a merit function that is inversely proportional to the χ 2 -function, so that the best match is defined by a peak in a merit value distribution for the library entries. The merit function can also include other criteria, such as the phase gap A(x, y, L) calculated in Eq. (9) from the complex correlation. As has been noted, in the ideal case, the phase gap A(x, y, L) measured in this way should be zero at the correct thickness L=L best ; therefore, a non-zero value is a measure of the mismatch between experiment and theory. In addition, a good match should have a large correlation peak at ζ fine . Thus a suitable merit function is, for example, 
                     Π   ⁡     (     x   ,   y   ,   L     )       =                    J   ⁡     (     x   ,   y   ,   L   ,     ζ   fine       )            2         χ   2     ⁡     (     x   ,   y   ,   L     )         ⁡     [     1   -       w   A     ⁢            A   ⁡     (     x   ,   y   ,   L     )       π              ]       2     .             (   15   )               
One can of course construct other merit functions to optimize the robustness of the algorithm, or to use other factors such as the signal strength as merit criteria.
 
     To determine the parameter characterizing surface structure, one evaluates the calculated values of the merit-function for the applied model signals.  FIG. 10  shows a distribution of merit values for the example signal of  FIG. 3 . If the model-signal library has a small enough thickness increment, then it is sufficient to simply identify the model signal at L=L best  that gives the highest merit-value. Otherwise, it may be useful and efficient to interpolate to L fine  by means of a 2 nd -order fit near the library value L best . Other possibilities include interpolating the model signal itself between neighboring values, or performing a “live” search that involves calculating the model signal in real time, rather than using stored library values. An additional option is to average the merit values over multiple pixels, to improve signal to noise. 
     The distribution of the merit values indicates the quality of the match between the model signals for a specific film thickness (SiO 2  over Si) and the experimental interferometry signal. In the case of  FIG. 10 , the best matching model signal has been modeled for a model parameter associated with a film thickness of 1008 nm. 
     In some embodiments, it may be straight forward to generate top-surface height profiles because one has already calculated the necessary information during the correlation procedure. A first estimate of surface height based on the coherence peak is
 
 h   Θ ( x,y )=ζ fine ( x,y,L   best ),  (16)
 
where the subscript Θ indicates that this height relates to the coherence or signal shape effect. A more refined estimate is given by
 
                         h   θ     ⁡     (     x   ,   y     )       =         h   Θ     ⁡     (     x   ,   y     )       +       1     K   0       ⁢     {             A   ⁢     (     x   ,   y   ,     L   best       )       -               2   ⁢   π   ⁢           ⁢     round   [         A   ⁡     (     x   ,   y   ,     L   best       )       -   α       2   ⁢   π       ]             }           ,           (   17   )               
where α is the field average of A(x, y, L best ) over the x and y coordinates, and K 0  is the nominal carrier-signal frequency defined by the centroid of the square magnitudes of the Fourier coefficients for a surface without a film:
 
     
       
         
           
             
               
                 
                   
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       FIGS. 11 and 12  illustrate the profiling capability of the methods and systems described herein.  FIG. 11  shows a 2D surface profile of a trench that has been etched into the 980-nm thick SiO 2  film to a depth of approximately 160 nm. Part of the trench has been coated with gold so that the top-surface profile may be measured without interference from the thin film effect. The line is the top-surface profile as measured by coating the trench with gold. The comparison in  FIG. 11  is between this top-surface height profile and the measured film thickness, with an offset to the height profile to line up the curves at the top surface. The result shows a slightly deeper trench depth, which may be real (a consequence of the gold pooling at the bottom of the trench) or an artifact of the modeling. In either case, the match is quite close and illustrates &lt;200 nm film thickness profiling to high lateral resolution. 
       FIG. 12  shows a 3D surface profile of a TFT area for a flat-panel display. The TFT area as shown in the 100× intensity image on the left, has a thickness range for a photoresist film in the horseshoe-shaped HT area that measures from 120 nm to 320 nm in the 3D profile on the right. 
     The disclosed embodiments do not depend on unwrapping the phase when one identifies the height-offset and are, therefore, generally not affected by the uncertainty that can be introduced by phase unwrapping. The uncertainty of phase unwrapping is explained in connection with  FIG. 13 . Some methods for analyzing an interferometry signal relay on phase unwrapping. For example, in one embodiment disclosed in U.S. Pat. No. 7,106,454, one removes the linear phase change by subtracting a linear fit to the difference in phase between the scanning interferometry signal and the model signal. Then, one analyzes the remaining non-linear phase spectrum. 
     Removing the phase slope by linear fitting requires that one unwraps or connects the phase data across the Fourier frequencies. Phase unwrapping removes the inevitable 2π phase uncertainties, which are generated when the phase values are calculated. However, phase unwrapping is not always easy, for example, with complex surface structures. Real phase nonlinearities associated with a thin film can have amplitudes of π for wavelengths and angles corresponding to an anti-reflection coating. 
     In  FIG. 13 , the Fourier magnitude and phase are plotted over the Fourier frequency (cycles/trace) for a scanning interferometry signal of test object having a thin film of a photoresist material over molybdenum with a thickness of 508 nm. 
     The example of  FIG. 13  illustrates the uncertainty that is present in phase unwrapping and that affects the quality of the analysis of the test object. One-cycle or 2π phase jumps are given between frequency bins  28  and  29  and between frequency bins  55  and  56 . A 2π phase jump is most likely a result of the overall phase slope wrapped into the ±π range. The 2π phase jump at frequency bins  55  and  56  can be repaired by subtracting 2π and continuing the phase at a value of −0.5 cycles. 
