CN1784588A - Characterizing and profiling complex surface structures using scanning interferometry - Google Patents

Abstract

A method including comparing information derivable from a scanning interferometry signal for a first surface location of a test object to information corresponding to multiple models of the test objects, wherein the multiple models are parametrized by a series of characteristics for the test object. The information corresponding to the multiple models may include information about at least one amplitude component of a transform (eg. A Fourier transform) of a scanning interferometry signal corresponding to each of the models of the test object. In a second aspect the models correspond to a fixed surface height and they are parametrized by a series of characteristics different from the fixed surface height. In a third aspect the comparing comprises accounting for systematic contributions to the interferometry signal.

Description

The profile that uses scanning interferometer to measure to form complex surface structures and to its sign

Cross reference to related application

This application requires the right of priority of following U.S. Provisional Patent Application under 35U.S.C 119 (e): on March 6th, 2003 submit to, title for " PROFILING COMPLEX SURFACESTRUCTURES USING HEIGHT SCANNING INTERFEROMETRY (using the profile of height scanning interferometer measurement formation complex surface structures) ", sequence number is 60/452,615 U.S. Patent application; In on March 6th, 2003 submit to, title for " PROFILING COMPLEX SURFACESTRUCTURES USING SIGNALS FROM HEIGHT SCANNINGINTERFEROMETRY (using the profile that forms complex surface structures from the signal of height scanning interferometer measurement) ", sequence number is 60/452,465 U.S. Patent application; And on January 26th, 2004 submit to, title for " SURFACE PROFILING USING AN INTERFERENCE PATTERNMATCHING TEMPLATE (using the surface profile of interference figure matching template) ", sequence number is 60/539,437 U.S. Patent application.At this, by reference with its whole merging.

Technical field

The present invention relates to use scanning interferometer measurement measurement have such as the discrete topology of film (a plurality of), dissimilar material or by the optical resolution of interference microscope below differentiating surfac topography (topography) and/or other characteristic of the object of the complex surface structures of the discrete topology of (underresolve).The sign (characterization) of such measurement and flat pannel display assembly, semiconductor wafer measures and (insitu) film is relevant with dissimilar material analysis on the spot.

Background technology

Usually, use interfere measurement technique to come the profile (profile) on the surface of measuring object.For this reason, interference measuring instrument will be from the measurement wavefront (wavefront) of surface reflection interested and the reference wavefront combination of reflecting from reference surface, to produce interferogram (interferogram).Spatial variations between striped in the interferogram (fringe) indication surface interested and the reference surface.

Scanning interferometer meter (interferometer) similar or greater than the scope of the coherent length before the interference wave on light path (optical path) length difference (OPD) between scanning this interferometric reference and measurement leg (leg), to produce the scanning interferometer measuring-signal for being used for measuring each camera (camera) pixel of interferogram.For example, by using white light source (it is called as scanning white light interferometry (SWLI)), can produce limited coherent length.Exemplary scanning white light interferometric (SWLI) signal is for being positioned near some stripeds of zero optical path difference (OPD) position.Typically, this signal is characterised in that: by the sinusoidal carrier modulation (" striped ") of bell striped contrast envelope.The traditional thought of measuring according to SWLI is: utilize the location of striped to come the surface measurements profile.

The SWLI treatment technology comprises two primary branches.First approach is to survey the peak value or the center of envelope, supposes that the zero optical path difference (OPD) of this position and two beam interference meters (wherein, a light beam reflects from object surface) is corresponding.Second approach is to transform the signal in the frequency domain, and calculates the change rate of phase place by wavelength, supposes that substantially linear slope and object's position are directly proportional.For example, referring to No. the 5th, 398,113, the United States Patent (USP) of authorizing Peterde Groot.The approach of back is called as frequency-domain analysis (FDA).

Unfortunately, such supposition may be false when being applied to have the tested object of film, and this is because the reflection of the film/substrate interface on top surface and basis.Recently, in the United States Patent (USP) 6,545,763 of authorizing S.W.Kim and G.H.Kim, disclose a kind of method and handled such structure.This method is used in the SWLI signal frequency-domain phase outline and the estimation frequency domain phase outline match mutually (fit) that is used for various film thicknesses and surface elevation of membrane structure.Optimize simultaneously and determined correct film thickness and surface elevation.

Summary of the invention

The inventor recognizes: have abundant information when the scanning interferometer signal, ignored many in these information in conventional process.Although for example the complex surface structures of film may destroy the traditional treatment technology based on the slope of the position of discerning the peak value in the striped contrast envelope or calculating frequency domain phase outline, new treatment technology described herein can extract the information of surface elevation information and/or related complicated surface structure.

For example, although do not suppose that surface elevation information is directly related with the peak value in the striped contrast envelope, but the change in the some embodiments of the present invention supposition surface elevation has been changed the scanning interferometer measuring-signal with respect to the reference scan position, but has kept the shape of scanning interferometer measuring-signal.Thus, the shape of scanning interferometer measuring-signal is particularly useful when characterizing complex surface structures, and this is irrelevant because of it and surface elevation.Similarly, in frequency domain, some embodiment supposition: the linear term in the frequency domain phase outline is introduced in the change in the surface elevation, even frequency domain profile self may not be linear.Yet the change in the surface elevation makes the frequency domain amplitude profile constant.Therefore, the frequency domain amplitude profile is particularly useful when characterizing complex surface structures.

