fig2 is a block diagram of a system 16 for performing scatterometry measurements on a sample 8 having a periodic structure . for the purposes of this disclosure , the periodic structure means any repeating feature , where the feature size is about the same or smaller than the beam of light probing the sample such that at least some of the light is scattered rather than specularly reflected . the system 16 includes a light source 20 . as noted above , scatterometry measurements are often made using a broad band light source generating a probe beam 22 having a plurality of wavelengths . as described in u . s . pat . no . 6 , 429 , 943 , cited above , the light can also be from a laser . in such a case , measurements would be taken as a function of angle of incidence , preferably without moving the sample . probe beam is directed to the sample . typically a lens ( not shown ) is used to focus the probe beam to a small spot on the sample . the reflected probe beam is captured and measured by detector 26 . the measured intensity of the probe beam will be affected by the amount of light scattered by the periodic structure . more specifically , the proportion of light diffracted into higher orders varies as a function of wavelength and angle of incidence such the amount of light redirected out of the path to the detector also varies thereby permitting the scattering effects to be observed . the configuration of the detector will be based on the type of measurement being made . for example , a single photodetector can be used to measure spectroscopic reflected intensity as long as a tunable wavelength selective filter ( monochrometer ) is located in the path of the probe beam . given the desire to minimize measurement time , a spectrometer is typically used which includes a wavelength dispersive element ( grating or prism ) and an array detector to measure multiple wavelengths simultaneously . an array detector can also be used to measure multiple angles of incidence simultaneously . if spectroscopic ellipsometry measurements are to be made , the detection system will include some combination of polarizers , analyzers and compensators . the various measurement modalities discussed above are well known in the art and will not be discussed herein . it should be noted that commercial devices are available that can make multiple measurements . examples of such devices are described in u . s . pat . no . 5 , 608 , 526 and pct application wo 99 / 02970 , both of which are incorporated herein by reference . the signals generated by the detector are supplied to a processor 30 . the processor 30 need not be physically located near the detector . it is only required that the measurements from the detector be supplied to the processor . preferably , there is an electrical connection between the detector and the processor , either directly or through a network . as is well known to those skilled in the art , the processor will also be supplied with other signals from the system 10 to permit normalization of the signals . for example , a detector ( not shown ) would be provided to measure the output power of the light source so that variations in the output power of the light source could be taken into account during the calculations . in the preferred embodiment , and as shown in fig3 , the architecture of the processor 30 consists of a plurality of microprocessor units linked by an ethernet connection . the operating software is arranged to set one of the processors as a master 32 and the remainder of the processors as slaves 34 . the master handles the higher level functions and distributes the tasks and retrieves the results from the slaves . one such high performance clustering system is available commercially from linux networx , inc ., of bluffdale , utah under the trade name evolocity ™ a registered trademark of linux networx , inc . in the system used to evaluate the subject invention , an eight processor configuration was selected with each processor operating at 1 . 3 gigahertz . when properly combined , the system will operate at a speed equivalent to about 10 gigahertz . the approach for distributing the processing tasks will be discussed below . as noted above , a wide variety of methods have been developed for evaluating characteristics of a periodic structure based on measured data . the approach described herein falls within the general class of procedures where an initial model is created and calculations are performed to determine the expected response of that sample to interaction with light . the model is then iteratively modified until the results of the calculation are close to the actual measured ( and normalized ) data . the subject approach can be contrasted with the earlier approaches which required the fabrication of many references samples , each of which would be measured , with the results stored for later comparison to the test sample . this subject approach is also different from the library approach where large numbers of configurations and their associated optical responses are created and stored for later comparison . the subject approach recognizes that the shape of the structure can be represented as a plurality of stacked layers . however , rather than evaluate the sample based on a preordained , fixed number of layers , the algorithm is designed to progressively add layers , while seeking a best fit at each model level . this progression allows a theoretical structure which is relatively close the actual structure to be efficiently determined . in this approach , only the minimum number of layers which is actually necessary to achieve the desired level of fitness must be analyzed . the subject method will be described with respect to the flow chart of fig4 and the drawings of fig5 to 7 . in the initial step 102 , a rectangular model 50 ( fig5 ) is created . typically , the model is created using seed values based on the expected