     The phase jump from bin  56  to bin  57  is more complicated because it is different from bin  56  by almost exactly π. The unwrapping procedure is chaotic across such a phase step, sometimes wrapping up by 2π, sometimes not. When the phase unwrapping is inconsistent, the result of the analysis is also inconsistent. 
     A number of embodiments have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention, some examples of which are described below. 
     In this disclosure, “interferometry signal” and “model signal” are often used for simplifying reasons but information derived thereof can be used in a like manner for many purposes. For example, the comparing of the interferometry signal and the model signal can be based on processed interferometry and/or model signals. For example, the interferometry signals can be digitally pre-processed, by noise suppression or correction, selection of a signal portion or a time window. Moreover, the comparing can be based on a library comparison of the interferometry signal in a frequency domain representation, e.g., a comparison of a frequency spectrum associated with the interferometry signal and a modeled frequency spectrum. 
     Although in the above described embodiments, the height compensation was achieved by modifying the scanning interferometry signal, one can also modify the model signal, or both, i.e., the scanning interferometry signal and the model signals. However, the modification should be such that comparing the interferometry signal and the model signal is based on signals that are associated with a common surface height. For example, the propagated optical path length in the model is adjusted to the optical path length of the interferometer such that the zero OPDs in the interferometer and the model are based on same condition for the measurement light and the test light. 
     Generally, the height-offset compensated merit value can be calculated based on a height-offset compensated, phase-compensated, and/or surface-height independent interferometry signal (or information derived thereof). For example, the height-offset compensated merit value can be derived in a phase-compensated spectral presentation of the interferometry signal, e.g. a Fourier spectrum. 
     For the comparison with a model signal, a library of model signals may be generated empirically, using sample artifacts. As another alternative, the library may use information from prior supplemental measurements of the object surface provided by other instruments, for example an ellipsometer, and any other input from a user regarding known properties of the object surface, so as to reduce the number of unknown surface parameters. Any of these techniques for library generation, theoretical modeling, empirical data, or theory augmented by supplemental measurements, may be expanded by interpolation to generate intermediate values, either as part of the library creation or in real time during a library search. 
     Comparing the model and the interferometry signals may be based on any of the following: a product of, or a difference between, magnitude and/or phase data in the frequency spectrum, including, e.g., the product of, or difference between, the average magnitude and the average phase, the average magnitude itself, and the average phase itself, the slope, width and/or height of the magnitude spectrum; interference contrast; data in the frequency spectrum at DC or zero spatial frequency; nonlinearity or shape of the magnitude spectrum; the zero-frequency intercept of the phase; nonlinearity or shape of the phase spectrum; and any combination of these criteria. 
     In some embodiments, a test object parameter is determined based on the calculated merit value. Specifically, the test object parameter can be based on a “best-matching” model signal having the best merit value, on one or more interpolated model signal derived from one or more “best-matching” model signals, and/or on interpolated model parameters associated with one or more “best-matching” model signals. 
     Examples of a test object parameter include parameters describing the surface structure. The surface structure can be characterized by surface-height features, which can be, for example, optically resolved with an interferometry microscope, and by features of a complex surface structure. In this specification complex surface structure includes inner structure of the test object and under-resolved surface structure that can not be optically resolved with the interferometry microscope. Examples for parameters of a surface height feature include the surface height itself. Examples for parameters of an inner structure include thin-film data (e.g., thickness, index of refraction, and number of thin film layers). Examples for parameters of an under-resolved surface structure include under-resolved feature data such as under-resolved diffraction grating structure, step height structure, and location of a step. 
     The test object parameter can be associated with the model signal. For example, a parameter characterizing the surface height can be determined through correlating the interferometry signal and a best matching model signal. Then, the correlation produces a peak at a scan coordinate associated to the surface height. Similarly, in the frequency domain, the surface height can be extracted using conventional FDA analysis. As an example for a complex surface feature, one can assign the thickness of a surface film that was used as a model parameter when modeling the best matching model signal as the determined thickness of a surface film of the test object. 
     In some cases, the comparison can be performed iteratively to further improve the results. In two dimensions, the comparison can be refined on a pixel-by-pixel or regional basis, by the creation of refined model signals relevant to the local surface type. For example, if it is found that the surface has a thin film of approximately 0.1 micron during a preliminary comparison, then the computer may generate a fine-grain library of example model parameters (thin film thickness) close to 1 micron to further refine the comparison. 
     In some embodiments, the analysis may be similar to that described in  FIG. 2  except that the height compensated comparison between the interferometry signal and the model signals is based on information in the scan coordinate domain. The experimental signal may be characterized by a quasi-periodic carrier oscillation modulated in amplitude by an envelope function with respect to the scan coordinate. Comparing the model and the interferometry signals may then be based on any of the following: average signal strength; the shape of the signal envelope, including e.g. deviation from some ideal or reference shape such as a Gaussian; the compensated phase of the carrier signal with respect to the envelope function; the relative spacing of zero crossings and/or signal maxima and minima; values for maxima and minima and their ordering; peak value of the correlation between the interferometry and model signals, after adjusting for optimal relative scan position; and any combination of these criteria. 
     Based on the comparison of the interferometry signal and model signals, one can determine one or more test object parameters. The computer may then display or transmit these test object parameters describing the surface structure (complex surface structure and height information) numerically or graphically to the user or to a host system for further analysis or for data storage. 
     For example, using the matching model and/or the correlation function, the computer determines surface height information in addition to characteristics of the identified complex surface structure. For the case of 2D imaging, the computer can generate, for example, a three-dimensional image constructed from the height data and corresponding image plane coordinates, together with graphical or numerical display of the complex surface structure. 