After characterizing complex surface structures, can determine surface elevation effectively.For example, the scanning interferometer signal and have and the model signals of the corresponding shape of complex surface structures between simple crosscorrelation can with the corresponding scanning coordinate of surface elevation on produce peak value.Similarly, in frequency domain, can from the frequency domain phase outline, deduct the phase place contribution that produces by complex surface structures, and, can use traditional FDA to analyze and extract surface elevation.

The example of complex surface structures comprises: simple film (in this case, for example, interested variable element can be the refractive index of film thickness, film, refractive index or its certain combination of substrate); Multilayer film; The sharpened edge and the surface characteristics of the interference effect that diffraction or generation are complicated; Unresolved (unresolved) surfaceness; Unresolved surface characteristics, for example, the wavelet length and width degree groove on other smooth surface; Dissimilar material (for example, the surface can comprise the combination of film and solid metal, and in this case, the storehouse can comprise described two surface structure types, and automatically discerns film or solid metal by mating corresponding frequency domain spectra); Produce the surface structure of optical activity (as fluorescence); Optical splitter (spectroscopic) attribute on surface is as color and the reflectivity that depends on wavelength; Depend on the surface properties of polarization; And the deflection on surface, vibration or motion or deformable surface feature, it causes the disturbance (perturbation) of interference signal.

In certain embodiments, with the limited coherent length of light that generates the scanning interferometer measuring-signal based on white light source, perhaps, more generally, wideband light source.In other embodiments, light source can be monochromatic, and, can use high-NA (NA) and produce limited coherent length, so that light is directed to tested object and/or receives light from tested object.High NA makes light engaged test surface on the scope of angle, and, when scanning OPD, in the signal that is write down, generate different spatial frequency components.In another embodiment, can produce limited relevant from the combination of described two effects.

The source of limited coherent length also is the physical basis that has information in the scanning interferometer measuring-signal.Especially, the scanning interferometer measuring-signal comprises the information of relevant complex surface structures, and this is because it is by by a lot of different wave lengths and/or a lot of different angles and the light on engaged test surface produces.

Here in the treatment technology of Miao Shuing, the information (comprising scanning interferometer measuring-signal self) that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations.For example, tested object can be modeled as film, and described series of characteristics can be a series of values that are used for film thickness.Although the information that is compared for example can comprise the information of relevant frequency domain phase outline, it also can comprise the information of shape of relevant scanning interferometer measurement data and/or the information of relevant frequency domain amplitude profile.In addition, for making the surface elevation that relatively is directed to complex surface structures rather than first surface position, a plurality of models can be all corresponding to the fixed surface height of the tested object of first surface position.Described comparison self can indicate from the information of actual scanning interferometry signal with from the quality function of the similarity between the information of each model based on calculating.For example, the quality function can indicate information and the match between the parameterized function by series of characteristics that can derive from the scanning interferometer measurement data.

In addition, in certain embodiments, series of characteristics is corresponding to the characteristic of the tested object at the second place place that is different from primary importance, and it comprises the diffractive surface structure of for example interface signals of first surface position being made contributions.Thus, although we often are called complex surface structures certain things outside the surface elevation with the corresponding first surface of scanning interferometer measuring-signal position, complex surface structures can corresponding to: and with first surface position surface elevation feature separately corresponding to the scanning interferometer measuring-signal.

Method described herein and technology can be used for internal process (in-process) tolerance (metrology) of semi-conductor chip and measure.For example, the measurement of scanning interferometer surveying can be used for: the noncontact surfac topography during the dielectric on the wafer was once carried out chemically mechanical polishing (CMP) is measured semiconductor wafer.CMP is used for creating the smooth surface that is used for dielectric layer, and it is suitable for accurate projection printing art.Based on the result of interferometry topographical method, scalable is used for the process condition (for example, stuffing pressure, brilliant polish become to grade) of CMP, so that surperficial heterogeneity remains in the acceptable limit.

Now, we sum up various aspects of the present invention and feature.

Usually, in one aspect, the invention is characterized in a kind of method, it comprises: the information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations.

Embodiments of the invention also can comprise any following feature.

This method also can comprise: the precise characteristics that is identified for tested object based on described comparison.

This method also can comprise: determine the apparent surface's height for the first surface position based on described comparison.In addition, the determining and to comprise of apparent surface height: determine which model corresponding to the precise characteristics in the characteristic of tested object based on described comparison, and use is calculated the apparent surface highly with the corresponding model of precise characteristics.

For example, the use with the corresponding model of precise characteristics can comprise: compensation is from the data of scanning interferometer measuring-signal, with the contribution that reduces to produce from this precise characteristics.Compensation to data can comprise: eliminate the phase place contribution that produces from this precise characteristics from the phase component for the conversion of the scanning interferometer measuring-signal of tested object, and, also can comprise with the use of the corresponding model of precise characteristics: after eliminating the phase place contribution that produces from this precise characteristics, calculate apparent surface's height according to the phase component of this conversion.

In another example, use is calculated apparent surface's height with the corresponding model of precise characteristics and can be comprised: determine to be used for and will be used for the information of tested object and the position of the peak value of the related function that is used for comparing with the information of the corresponding model of precise characteristics.

This method also can comprise: the information that can derive from the scanning interferometer measuring-signal that is used for the additional surfaces position and compare with a plurality of model information corresponding.And this method also can comprise: the surface elevation profile that is identified for tested object based on described comparison.