characteristics of the sample . for example , the model will include information such as the index of refraction and extinction coefficient of the material . it is possible that this information can be obtained by measuring a region of the wafer which is not patterned . the model will also have a value for the height h 1 and width w 1 . the processor will then calculate the expected intensity that would be measured from a sample having a periodic structure with these initial characteristics ( step 104 ). for the purposes of this example , it will be assumed that the measured data is obtained from a spectroscopic reflectometer . accordingly , for each of the measured wavelengths , the processor will determine , using maxwell &# 39 ; s equations and a rigorous coupled wave theory , the expected normalized measured intensity of light reflected from the theoretical model . in a typical example , a measurement might include 50 to 100 wavelengths . once this calculation has been performed for each of the wavelengths , the results are compared to the normalized measurements obtained from the sample ( step 106 ). this comparison can be done with a conventional least squares fitting algorithm , for example levenberg - marquardt . the result of this comparison will be used to modify the parameters of the model , in this case the starting height and width ( step 108 ). the processor then calculates the expected intensity of reflected light at each wavelength from a structure with the modified attributes ( step 110 ). these new values are compared to measured values and , if necessary , the model is once again modified . in practice , the iterative process usually needs to be repeated some 4 to 8 times before a suitable fit is achieved . the operator can define the desired level of fitness , i . e ., when the differences between the model and the actual measurements as represented by the result of the comparison drops below a predetermined value . the best fit result will be a rectangle which most closely approximates the actual periodic structure . the next step is to modify the model by increasing its complexity ( step 114 ). more specifically , the shape or grating profile is changed from a simple rectangle to a trapezoid ( see 54 in fig6 ) having a top width w 1 and an independent bottom width w 2 . in addition , the structure will be divided in a plurality of rectangular layers , in this case preferably three . rather than using a polynomial expansion , the modification of this model is done using spline coefficients . the starting point for the modification is the best fit rectangle determined in the previous step . the grating profile is defined using a class of spline algorithms including the well known cubic spline , bezier - curves , b - spline and its more generalized form of non - uniform rational b - splines ( the nurbs book , by les piegl and wayne tiller , springer , 1995 ). the benefits of such an approach are that 1 ) the curves are controlled by a set of control points and 2 ) the shape described by splines is more flexible than that described by polynomial expansions . a b - spline curve is described as where p j are the control points which can be scalars or vectors depending the desired flexibility . to minimize the number of fitting parameters for cubic splines , a user has the flexibility to choose different ways to allocate the spline points , in the vertical direction . assuming that the grating height is scaled between 0 and 1 and assuming that the points t j are evenly distributed between 0 and 1 , we then use a sigmoid function of the following form to transform t j to u j : as ( david elliot , j . australian math . soc . b40 ( e ), p e77 – e137 , 1998 ): u = f n ( t )/( f n ( t )+ f n ( 1 − t )) f ( t )= t ( 1 + c ( 1 − t )), the effect of this transformation is that the spline points are more densely allocated at the two ends when n & gt ; 1 . this is very close to how the nodes are distributed in gaussian integration . it also corresponds to the more common periodic profiles which have more curvature near the top and bottom of the structure . another aspect of our algorithm involves how the system is divided into slices or layers ( discretized ). the simplest approach is to divide the grating evenly in each material . however , similar to how spline points are allocated , we can also discretize the system similar to the gaussian integration which is again similar to the sigmoid function described earlier . significantly , we also use the idea of adaptively discretizing the system according to the curvature of the curve . in this approach , we allow the assignment of layers to be actively varied , along with the other characteristics of the model , during the fitting process . in this process , we define d =∫ du | dw / du |, then each segment ( between spline points ) should have d / n slices , where w is the width as a function of height u , and n is the total number of slices in the model . once the starting parameters are defined , the processor will calculate the expected intensity for this new structure at each of the measured wavelengths ( step 116 ). the results are compared to the measured values ( step 118 ) and if the fit is not acceptable , the model is modified ( 120 ). in accordance with the subject method , the algorithm is free to modify the widths and layer thicknesses regardless of the values obtained in previous steps . the algorithm is also designed to adjust the layer thickness such that the greater number of layers will be used to define regions where the width is changing the fastest . this procedure is repeated until a trapezoid which most closely approximates the actual periodic structure is determined . once the best fit trapezoid is defined , the complexity of the model is again increased to include one or more widths and layers ( step 130 , and fig7 ). in the preferred embodiment , the model is modified by adding a