     In some embodiments, the user may only be interested in the complex surface structure modeled by the model signals, but not in the surface height, in which case the steps for determining surface height are not performed. Conversely, the user may only be interested in surface height, but not the complex surface structure modeled by the model signals, in which case the computer compensates the experimental interferometry signal (or information derived thereof), and/or the model signal (or information derived thereof) for the contributions of the linear phase when comparing the interferometry signal and the model signal, so that the matching model and consecutively the surface height may be more accurately and more efficiently determined, but the computer needs not explicitly determine the complex surface structure or display it. 
     The above described analysis may be applied to a variety of surface analysis problems, including: simple thin films (in which case, for example, the variable parameter of interest may be the film thickness, the refractive index of the film, the refractive index of the substrate, or some combination thereof); multilayer thin films; sharp edges and surface features that diffract or otherwise generate complex interference effects; under-resolved surface roughness; under-resolved surface features, for example, a sub-wavelength width groove on an otherwise smooth surface; dissimilar materials (for example, the surface may comprise a combination of thin film and a solid metal, in which case the library may include both surface structure types and automatically identify the film or the solid metal by a match to the corresponding frequency-domain spectra); optical activity such as fluorescence; spectroscopic properties of the surface, such as color and wavelength-dependent reflectivity; polarization-dependent properties of the surface; deflections, vibrations or motions of the surface or deformable surface features that result in perturbations of the interference signal; and data distortions related to the data acquisition procedure, e.g. a data acquisition window that does not fully encompass the interferometry signal. 
     Thus, test object parameters characterizing related features can be determined and model signals can be parameterized with model parameters describing these features in the modeling process. 
     In some cases, the analysis may also include a system characterization, which includes, e.g. measuring one or more reference artifacts having a known surface structure and surface topography, so as to determine parameters such as system wave front error, dispersion, and efficiency that may not be included in the theoretical model. 
     Furthermore, the analysis may include an overall calibration, which includes e.g., measuring one or more reference artifacts to determine the correlation between measured surface parameters, such as film thickness as determined by the library search, and the values for these parameters as determined independently, e.g. by ellipsometric analysis. 
     The interferometry system may include any of the following features: a spectrally narrow-band light source with a high numerical aperture (NA) objective; a spectrally broad band light source; a combination of a high-NA objective and a spectrally broadband source; an interferometric microscope objective, including oil/water immersion and solid immersion types, in e.g. Michelson, Mirau or Linnik geometries; a sequence of measurements at multiple wavelengths; unpolarized light; and polarized light, including linear, circular, or structured. For example, structured polarized light may involve, for example, a polarization mask, generating different polarizations for different segments of the illumination or imaging pupils, so as to reveal polarization-dependent optical effects attributable to surface characteristics. The interferometer may also include the overall system calibration, described above. 
     In other embodiments, a source module may include an arrangement in which a spatially extended light source is imaged directly onto the test object, which is known as critical imaging. 
     In some embodiments, the limited coherence length of the light used to generate the scanning interferometry signal is based on a white light source, or more generally, a broadband light source. In other embodiments, the light source may be monochromatic, and the limited coherence length can result from using a high numerical aperture (NA) for directing light to, and/or receiving light from, the test object. The high NA causes light rays to contact the test surface over a range of angles, and generates different spatial frequency components in the recorded signal when the OPD is scanned. In yet further embodiments, the limited coherence can result from a combination of both effects. 
     The origin of the limited coherence length may also be a physical basis for there being information in the scanning interferometry signal. Specifically, the scanning interferometry signal contains information about complex surface structure because it is produced by light rays contacting the test surface with many different wavelengths and/or at many different angles. 
     To provide ellipsometry measurements, the interferometry system may include a fixed or variable polarizer in the pupil plane. Referring to  FIG. 1 , the Mirau-type interferometry system  100  can include polarization optics  197  in the pupil plane to select a desired polarization for the light incident on, and emerging from the test sample. Furthermore, the polarization optics may be reconfigurable to vary the selected polarization. The polarization optics may include one or more elements including polarizers, waveplates, apodization apertures, and/or modulation elements for selecting a given polarization. Furthermore, the polarization optics may be fixed, structured or reconfigurable, for the purpose of generating data similar to that of an ellipsometer. For example, a first measurement with a radially-polarized pupil for s polarization, followed by a radially-polarized pupil for p polarization. In another example, one may use an apodized pupil plane with linearly polarized light, e.g., a slit or wedge, which can be rotated in the pupil plane so as to direct any desired linear polarization state to the object, or a reconfigurable screen such as a liquid crystal display. 
     In further embodiments, polarization optics may be positioned elsewhere in the apparatus. For example, linear polarization can be achieved anywhere in the system. 
     Alternative configurations may allow the use of apertures, polarizers, wavelength filters, or other devices at or near the pupil plane  195  of the interferometry system so as to isolate various azimuthal angles, positions within the pupil plane, polarizations etc., either statically or dynamically. 
     For example, to analyze the test object with various polarization states, one can use polarizing elements e.g. in the illumination or imaging planes. These elements may be electro-optically actuated and operate at high speed, again providing hundreds of measurements per second because of the high-speed data acquisition afforded by the single-detector geometry. 
     Alternatively, or in addition, one can apply or select multiple wavelengths by using a filtered light source and multiple data acquisitions. The filtering of wavelengths may be performed by spectroscopic means, tunable-wavelength interference filters, a second interferometer, an acousto-optic tunable filter, switchable light sources such as multiple lasers operated in sequence, or any other device or combination of devices. 