Described comparison can comprise: calculate indicate the information that can derive from the scanning interferometer measuring-signal and and each model information corresponding between one or more quality functions of similarity.

Described comparison can comprise: information that can derive from the scanning interferometer measuring-signal and expression match mutually corresponding to the information of model.

Can comprise with a plurality of model information corresponding: with the information of at least one amplitude components of the conversion (for example, Fourier transform) of corresponding, the relevant scanning interferometer measuring-signal of the model of each tested object.Equally, can comprise from the information that the scanning interferometer measuring-signal is derived: the information of at least one amplitude components of the conversion of relevant scanning interferometer measuring-signal for tested object.

Described comparison can comprise: the relative intensity of at least one amplitude components of tested object is compared with the relative intensity of at least one amplitude components of each model.

Can be the function of the coordinate that is used for conversion with a plurality of model information corresponding.For example, can comprise with a plurality of model information corresponding: the amplitude profile that is used for the conversion of each model.In addition, described comparison can comprise: will for the amplitude profile of the conversion of the scanning interferometer measuring-signal of tested object with for each amplitude profile phase of model relatively.

Describedly more also can comprise: will for the information in the phase outline of the conversion of the scanning interferometer measuring-signal of tested object with compare for the information in the phase outline of the conversion of each model.For example, the information in the phase outline can comprise: relevant phase outline is with respect to the nonlinear information of coordinate transforming and/or the information of relevant phase place gap width.

Can be a number from the information that the scanning interferometer measuring-signal is derived and just is being compared.Replacedly, can be function from the information that the scanning interferometer measuring-signal is derived and just is being compared.For example, it can be the function of scanning position or the function of spatial frequency.

Can be spatial frequency domain from the scanning interferometer measuring-signal conversion (for example, Fourier transform) that will be used for tested object, and derive the information that is used for tested object.The information that is used for tested object can comprise the information of the phase outline of the amplitude profile of relevant conversion and/or conversion.

The information that is used for tested object can be relevant with the shape at the scanning interferometer measuring-signal that is used for tested object at primary importance place.For example, being used for the information of tested object can be relevant with the striped contrast amplitude of the shape of scanning interferometer measuring-signal.It also can be relevant with the relative spacing between the zero crossing in the shape of scanning interferometer measuring-signal.Also it can be expressed as the function of scanning position, wherein, derive this function from the shape of scanning interferometer measuring-signal.

Described comparison can comprise: calculate and to be used for the information of tested object and to be used for related function (for example, compound correlative function) between the information of each model.Describedly more also can comprise: determine the one or more peak values in each related function.So, this method also can comprise: based on the precise characteristics of determining tested object with the parametrization of the corresponding model of peak-peak.Replacedly, maybe can add ground, this method also can comprise: apparent surface's height of determining the tested object of first surface position based on the coordinate of at least one peak value in the related function.

A plurality of models can be corresponding to the fixed surface height of the tested object at primary importance place.

Series of characteristics can comprise a series of values of at least one physical parameter of tested object.For example, tested object can comprise the thin layer with thickness, and physical parameter can be the film thickness at primary importance place.

Series of characteristics can comprise the series of characteristics of tested object in the second surface position that is different from the first surface position.For example, tested object can comprise: the structure in the second surface position of making contributions with optical diffraction and to the scanning interferometer measuring-signal that is used for the first surface position.In an example, the series of characteristics of second surface position can comprise: in the arrangement (permutation) of the amplitude of the step height at second place place and the location of the second place.In another example, the series of characteristics of second surface position can comprise: be used for the arrangement of depth of modulation of grating and the deviation post of grating, wherein, grating extends on the second place.

Series of characteristics can be a series of surfacings that are used for tested object.

Series of characteristics can be a series of superficial layer configurations that are used for tested object.

Can produce the scanning interferometer measuring-signal by the scanning interferometer measuring system, and described comparison can comprise: consider to produce by the scanning interferometer measuring system, to system's contribution of scanning interferometer measuring-signal.For example, system contribution can comprise: relevant information for the deviation from the phase change of the reflection of the assembly of scanning interferometer measuring system.In addition, this method also can comprise: will compare for the information additional surface position, that can derive from the scanning interferometer measuring-signal with a plurality of model information corresponding, in this case, can be for a plurality of in the surface location and the contribution of resolution system.This method also can comprise: use another tested object with known attribute, and the contribution of the system of calibration scan interferometer measuration system.

Can be by to sending from tested object with on the detecting device and the reference light test light of interfering is carried out imaging and the optical path length from the common source to the detecting device that changes between the interference portion of test and reference light is poor, produce the scanning interferometer measuring-signal, wherein, from common source (for example, the source that extend in the space) derive test and reference light, and, wherein, the scanning interferometer measuring-signal corresponding to: when the optical path length difference changes, by the interference strength of detectors measure.

Test and reference light can have greater than test and about 5% bands of a spectrum of the centre frequency of reference light wide.

Common source can have the spectrum coherent length, and the optical path length difference changes on the scope greater than the spectrum coherent length, to produce the scanning interferometer measuring-signal.

Be used for being directed to test light on the tested object and with its optical device definable that is imaged onto detecting device greater than numerical aperture about 0.8, that be used for test light.

This method also can comprise: produce the scanning interferometer measuring-signal.

In one aspect of the method, the invention is characterized in a kind of equipment, it comprises: computer-readable medium, it has information that the processor that makes in the computing machine can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and the program of comparing with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations.

This equipment can comprise above-mentioned combining with this method and the arbitrary characteristics described.