single extra width . the number of layers is also increased . preferably , the number of layers at each iteration is at least 2y − 1 ( where y is the number of widths ) but no greater than 2y + 1 . the processor will then calculate the expected intensity for this new structure at each of the measured wavelengths ( step 134 ). the results are compared to the measured values ( step 136 ) and if the fit is not acceptable , the model is modified ( 140 ). this procedure will repeat in an iterative fashion until the model with the selected number of widths and layers best fits the data . if that structure meets the overall predetermined level of fitness , the process is complete and the model will suitably match the actual periodic structure ( step 142 ). if not , the processor will loop back ( along path 144 ) to create a new model with additional widths and layers ( see 56 in fig7 ). in initial experiments , the average number of widths and layers needed to adequately characterize a structure was about 7 to 9 widths and 13 to 17 layers . with these additional widths and layers , structures with various wall profiles can be analyzed . as noted above , one feature of the subject method is its ability to permit the thickness and density of the theoretical layers to vary during each iteration . it should be noted , however , that some periodic structures under investigation will include actual physical layers . if so , these physical layers can be used as boundaries to further define or constrain the model . the calculations required to determine the response of a sample to incident radiation are complex . as the number of widths and layers increases , the time required to make the calculations increases dramatically . accordingly , in a second aspect of the subject invention , the processing tasks are distributed to a parallel processor system . in the preferred embodiment , one of the eight processors ( fig3 ) is configured as the master processor 32 and the remaining seven processors are slaves 34 . the master processor controls the overall analysis and distributes certain of the functions to the slave processors . as noted above , the most time consuming portion of the calculation is the determination of the optical response of the model to each of the different measured wavelengths or angles of incidence . the comparison of these theoretical results with the measured signals and the modification of the model can , by comparison , be handled relatively quickly . therefore , in the preferred embodiment of the subject invention , the master processor is responsible for distributing the calculations of theoretical data to the slave processors ( such calculations being shown as steps 104 , 110 , 116 and 134 in fig4 ). in the preferred embodiment , the master processor would also participate in these calculations . a maximum reduction in computational time can be achieved if the workload is evenly distributed . the preferred approach to achieve uniformity is to distribute the wavelength or angle of incidence information serially across the processors . thus , the first slave processor ( in an eight processor system ) would be responsible for calculating the first ( shortest ) wavelength as well as the ninth , seventeenth , etc . ( n + 8 ). the second slave processor would be responsible for the second ( next shortest ) wavelength as well as the tenth , eighteenth , etc . this approach can be used for both spectrophotometry and spectroscopic ellipsometry . a similar approach can be used with multiple angle of incidence measurements wherein the first , eighth , seventeenth measured angle would be calculated by the first slave processor , etc . once each of the calculations is made at each of the wavelengths ( or angles ), the master processor will compare the results at each of the wavelengths to the normalized measurements at the corresponding wavelengths . the difference will define the level of fitness of the result and will be used to determine if another iteration is required . the calculations necessary for each iteration of the model are again distributed to the slave processors in the manner discussed above . while the subject invention has been described with reference to a preferred embodiment , various changes and modifications could be made therein , by one skilled in the art , without varying from the scope and spirit of the subject invention as defined by the appended claims . for example , it should be apparent that the inventions described herein are not specifically dependent upon the particular scatterometry approach used to collect the data . data can be obtained from spectroscopic reflectometers or spectroscopic ellipsometers . it should be noted that spectroscopic reflectometers can obtain data from probe beams directed either at normal incidence or off - axis to the sample . similarly , spectroscopic ellipsometers can obtain data from probe beams directed either at normal incidence or off - axis to the sample . data can also be obtained from multiple angle of incidence devices . as noted in u . s . pat . no . 6 , 429 , 943 , applicant has developed a variety of simultaneous multiple angle of incidence devices that would be suitable . detailed descriptions of assignee &# 39 ; s simultaneous multiple angle of incidence devices can be found in the following u . s . pat . nos . 4 , 999 , 014 ; 5 , 042 , 951 ; 5 , 181 , 080 ; 5 , 412 , 473 and 5 , 596 , 411 , all incorporated herein by reference . it should also be understood , that data from two or more of the devices can be combined to reduce ambiguities in the analysis . such additional data can be combined in the regression analysis discussed above . see also , u . s . pat . no . 5 , 889 , 593 incorporated by reference . in this patent , a proposal is made to include an optical imaging array for breaking up the coherent light bundles to create a larger spot to cover more of the periodic structure .