     Alternative configurations may allow high-speed data acquisition, which is made possible by a single or small number of detector elements near the image plane, allowing for rapid, repetitive measurements as needed to accommodate averaging or sequential changes in the instrument configuration, e.g., sequencing through a range of wavelengths. 
     Among other applications, the techniques described above can be applied to process control in semiconductor manufacturing. An example of this is in-process monitoring of critical dimensions (CDs), which is central to the fabrication of many high-technology components on the micron and nanometer scales. Examples include semiconductor IC processes such as transistor and logic creation, as well as copper-damascened connections. Defined broadly, CDs include lateral dimensions, etch depth, film thickness, step height, sidewall angle and related physical dimensions that influence the performance of semiconductor devices. CD metrology provides process control and defect detection that occur in the course of manufacturing, especially as a result of processes such as etching, polishing, cleaning and patterning. In addition, the basic measurement capabilities implied by CD metrology have broad application outside of Semiconductor IC manufacturing, including e.g. displays, nanostructures, and diffractive optics. 
     For example, scanning interferometry measurements can be used for non-contact surface topography measurements semiconductor wafers during chemical mechanical polishing (CMP) of a dielectric layer on the wafer. CMP is used to create a smooth surface for the dielectric layer, suitable for precision optical lithography. Based on the results of the interferometric topography methods, the process conditions for CMP (e.g., pad pressure, polishing slurry composition, etc.) can be adjusted to keep surface non-uniformities within acceptable limits. 
     It is to be further understood that, because some of the constituent system components and method steps depicted in the accompanying figures can be implemented in software, the actual connections between the systems components (or the process steps) may differ depending upon the manner in which the disclosed method is programmed. Given the teachings provided herein, one of ordinary skill in the related art will be able to contemplate these and similar implementations or configurations of the disclosed systems and methods. 
     For example, the numerical and symbolic steps described herein can be converted into a digital program executed, e.g., on a digital signal processor according to methods well known in the art. The digital program can be stored on a computer readable medium such as a hard disk and can be executable by a computer processor. Alternatively, the appropriate steps can be converted into a digital program that is hardwired into dedicated electronic circuits within the processor that executes the steps. Methods for generating such dedicated electronic circuits based on a given numerical or symbolic analysis procedure are also well known in the art. 
     Exemplary Applications 
     As discussed previously, the above-described systems and methods can be applied to a variety of surface analysis problems. A description of certain exemplary applications follows. 
     Semiconductor Processing 
     The systems and methods described above can be used in a semiconductor process for tool specific monitoring or for controlling the process flow itself. In the process monitoring application, single/multi-layer films are grown, deposited, polished, or etched away on unpatterned Si wafers (monitor wafers) by the corresponding process tool and subsequently the thickness and/or optical properties are measured using the interferometry system employing the scan error correction technique disclosed herein. The average, as well as within wafer uniformity, of thickness (and/or optical properties) of these monitor wafers are used to determine whether the associated process tool is operating with targeted specification or should be retargeted, adjusted, or taken out of production use. 
     In the process control application, latter single/multi-layer films are grown, deposited, polished, or etched away on patterned Si, production wafers by the corresponding process tool and subsequently the thickness and/or optical properties are measured with the interferometry system employing the scan error correction technique disclosed herein. Production measurements used for process control typical include a small measurement site and the ability to align the measurement tool to the sample region of interest. This site may consists of multi-layer film stack (that may itself be patterned) and thus requires complex mathematical modeling in order to extract the relevant physical parameters. Process control measurements determine the stability of the integrated process flow and determine whether the integrated processing should continue, be retargeted, redirected to other equipment, or shut down entirely. 
     Specifically, for example, the interferometry systems disclosed herein can be used to monitor the following equipment: diffusion, rapid thermal anneal, chemical vapor deposition tools (both low pressure and high pressure), dielectric etch, chemical mechanical polishers, plasma deposition, plasma etch, lithography track, and lithography exposure tools. Additionally, the interferometry system disclosed herein can be used to control the following processes: trench and isolation, transistor formation, as well as interlayer dielectric formation (such as dual damascene). 
     Copper Interconnect Structures and Chemical Mechanical Polishing 
     It is becoming common among chip makers to use the so-called ‘dual damascene copper’ process to fabricate electrical interconnects between different parts of a chip. This is an example of a process which may be effectively characterized using a suitable surface topography system. The dual damascene process may be considered to have six parts: (1) an interlayer dielectric (ILD) deposition, in which a layer of dielectric material (such as a polymer, or glass) is deposited onto the surface of a wafer (containing a plurality of individual chips); (2) chemical mechanical polishing (CMP), in which the dielectric layer is polished so as to create a smooth surface, suitable for precision optical lithography, (3) a combination of lithographic patterning and reactive ion etching steps, in which a complex network is created comprising narrow trenches running parallel to the wafer surface and small vias running from the bottom of the trenches to a lower (previously defined) electrically conducting layer, (4) a combination of metal deposition steps which result in the deposition of copper trenches and vias, (5) a dielectric deposition step in which a dielectric is applied over the copper trenches and vias, and (6) a final CMP step in which the excess copper is removed, leaving a network of copper filled trenches (and possibly vias) surrounded by dielectric material. 
     Referring to  FIG. 14A , a device  500  is exemplary of the film structure resulting from the deposition of a dielectric  504  over copper features  502  deposited on a substrate  501 . The dielectric  504  has a non-uniform outer surface  506  exhibiting height variations therealong. Interference signals obtained from device  500  can include interference patterns resulting from surface  506 , an interface  508  between copper features  502  and dielectric  504 , and an interface  510  between substrate  501  and dielectric  504 . The device  500  may include a plurality of other features that also generate interference patterns. 