In one aspect of the method, the invention is characterized in a kind of equipment, it comprises: the scanning interferometer measuring system, and it is configured to produce the scanning interferometer measuring-signal; And electronic processors, it is coupled to the scanning interferometer measuring system, to receive the scanning interferometer measuring-signal, and be programmed to the information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations.

This equipment can comprise above-mentioned combining with this method and the arbitrary characteristics described.

Usually, in one aspect of the method, the invention is characterized in a kind of method, it comprises: tested object is carried out chemically mechanical polishing; Collection is used for the scanning interferometer measurement data of test object surface configuration; And based on the information that can derive from the scanning interferometer measurement data, and regulate the process condition that is used for chemically mechanical polishing.For example, process condition can be stuffing pressure and/or brilliant polish composition.In a preferred embodiment, based on the information that derives from the scanning interferometer measurement data and the adjustment process condition can comprise: can be for the position of first surface at least of tested object and the information that derives from the scanning interferometer measuring-signal and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations.Analysis to the scanning interferometer measuring-signal also can comprise the arbitrary characteristics of describing for the method for at first mentioning.

Unless otherwise defined, all technology used herein and scientific terminology have with the present invention under the field the technician the common same meaning of understanding.Under the afoul situation of other reference documents that merges, will be as the criterion by reference with publication, patented claim, patent and at this with this instructions that comprises qualification.

From following detailed, it is clear that further feature of the present invention, purpose and advantage will become.

Description of drawings

Fig. 1 is the process flow diagram of interferometric method.

Fig. 2 is the process flow diagram of variation that the interferometric method of Fig. 1 is shown.

Fig. 3 is the synoptic diagram of Linnik type scanning interferometer meter;

Fig. 4 is the synoptic diagram of Mirau type scanning interferometer meter;

Fig. 5 illustrates the figure that test sample book sees through the illumination of objective lens.

Fig. 6 shows the theoretic Fourier amplitude frequency spectrum that is used for the scanning interferometer measurement data under two kinds of restrictions.

Fig. 7 shows two kinds of surface types that have and do not have film.

Fig. 8 shows and is used for quality function (merit function) search procedure that film thickness is the emulation of the silicon dioxide film on 0 the silicon substrate.

Fig. 8 shows and is used for the quality function search procedure that film thickness is the emulation of the silicon dioxide film on 0 the silicon substrate.

Fig. 9 shows the quality function search procedure of the emulation that is used for the silicon dioxide film on the silicon substrate that film thickness is 50nm.

Figure 10 shows the quality function search procedure of the emulation that is used for the silicon dioxide film on the silicon substrate that film thickness is 100nm.

Figure 11 shows the quality function search procedure of the emulation that is used for the silicon dioxide film on the silicon substrate that film thickness is 300nm.

Figure 12 shows the quality function search procedure of the emulation that is used for the silicon dioxide film on the silicon substrate that film thickness is 600nm.

Figure 13 shows the quality function search procedure of the emulation that is used for the silicon dioxide film on the silicon substrate that film thickness is 1200nm.

Figure 14 shows the definite surface and the substrate profile of emulation of the mode that increases progressively with every pixel 10nm for the film thickness silicon dioxide from the silicon thin film of 0 to 1500nm even variation, and wherein top surface is in 0 always.

Figure 15 shows and is emulation definite surface and the substrate profile identical with the emulation among Figure 14 except having added random noise (from 2 rms of average 128 intensity positions).

Figure 16 show for the actual peak with 120nm to the grating (grating) of every millimeter 2400 line of paddy depth of modulation, use traditional FDA to analyze that (Figure 16 a) and the library searching method of describing (Figure 16 b) and the surface elevation profile determined here.

The distortion that causes when Figure 17 shows near scanning and step height the interference signal of the corresponding pixel of each surface location, by the step height below resolution.

Figure 18 shows that (Figure 18 a) and the nonlinear distortion in the frequency domain phase spectrum of the corresponding pixel of surface location of the right (Figure 18 b) with the left side of the step height below differentiating of Figure 17.

Figure 19 shows for differentiating following step height, using traditional FDA to analyze that (Figure 19 a) and the library searching method of describing (Fig. 1 b) and definite surface elevation profile here.

Figure 20 shows the actual scanning interferometry signal of the basic silicon substrate of no film.

Figure 21 and 22 shows the interference die plate pattern of membrane structure that is used for the bare silicon substrate and has 1 micron silicon dioxide on silicon respectively.

Figure 23 and 24 shows the quality function as the function of the scanning position of the stencil function in Figure 21 and 22 respectively.

In different accompanying drawings, identical Reference numeral is represented identical element.

Embodiment

Fig. 1 shows the process flow diagram of describing one embodiment of the present of invention prevailingly, wherein, carries out the analysis of scanning interferometer measurement data in spatial frequency domain.

With reference to Fig. 1,, use the optical path difference (OPD) between interferometer machinery or electric light formula ground scan reference and the measuring route, wherein measuring route point at objects surface for measuring data from test object surface.OPD when the scanning beginning is the function of the local height of object surface.For with the corresponding a plurality of camera pixel in the different surfaces position of object surface in each and carry out OPD scanning during, computer recording interference strength signal.Next, storing as after for each the interference strength signal of function of OPD scanning position in the different surfaces position, computing machine is carried out conversion (for example, Fourier transform), to generate the signal frequency-domain frequency spectrum.This frequency spectrum comprises: as amplitude and the phase information at the function of the spatial frequency of the signal of scanning in the dimension.For example, the title that generally has at Peter de Groot is that the United States Patent (USP) of " Method and Apparatus for Surface Topography Measurements bySpatial-Frequency Analysis of Interferograms (being used for the method and apparatus by the surfac topography that spatial-frequency analysis the carried out measurement of interferogram) " discloses the suitable frequency-domain analysis (FDA) that is used to generate such frequency spectrum for No. 5398113, at this, by reference its content is merged.

In independent process, computing machine generates the storehouse of the theoretical prediction of the frequency domain frequency spectrum that is used for kinds of surface parameter and interferometric model.For example, these frequency spectrums can cover the scope of possible film thickness, surfacing and superficial makings.In a preferred embodiment, computing machine generates the storehouse frequency spectrum that is used for constant surface elevation (for example, highly=0).Thus, in such embodiments, the storehouse does not comprise the information of relevant surfac topography, only comprise with the type (type) of surface structure and when generating the distinguishing characteristics of frequency domain frequency spectrum the relevant information of interaction of this surface structure, optical system, illuminance (illumination) and detection system.As an alternative, can use sampling pseudomorphism (artifact) and generate the prediction storehouse according to experiment.Replace as another kind, the storehouse by Other Instruments (for example can be used, ellipsometer (ellipsometer)) any other input from the user from the known attribute of the information of the additional survey of previous object surface and relevant object surface that provides is so that reduce the number of unknown surface parameter.Any technology of these technology of the theory that is used for storehouse establishment, theoretical modeling, experimental data or is expanded by additional survey can expand by interpolation, to generate intermediate value, it both can be used as the part that create in the storehouse, also can carry out in real time during library searching.

In following step, by the library searching of surface structure parameter is provided, and experimental data is compared with the prediction storehouse.In the sample situation of the film of unknown thickness, be used for single surface type (for example, silicon dioxide (SiO on the silicon ₂On Si)) a lot of possible film thicknesses will be contained in storehouse, and wherein the top surface height equals 0 always.Another sample situation is a surfaceness, and the adjustable parameters that is used for surfaceness can be the roughness degree of depth and/or spatial frequency.Library searching causes with those characteristics that are independent of the FDA frequency spectrum of surface elevation and is complementary, described characteristic is promptly: for example, the mean value of the amplitude spectrum relevant with the total reflectivity on surface, or relevant with catoptrical scattering angle in the high NA of monochrome system, as the variation in the amplitude of the function of spatial frequency.

This analysis also can comprise characterized systematically, it comprises: for example, measurement has the one or more with reference to pseudomorphism of known surface structure and surfac topography, so that definite parameter such as systematic wavefront, chromatic dispersion (dispersion) and efficient that does not comprise in theoretical model.

In addition, this analysis can comprise overall calibration, it comprises: for example, measure one or more with reference to pseudomorphism, to determine being correlated with between the measured surface parameter, described surface parameter is promptly: for example, and the definite film thickness and as analyzing and independent definite value that is used for these parameters by library searching by for example ellipsometric measurement (ellipsometric).

Based on the comparison of experimental data with the prediction storehouse, the corresponding surface model of computer Recognition and optimum matching.Subsequently, can to the user or host computer system quantize or show graphically or transmit the surface parameter result, be used for further analysis or be used for data storage.Computing machine can use the surface parameter result to determine surface elevation information except the characteristic of discerning by library searching subsequently.In certain embodiments, for example, computing machine is composed by directly deducting corresponding notional phase from the experiment phase spectrum, and generates the compensation of phase spectrum.Subsequently, for example, the coefficient that computing machine is generated by linear fit (linear fit) by analysis is analyzed the compensation of phase as the function of spatial frequency, and determines the local surfaces height for one or more surface points.Afterwards, computing machine generates by the complete 3-D view of altitude information structure and corresponding plane of delineation coordinate, shows together with figure or numerical value as definite character of surface by library searching.

In some cases, can repeat library searching and data aggregation, with the further result that improves.Particularly, can be by creating the storehouse that become more meticulous relevant with the local surfaces type, based on the individual element or the regional and library searching that becomes more meticulous.For example, if during initial library searching have an appointment 1 micron film of table of discovery mask, so, computing machine can generate fine granulation (fine-grain) storehouse near 1 micron example value, with the search that further becomes more meticulous.

In another embodiment, the user may in this case, not carry out the step that is used for determining surface elevation only to interested by the character of surface rather than the surface elevation of prediction storehouse modeling.On the contrary, the user may be only to surface elevation rather than interested by the character of surface of prediction storehouse modeling, in this case, computing machine uses the experimental data that relatively compensates the contribution that is used for character of surface between experimental data and the prediction storehouse, so that determine surface elevation more accurately, and do not need explicitly to determine character of surface or show them.

Can be with this analytical applications in the kinds of surface problem analysis, it comprises: simple film (in this case, for example, interested variable element can be the refractive index of film thickness, film, the refractive index of substrate or their some combination); Multilayer film; The steep edge and the surface characteristics of the interference effect that diffraction or generation are complicated; Do not differentiate the surfaceness of (unresolved); Unresolved surface characteristics, for example, the wavelet length and width degree groove (groove) on the other smooth surface; Dissimilar material (for example, this surface can comprise the combination of film and solid metal, and in this case, the storehouse can comprise described two surface structure types, and discerns film or solid metal automatically by matching corresponding frequency domain frequency spectrum); Optical activity (optical activity) is as fluorescence; Spectrum (spectroscopic) attribute on surface closes the reflectivity that depends on wavelength as color; Depend on the surface properties of polarization; The deflection on surface, vibration or motion or deformable surface feature, it causes the disturbance (perturbation) of interference signal; And with the relevant data distortion of data acquisition (for example, data obtain window do not contain the interference strength data fully).