     Referring to  FIG. 14B , a device  500 ′ illustrates the state of device  500  after the final CMP step. The upper surface  506  has been planarized to a surface  506 ′, and interface  508  may now be exposed to the surroundings. Interface  510  at the substrate surface remains intact. Device performance and uniformity depends critically on monitoring the planarization of surface  504 . It is important to appreciate that the polishing rate, and therefore the remaining copper (and dielectric) thickness after polishing, depends strongly and in a complex manner on the polishing conditions (such as the pad pressure and polishing slurry composition), as well as on the local detailed arrangement (i.e., orientation, proximity and shape) of copper and surrounding dielectric regions. Hence, portions of surface  506  over copper elements  502  may etch at different rates than other portions of surface  506 . Additionally, once interface  508  of copper elements  502  is exposed, the dielectric and copper elements may exhibit different etch rates. 
     This ‘position dependent polishing rate’ is known to give rise to variable surface topography on many lateral length scales. For example, it may mean that chips located closer to the edge of a wafer on aggregate are polished more rapidly than those located close to the center, creating copper regions which are thinner than desired near the edges, and thicker than desired at the center. This is an example of a ‘wafer scale’ process nonuniformity—i.e., one occurring on length scale comparable to the wafer diameter. It is also known that regions which have a high density of copper trenches polish at a higher rate than nearby regions with low copper line densities. This leads to a phenomenon known as ‘CMP induced erosion’ in the high copper density regions. This is an example of a ‘chip scale’process non-uniformity—i.e., one occurring on a length scale comparable to (and sometimes much less than) the linear dimensions of a single chip. Another type of chip scale nonuniformity, known as ‘dishing’, occurs within single copper filled trench regions (which tend to polish at a higher rate than the surrounding dielectric material). For trenches greater than a few microns in width dishing may become severe with the result that affected lines later exhibit excessive electrical resistance, leading to a chip failure. 
     CMP induced wafer and chip scale process nonuniformities are inherently difficult to predict, and they are subject to change over time as conditions within the CMP processing system evolve. To effectively monitor, and suitably adjust the process conditions for the purpose of ensuring that any nonuniformities remain within acceptable limits, it is important for process engineers to make frequent non-contact surface topography measurements on chips at a large number and wide variety of locations. This is possible using embodiments of the interferometry methods and systems described above. 
     In some embodiments one or more spatial properties, e.g., the topography of surface  506  and/or the thickness of dielectric  504 , are monitored by obtaining low coherence interference signals from the structure before and/or during CMP. Based on the spatial properties, the polishing conditions can be changed to achieve the desired planar surface  506 ′. For example, the pad pressure, pad pressure distribution, polishing agent characteristics, solvent composition and flow, and other conditions can be determined based on the spatial properties. After some period of polishing, the spatial property can again be determined and the polishing conditions changed as needed. The topography and/or thickness is also indicative of the end-point at which, e.g., surface  504 ′ is achieved. Thus, the low coherence interference signals can be used to avoid depressions caused by over polishing different regions of the object. The low coherence interference methods and systems are advantageous in this respect because spatial properties of the device, e.g., the relative heights of the surface of the dielectric (a) over copper elements  502  and (b) over substrate surface  510  but adjacent copper elements  502  can be determined even in the presence of the multiple interfaces. 
     Photolithography 
     In many microelectronics applications, photolithography is used to pattern a layer of photoresist overlying a substrate, e.g., a silicon wafer. Referring to  FIGS. 15A and 15B , an object  30  includes a substrate, e.g., a wafer,  32  and an overlying layer, e.g., photoresist layer  34 . Object  30  includes a plurality of interfaces as occur between materials of different refractive index. For example, an object-surroundings interface  38  is defined where an outer surface  39  of photoresist layer  34  contacts the environment surrounding object  30 , e.g., liquid, air, other gas, or vacuum. A substrate-layer interface  36  is defined between a surface  35  of wafer  32  and a bottom surface  37  of photoresist layer  34 . Surface  35  of the wafer may include a plurality of patterned features  29 . Some of these features have the same height as adjacent portions of the substrate but a different refractive index. Other features may extend upward or downward relative to adjacent portions of the substrate. Accordingly, interface  36  may exhibit a complex, varying topography underlying the outer surface of the photoresist. 
     A photolithography apparatus images a pattern onto the object. For example, the pattern may correspond with elements of an electronic circuit (or the negative of the circuit). After imaging, portions of the photoresist are removed revealing the substrate underlying the removed photoresist. The revealed substrate can be etched, covered with deposited material, or otherwise modified. Remaining photoresist protects other portions of the substrate from such modification. 
     To increase manufacturing efficiencies, more than one device is sometimes prepared from a single wafer. The devices may be the same or different. Each device requires that a subset of the wafer be imaged with a pattern. In some cases, the pattern is sequentially imaged onto different subsets. Sequential imaging can be performed for several reasons. Optical aberrations can prevent achieving adequate pattern focus quality over larger areas of the wafer. Even in the absence of optical aberrations, the spatial properties of the wafer and photoresist may also prevent achieving adequate pattern focus over large areas of the wafer. Aspects of the relationship between the spatial properties of the wafer/resist and focus quality are discussed next. 