Interferometer can comprise the arbitrary characteristics in the following feature: the spectrum narrow-band light source with high-NA (NA) object lens; Spectrum width band light source; The combination of high NA object lens and spectrum width band light source; The interferometry micro objective, it comprise for example Michelson, Mirau or Linnik geometric configuration (geometry), oil/water immerses (immersion) and solid immersion type; Measurement sequence on multi-wavelength; Nonpolarized light; And the polarized light that comprises linearity, circle or structuring (structured).For example, structurized polarized light can relate to: for example, polarization mask (polarization mask), its different fragments for irradiation or imaging pupil (pupil) generates different polarization, so that show the optical effect that depends on polarization that is attributable to character of surface.Interferometer also can comprise above-mentioned overall system calibration.

When comparing gross data and experimental data, library searching can be based on the arbitrary standards in the following standard: the amplitude in the frequency spectrum and/or the product of phase data or poor between them, and it comprises the product of for example average amplitude and average phase, average amplitude self and average phase self or poor between them; The slope of amplitude spectrum, width and/or height; Interference contrast (interference contrast); Be in the data in the frequency spectrum of DC or 0 spatial frequency; Non-linear or the shape of amplitude spectrum; 0 frequency intercept (intercept) of phase place; Non-linear or the shape of phase spectrum; And the combination in any of these standards.Note, as used herein, use amplitude (magnitude) and amplitude (amplitude) interchangeably.

Fig. 2 shows the process flow diagram of describing another embodiment that is used for the analysis scan interferometry data prevailingly.Except experimental data and prediction the information in the scanning coordinate territory of relatively being based between the storehouse, this analysis classes is similar to the described analysis to Fig. 1.Can utilize the quasi-periodic carrier oscillation of the which amplitude modulation of the envelope function by being directed to scanning coordinate, and characterize test signal.When relatively more theoretical and experimental data, library searching can be based on the arbitrary standards in the following standard: average signal strength; The shape of signal envelope, it for example comprises the deviation with certain ideal or reference figuration (as Gaussian); Carrier signal is with respect to the phase place of envelope function; The relative spacing of zero crossing and/or signal maximum and minimum value; Be used for the value of maximal value and minimum value and their order; After having carried out for best relative scanning position adjusting, the relevant peak value between storehouse and the measured signal; And the combination in any of these standards.

Below, we provide the detailed mathematical description to described analysis, and example is provided.The first, we describe example scanning interferometer meter.The second, we are identified for the mathematical model of scanning interferometer measurement data.The 3rd, we describe the optical properties on surface and how to use such information to generate the accurate model of the interferometry data that is used for the different surfaces characteristic.The 4th, we describe and can how the experiment interferometry data be compared with the prediction storehouse, so that the information of relevant tested object to be provided.At first, we will describe film and use, and afterwards, we will describe the application to other complex surfaces structure, particularly, and (under-resolved) step height and grating pattern below optical resolution.And we at first are directed to the analysis in the spatial frequency domain, and afterwards, we will describe the analysis in the scanning coordinate territory.

Fig. 3 shows Linnik type scanning interferometer meter.Partly send the irradiates light 102 from the source (not shown) by beam splitter 104, with definition reference light 106, and the reflected illumination light 102 partly by beam splitter 104 is with definition measuring light 108.By measuring object lens 110 measuring light is focused on the test sample book 112 (sample that for example, comprises the single or multiple lift film of one or more dissimilar materials).Similarly, by reference objective lens 114 reference light is focused on the reference mirror 116.Preferably, measurement and reference objective lens have common optical properties (numerical aperture that for example, is complementary).(perhaps, scattering or diffraction) measuring light is propagated by measuring object lens 110, transmits by beam splitter 104, and is imaged onto on the detecting device 120 by imaging lens 118 from test sample book 112 reflections.Similarly, propagate by reference objective lens 114, reflect by beam splitter 104, and be imaged onto on the detecting device 120 by imaging lens 118 from the reference light of reference mirror 116 reflection, wherein, it and measuring light interference.

For simplicity, Fig. 3 shows the measurement and the reference light of on the specified point that focuses on respectively on test sample book and the reference mirror, also interfering subsequently on the corresponding point of detecting device.Such light corresponding to interferometric measurement and and those parts irradiates light propagated perpendicular with reference to the pupil plane of leg (reference leg).The other parts of irradiates light are shone other point on test sample book and the reference mirror at last, are imaged onto subsequently on the corresponding point on the detecting device.In Fig. 3, use with the corresponding point that are imaged onto on the detecting device on, the corresponding dotted line 122 of chief ray that difference on test sample book sends is its diagram.Chief ray intersects with the center of the pupil plane 124 of the measurement leg (measurement leg) of the back focussing plane of conduct measurement object lens 110.The light that sends from test sample book on the angle different with the angle of chief ray intersects at the diverse location of pupil plane 124.

In a preferred embodiment, difference on independent measurement and test sample book and the reference mirror is corresponding in order to be used for for detecting device 120, the interference between measurement and the reference light (promptly, be used to provide spatial resolution to interference figure) multicomponent (that is many pixels) camera.

As represented with scanning coordinate ζ in Fig. 3, the scanning stage 26 sweep test samples that are couple to test sample book 112 are with respect to the position of measuring object lens 110.For example, scanning stage can be based on piezoelectric transducer (PZT).When the relative position to test sample book scans, the intensity of the optical interference at one or more pixels place of detecting device 120 detector for measuring, and this information is sent to computing machine 128 be used for analyzing.

Because in measuring light is focused zone on the test sample book, scan, so, depend on the angle of the measuring light that incides on the test sample book and send from test sample book, this scanning differently changes the optical path length of measuring light from the source to the detecting device.As a result, depend on the angle of the measuring light that incides on the test sample book and send from test sample book, measure and the interference portion of reference light between the optical path length from the source to the detecting device poor (OPD) differently change in proportion according to scanning coordinate ζ.In other embodiments of the invention, can realize identical result with respect to the position of reference objective lens 114 (rather than come sweep test sample 112 with respect to measuring object lens 110) by scan reference mirror 116.

This difference how OPD changes along with scanning coordinate ζ has been introduced: the limited coherent length in the interference signal that each pixel place of detecting device is measured.For example, typically, by having about λ/2 (NA) ²The envelope of spatial coherence length and interferometric modulator signal (as the function of scanning coordinate), wherein, λ is the nominal wavelength of irradiates light, and NA is for measuring and the numerical aperture of reference objective lens.As be further described below, the modulation of interference signal is provided the information that depends on angle of the reflectivity of relevant test sample book.For increasing limited spatial coherence, preferably, object lens in scanning interferometer meter definition large-numerical aperture, for example, greater than about 0.7 (perhaps, more preferably, greater than about 0.8, or greater than about 0.9).Also can be by coming the interferometric modulator signal with the wide limited temporal coherent length that is associated of the bands of a spectrum of irradiation source.Depend on interferometric configuration, in these limited coherent length effects one or other can be preponderated, and perhaps, they all can substantially be made contributions to total coherent length.

Another example of scanning interferometer meter is the Mirau type interferometer shown in Fig. 4.

With reference to Fig. 4, source module 205 provides irradiates light 206 to beam splitter 208, and beam splitter 208 is directed to Mirau interferometry objective lens unit 210 with it.Assembly 210 comprises objective lens 211; Reference plate 212 has reflectance coating on the very little core of its definition reference mirror 215; And beam splitter 213.During operation, objective lens 211 is by reference plate 212, focus on irradiates light towards test sample book 220.Beam splitter 213 reflexes to reference mirror 215 with the first of focused light, with definition reference light 222, and the second portion of focused light is sent to test sample book 220, with definition measuring light 224.Subsequently, beam splitter 213 will reconfigure with the reference light that reflects from reference mirror 215 from the measuring light of test sample book 220 reflections (or scattering), and 230 pairs of light that made up of object lens 211 and imaging lens carry out imaging, on detecting device (for example, many pixel camera) 240, to interfere.As in the system of Fig. 3, the measuring-signal (a plurality of) of self-detector sends to the computing machine (not shown) in the future.

Scanning among the embodiment of Fig. 4 relates to the piezoelectric transducer (PZT) 260 that is couple to Mirau interferometry objective lens unit 210, it is configured to along the optical axis of object lens 211 and with respect to test sample book 220 scannings assembly 210 as a whole, with the scanning interferometer measurement data I at each pixel place that camera is provided (ζ, h).Replacedly, PZT can be coupled to test sample book rather than assembly 210, so that relative motion therebetween to be provided, as specified by PZT actuator 270.In another embodiment, can be by along the optical axis of object lens 211, move in reference mirror 215 and the beam splitter 213 one or whole two with respect to object lens 211, and this scanning is provided.

The diaphragm (stop) 204 on telescope (telescope) that source module 205 comprises the source 201 of extending in the space, formed by camera lens 202 and 203 and the prefocusing plane (its back focussing plane with camera lens 203 is consistent) that is positioned at camera lens 202.The source that the space is extended in this configuration is imaged onto on the pupil plane 245 of Mirau interferometry objective lens unit 210, and this is the example of Koehler imaging.The size of the irradiation area on the size control test sample book 220 of diaphragm.In other embodiments, source module can comprise a kind of configuration, and wherein, the source direct imaging that the space is extended is on test sample book, and this is called as critical imaging (critical imaging).The source module of arbitrary type all can be used for the Linnik type scanning interferometer measuring system of Fig. 1.

In other embodiments of the invention, can use the scanning interferometer measuring system to determine scattering relevant test sample book, that depend on angle or diffraction information, that is, be used for scatterometry.For example, can use the scanning interferometer measuring system (for example to come by the test incident on the incident angle of very narrow scope only, the incident of perpendicular (normal) or parallel (collimated)) shine test sample book, but test specimen is come scattering or the described test incident of diffraction subsequently.The photoimaging that will send from this sample is to camera, so as mentioned above and with the reference light interference.The spatial frequency of each component in the scanning interferometer measuring-signal will depend on the variation of the angle of accompanying or follow the test light that test sample book sends.Thus, the vertical scanning (that is, along the scanning of the optical axis of object lens) of following Fourier analysis allows as the diffraction of the function that sends angle and/or the measurement of scattered light, and directly the back focussing plane of object lens is not conducted interviews or imaging.For the incident irradiation of perpendicular is provided, for example, source module can be configured to: point source is imaged onto on the pupil plane, or reduces the degree that irradiates light is filled the numerical aperture of measuring object lens.The scatterometry technology for differentiate may with optical diffraction and/or scatter to discrete topology in the sample surface of higher angle (as grid stroke, edge or surfaceness) may be useful.