     Referring back to  FIG. 15B , object  30  is shown with a number N subsets  40   i , each smaller than a total area  41  the object to be imaged. Within each subset  40   i , spatial property variations, e.g., height and slope variations of the wafer or photoresist, are typically smaller than when taken over the total area  41 . Nonetheless, the wafer or photoresist of different subsets  40   i  typically have different heights and slopes. For example, layer  34  exhibits thicknesses Δt 1  and Δt 2 , which vary the height and slope of surface  39 . Thus, each subset of the object may have a different spatial relationship with the photolithography imager. The quality of focus is related to the spatial relationship, e.g., the distance between the object and the photolithography imager. Bringing different subsets of the object into proper focus may require relative repositioning of the object and imager. Because of the object height and slope variations, proper subset focus cannot be achieved solely by determining the position and orientation of the object with respect to a portion of the object that is remote to the imaged subset, e.g., a side  43  of the object. 
     Proper focus can be achieved by determining a spatial property of an object within a subset of the object to be imaged (or otherwise processed). Once the position of the subset has been determined, the object (and/or a portion of the photolithography imager) can be moved, e.g., translated, rotated, and/or tilted, to modify the position of the subset with respect to a reference, e.g., a portion of the photolithography imager. The determination and movement (if necessary) can be repeated for each subset to be imaged. 
     The determination of the spatial property of the subset can include determining a position and/or height of one or more points of an outer surface of a thin layer of the object, the one or more points lying within the subset of the object to be imaged. For example, the position and orientation of the outer surface  39  of subset  40   2  ( FIG. 15A ) can be determined based upon the positions of points  42   1 - 42   3  within the subset. The determination of the spatial property of the subset to be imaged can include using an interferometer to illuminate the subset with light and detecting an interference signal including light reflected from the illuminated subset. In some embodiments, a plurality of subsets are simultaneously imaged with light to obtain a plurality of interference signals. Each interference signal is indicative of one or more spatial properties of a subset. Thus, the interference signals can be used to prepare an image indicative of the topography of the object over a plurality of the subsets. During photolithography of the subsets, the wafer is positioned based upon the topography of the individual subsets as determined from the plurality of interference signals. Hence, each subset can be positioned for optimum focus with respect to the photolithography apparatus. 
     Detecting an interference signal from each subset of an object to be imaged can include detecting light reflected from the subset and reference light over an OPD range that is at least as large as a coherence length of the detected light. For example, the light may be detected at least over its coherence length. In some embodiments, the interferometer is configured so that the light reflected from the illuminated subset is dominated by light reflected from either an outer interface (such as outer surface  39 ) or an inner interface (such as interface  36 ). In some embodiments, a spatial property of an object is determined based on only a portion of the interference signal. For example, if the interference signal includes two or more overlapping interference patterns, a spatial property of the object can be determined based upon a portion of one of the interference patterns that is dominated by contributions from a single interface of the object. 
     Solder Bump Processing 
     Referring to  FIGS. 16A and 16B , a structure  1050  is exemplary of a structure produced during solder bump processing. Structure  1050  includes a substrate  1051 , regions  1002  non-wettable by solder, and a region  1003  wettable by solder. Regions  1002  have an outer surface  1007 . Region  1003  has an outer surface  1009 . Accordingly, an interface  1005  is formed between regions  1002  and substrate  1001 . 
     During processing a mass of solder  1004  is positioned in contact with wettable region  1003 . Upon flowing the solder, the solder forms a secure contact with the wettable region  1003 . Adjacent non-wettable regions  1002  act like a dam preventing the flowed solder from undesirable migration about the structure. It is desirable to know spatial properties of the structure including the relative heights of surfaces  1007 ,  1009  and the dimensions of solder  1004  relative to surface  1002 . As can be determined from other discussions herein, structure  1050  includes a plurality of interfaces that may each result in an interference pattern. Overlap between the interference patterns prevents accurate determinate of the spatial properties using known interference techniques. Application of the systems and methods discussed herein allow the spatial properties to be determined. 
     Spatial properties determined from structure  1050  can be used to change manufacturing conditions, such as deposition times for layers  1002 ,  1003  and the amount of solder  1004  used per area of region  1003 . Additionally, heating conditions used to flow the solder can also be changed based on the spatial properties to achieve adequate flow and or prevent migration of the solder. 
     Flat Panel Displays 
     The interferometry systems and methods disclosed herein can be used in the manufacture of flat panel displays such as, for example, liquid crystal displays (LCDs). 
     In general, a variety of different types of LCDs are used in many different applications, such as LCD televisions, desktop computer monitors, notebook computers, cell phones, automobile GPS navigation systems, automobile and aircraft entertainment systems to name a few. While the specific structure of LCDs can vary, many types of LCD utilize a similar panel structure. Referring to  FIG. 17A , for example, in some embodiments, a LCD panel  450  is composed of several layers including two glass plates  452 ,  453  connected by an edge seal  454 . Glass plates  452  and  453  are separated by a gap  464 , which is filled with a liquid crystal material. Polarizers  456  and  474  are applied to the outer surfaces of glass plates  453  and  452 , respectively. When integrated into a LCD, one of the polarizers operates to polarize light from the display&#39;s light source (e.g., a backlight, not shown) and the other polarizer serves as an analyzer, transmitting only that component of the light polarized parallel to the polarizer&#39;s transmission axis. 
     An array of color filters  476  is formed on glass plate  453  and a patterned electrode layer  458  is formed on color filters  476  from a transparent conductor, commonly Indium Tin Oxide (ITO). A passivation layer  460 , sometimes called hard coat layer, commonly based on SiOx is coated over the electrode layer  458  to electrically insulate the surface. An alignment layer  462  (e.g., a polyimide layer) is disposed over the passivation layer  460  to align the liquid crystal material in gap  464 . 
     Panel  450  also includes a second electrode layer  472  formed on glass plate  452 . Another hard coat layer  470  is formed on electrode layer  472  and another alignment layer  468  is disposed on hard coat layer  470 . In active matrix LCDs (AM LCDs), one of the electrode layers generally includes an array of thin film transistors (TFTs) (e.g., one or more for each sub-pixel) or other integrated circuit structures. A 3D surface profile of a TFT is shown in  FIG. 12 , for example. 
     The liquid crystal material is birefringent and modifies the polarization direction of light propagating through the LCD panel. The liquid crystal material also has a dielectric anisotropy and is therefore sensitive to electric fields applied across gap  464 . Accordingly, the liquid crystal molecules change orientation when an electric field is applied, thereby varying the optical properties of the panel. By harnessing the birefringence and dielectric anisotropy of the liquid crystal material, one can control the amount of light transmitted by the panel. 
     The cell gap Δg, i.e., thickness of the liquid crystal material, is determined by spacers  466 , which keep the two glass plates  452 , 453  at a fixed distance. In general, spacers can be in the form of preformed cylindrical or spherical particles having a diameter equal to the desired cell gap or can be formed on the substrate using patterning techniques (e.g., conventional photolithography techniques). The cell gap affects both the amount of optical retardation of light traversing the panel and the viscoelastic response of molecular alignment of the liquid crystal material, and therefore an important parameter to accurately control in LCD panel manufacturing. 
     In general, LCD panel manufacturing involves multiple process steps in forming the various layers. For example, referring to  FIG. 17B , a process  499  includes forming the various layers on each glass plate in parallel, and then bonding the plates to form a cell. As illustrated, initially, TFT electrodes are formed (step  499 A 1 ) on a first glass plate. A passivation layer is formed (step  499 A 2 ) over the TFT electrodes, and then an alignment layer is formed (step  499 A 3 ) over the passivation layer. Next, spacers are deposited (step  499 A 4 ) on the alignment layer. Processing of the second glass plate typically involves forming color filters (step  499 B 1 ) and forming a passivation layer over the color filters (step  499 C 1 ). Then, electrodes (e.g., common electrodes) are formed (step  499 B 3 ) on the passivation layer, and an alignment layer is then formed (step  499 B 4 ) on the electrodes. 
     The cell is then formed by bonding the first and second glass plates together (step  499 C 1 ), and the cell is then filled with the liquid crystal material and sealed (step  499 C 2 ). After sealing, the polarizers are applied to the outer surface of each of the glass plates (step  499 C 3 ), providing the completed LCD panel. The combination and ordering of the steps shown in the flow chart are illustrative and, in general, other step combinations and their relative ordering can vary. 
     Furthermore, each step illustrated in the flow chart in  FIG. 17B  can include multiple process steps. For example, forming the TFT electrodes (commonly referred to as “pixel electrodes”) on the first glass plate involves many different process steps. Similarly, forming the color filters on the second glass plate can involve numerous process steps. Typically, forming pixel electrodes, for example, includes multiple process steps to form the TFTs, ITO electrodes, and various bus lines to the TFTs. In fact, forming the TFT electrode layer is, in essence, forming a large integrated circuit and involves many of the same deposition and photolithographic patterning processing steps used in conventional integrated circuit manufacturing. For example, various parts of the TFT electrode layer are built by first depositing a layer of material (e.g., a semiconductor, conductor, or dielectric), forming a layer of photoresist over the layer of material, and exposing the photoresist to patterned radiation. The photoresist layer is then developed, which results in a patterned layer of the photoresist. Next, portions of the layer of material lying beneath the patterned photoresist layer are removed in a etching process, thereby transferring the pattern in the photoresist to the layer of material. Finally, the residual photoresist is stripped from the substrate, leaving behind the patterned layer of material. These process steps can be repeated many times to lay down the different components of the TFT electrode layer, and similar deposition and patterning steps are often used to form color filters as well. 
     In general, the interferometry techniques disclosed herein can be used to monitor production of LCD panels at a variety of different stages of their production. For example, the interferometry techniques can be used to monitor the thickness and/or uniformity of photoresist layers used during LCD panel production. As explained previously, photoresist layers are used in lithographic patterning of TFT components and color filters, for example. For certain process steps, a layer of photoresist can be studied using a low coherence interferometry system prior to exposing the photoresist to patterned radiation. The low coherence interferometry systems can measure a thickness profile of the photoresist layer at one or more locations of the glass plate. Alternatively, or additionally, the techniques can be used to determine a surface profile of the photoresist layer. In either case, where the measured photoresist layer characteristics is within specified tolerance windows, the photoresist layer can be exposed to the desired patterned radiation. Where the photoresist layer is not within the specified window, it can be stripped from the glass plate and a new photoresist layer deposited. 
     In some embodiments, the interferometry techniques are used to monitor characteristics of a patterned photoresist layer. For example, critical dimensions (e.g., line widths) of patterned features can be studied. Alternatively, or additionally, the interferometry techniques can be used to determine overlay error between the features in the patterned resist and features beneath the photoresist layer. Again, where measured critical dimensions and/or overlay error are outside process windows, the patterned photoresist can be stripped from the substrate and a new patterned photoresist layer formed. 
     In certain embodiments, the interferometry techniques can be used in conjunction with half-tone photolithography. Increasingly, half-tone photolithography is used where specific thickness variations in the features of a patterned resist layer are desired. The low coherence interferometry techniques disclosed herein can be used to monitor thickness profiles of photoresist patterns in half-tone regions. In addition, the techniques can be used to determine both overlay and critical dimensions of these features. 
     In some embodiments, the interferometry techniques can be used to detect contaminants (e.g., foreign particles) at different stages on the glass plates at different stages of the production process. Such contaminants can give rise to visual defects (i.e., mura defects) in display panels, ultimately affecting the manufacturer&#39;s yield. Often, such defects are only detected by visual inspection, usually performed after the panel has been assembled. The interferometry techniques disclosed herein can be used to perform automated inspection of the glass plates at one or more points during the production process. Where particles are detected, the contaminated surface of the glass plate can be cleaned before the next production step. Accordingly, use of the techniques can reduce the occurrence of mura defects in panels, improving panel quality and reducing manufacturing costs. 
     Among other factors, the electrooptic properties (e.g., the contrast ratio and brightness) are dependent on the cell gap Δg. Cell gap control during manufacturing is often critical to obtaining uniform, quality displays. In certain embodiments, the disclosed interferometry techniques can be used to ensure that cell gap has desired uniformity. For example, the techniques can be used to monitor the height and/or position of spacers on a glass plate. Monitoring and controlling spacer height, for example, can reduce cell gap variations across a display. 
     In some cases, the actual cell gap may differ from the dimensions of spacers because, during assembly, pressure or vacuum is applied to introduce the liquid crystal medium, the edge seals cure and may change dimensions, and the added liquid crystal material can generates capillary forces between the glass plates. Both before and after adding the liquid crystal matter, the surfaces of the exposed layers on the glass plates reflect light that results in an interference pattern indicative of the cell gap Δg. The low coherence nature of the interference signal either itself or in combination with the described interference signal processing techniques can be used to monitor properties of the cell including the cell gap Δg during manufacture even in the presence of interfaces formed by other layers of the cell. 
     An exemplary method can include obtaining a low coherence interference signal including interference patterns indicative of the cell gap Δg prior to adding the liquid crystal material. The cell gap (or other spatial property of the cell) is determined from the interference patterns and can be compared to a specified value. Manufacturing conditions, e.g., a pressure or vacuum applied to the glass plates can be changed to modify the cell gap Δg if a difference between the specified value and the determined cell gap exceeds tolerances. This process can be repeated until achieving the desired cell gap. Liquid crystal material is then introduced into the cell. The amount of liquid crystal medium to be added can be determined from the measured spatial property of the cell. This can avoid over- or underfilling the cell. The filling process can also be monitored by observing interference signals from the surfaces of the exposed layers on the glass plates. Once the cell has been filed, additional low coherence interference patterns are obtained to monitor the cell gap Δg (or other spatial property). Again, the manufacturing conditions can be changed so that the cell gap is maintained or brought within tolerances. 
     In certain LCDs, the alignment layers include protruding structures that provide desired alignment characteristics to the liquid crystal material. For example, some LCDs have more than one alignment domain for each pixel of the display where protruding alignment structures provide the different align domains. Low coherence interferometry can be used to measure various properties of the protrusions, such as, for example, their shape, line width, height, and/or overlay error with respect to underlying features of the LCD panel. Where the protrusions are determined to be unsatisfactory, they can be repaired or removed and rebuilt as necessary. 
     In general, low coherence interferometry systems can be set up to monitor various stages of LCD panel production as desired. In some embodiments, inspection stations including an interferometry system can be set up in the manufacturing line itself. For example, monitoring stations can be installed in the clean manufacturing environment where the photolithography steps are performed. Delivery of the glass plates to and from the inspection stations can be entirely automated, being performed robotically. Alternatively, or additionally, inspection stations can be established removed from the manufacturing line. For example, where only a sampling of the displays are to be tested, the samples can be retrieved from the manufacturing line and taken offline for testing. 
     Referring to  FIG. 17C , an exemplary inspection station  4000  includes a table  4030 , which includes a gantry  4020  on which an interferometric sensor  4010  (e.g., an interferometric microscope, such as disclosed previously) is mounted. Table  4030  (which can include vibration isolation bearings) supports a LCD panel  4001  (or glass plate) and positions panel  4001  with respect to sensor  4010 . Sensor  4010  is mounted to gantry  4020  via a rail that allows the sensor to move back and forth in the direction of arrow  4012 . Gantry  4020  is mounted on table  4030  on rails that allows the gantry to move back and forth in the direction of arrow  4014 . In this way, inspection station  4000  can move sensor  4010  to inspect any location on display panel  4001 . 
     Station  4000  also includes control electronics  4050  which controls the positioning system for sensor  4010  and acquires the signals from sensor  4010  that include information about panel  4001 . In this way, control electronics  4050  can coordinate sensor positioning with data acquisition. 
     Laser Scribing and Cutting 
     Lasers can be used to scribe objects in preparation for separating different, concurrently manufactured structures, e.g., microelectronics structures. The quality of separation is related to the scribing conditions, e.g., laser focus size, laser power, translation rate of the object, and scribe depth. Because the density of features of the structure may be large, the scribe lines may be adjacent thin film or layers of the structures. Interfaces associated with the thin film or layers may create interference patterns that appear when interferometry is used to determine the scribe depth. The methods and systems described herein can be used to determine the scribe depth even in the presence of such adjacent films or layers. 
     An exemplary method can include scribing one or more electronic structures and separating the structures along the scribe lines. Before and/or after separation, low coherence interference signals can be used to determine the depth of scribe. Other scribing conditions are known, e.g., laser spot size, laser power, translation rate. The scribe depth can be determined from the interference signals. The quality of separation as a function of the scribing conditions, including the scribe depth, can be determined by evaluating the separated structures. Based on such determinations, the scribing conditions necessary to achieve a desired separation quality can be determined. During continued manufacturing, low coherence interference signals can be obtained from scribed regions to monitor the process. Scribing conditions can be changed to maintain or bring the scribe properties within tolerances. 
     A number of embodiments of the invention have been described. Other embodiments are in the claims.