Here, in a lot of the analysis, suppose that the polarized state of light in the pupil plane is at random, that is, and by s (with the plane of incidence quadrature) polarization and p (with the plane of incidence quadrature) polarization of approximately equal quantity.Interchangeable polarization is possible, it comprises pure s polarization, as can by the radial polarisation device that is arranged in pupil plane realize (for example, in the interferometric situation of Linnik, in the back focussing plane of measuring object, and in the interferometric situation of Mirau, in the back focussing plane of public object lens).Other possible polarization comprises radially p polarization, circular polarization and is used for the modulating polarization of ellipsometric measurement (for example, two states of following another).In other words, can not only be directed to the dependence of the optical properties of test sample book, also can be directed to their polarization dependence or be directed to selected polarization, differentiate the optical properties of test sample book angle or wavelength.Also can use such information to improve the precision that membrane structure characterizes.

For such ellipsometric measurement is provided, the scanning interferometer measuring system can comprise: the fixing or variable polarizer in pupil plane.Referring again to Fig. 4, for example, Mirau type interferometer measuration system comprises polarization optics device (optics) 280 in the pupil plane, is used for selecting inciding test sample book and the expectation polarization of the light that sends from test sample book.In addition, the polarization optics device can be reconfigured, to change selected polarization.The polarization optics device can comprise: comprise polarizer, wave plate, apodized aperture (apodization apertures) and/or be used to select one or more elements of the modulator element of given polarization.In addition, in order to generate the purpose of the data that are similar to ellipsometer, fixing, structuring or reconfigurable that the polarization optics device can be.For example, have the radial polarisation pupil that is used for the s polarization, be first the measuring of radial polarisation pupil that is used for the p polarization afterwards.In another example, can use the pupil plane that becomes mark (apodized) by linearly polarized photon, for example, otch that can rotate in pupil plane (slit) or wedge shape (wedge) are so that be incorporated into this object or the reconfigurable screen of LCD for example with the linear polarization state of any desired.

In addition, the polarization optics device can provide the variable polarization of crossing pupil plane (for example, by comprising a plurality of polarizers or spatial modulator).Thus, for example, can be by for to provide different polarizations than the high incident angle of shallow angle (shallow angle), and come " mark (tag) " polarization state according to spatial frequency.

In another embodiment, optional polarization can be merged mutually with phase shift as the function of polarization.For example, the polarization optics device can comprise linear polarization, and it is arranged in pupil plane, and two wave plates (for example, 1/8th wave plates) in the relative sector of pupil plane are followed in the back.Linear polarization is directed to the plane of incidence of object lens and produces FR polarization angle.For example, if being arranged as feasible dominant in advance s polarized light, wave plate has fixing phase shift, so, with phase shift simultaneously but each other for example pi mode and present the radially light of s polarization and p polarization, so that interferometer detects poor between these two polarization states effectively, as basis signal.

In another embodiment, the polarization optics device can be positioned other position in the equipment.For example, can realize linear polarization in the optional position in system.

Now, we describe the physical model that is used for the scanning interferometer measuring-signal.

Object surface has altitude feature h, and we wish its configuration (profile) by horizontal ordinate (1ateralcoordinate) x, on the zone of y indication.This level (stage) provides level and smooth, the continuous scanning ζ of arbitrary interference objective or object self (going out as shown).In scan period, computing machine is each picture point or the camera pixel record intensity data I in the continuous camera frame _{ζ, h}Note, indicated intensity I by subscript (mark that we can adopt in the text) _{ζ, h}Crucial dependence for scanning position and surface elevation.

Consider partial coherence, the polarization mixing in the interferometer, the imaging attribute of high NA object lens and the interaction between high incident angle and the electric-field vector when having discontinuous surface characteristics of light source, the suitable physical model of optical device may be very meticulous.For convenience's sake, we pass through supposition random polarization and diffusion, hang down relevant source of extending, and simplify this model.Modeling is reduced to interference signal: as shown in Figure 5, will with incident angle ψ by object lens pupil plane and from the contribution addition of all light shafts (bundle) of object surface reflection.

The interference contribution of the single light shafts by optical system is proportional with following equation:

$g_{\mathrm{\β},k,\mathrm{\ζ},h}=R_{\mathrm{\β},k}+Z_{\mathrm{\β},k}+2\sqrt{R_{\mathrm{\β},k}{Z}_{\mathrm{\β},k}}\cos [2\mathrm{\β}{\mathrm{kn}}_0(h-\mathrm{\ζ})+({\mathrm{\υ}}_{\mathrm{\β},k}-{\mathrm{\ω}}_{\mathrm{\β},k})]...\left(1\right)$

Wherein, Z _{β, k}Be effective object intensity reflectivity, it comprises: for example, and the influence of beam splitter, and, R _{β, k}With reference to reflectivity, it comprises beam splitter and reference mirror for effectively.The index of surrounding medium (index) is n ₀, the direction cosine of incident angle ψ are

β＝cos(ψ) (2)

And the wave number of source irradiation is

k＝(2π/λ) (3)

The notation convention of phase place makes the just change of the increase of surface elevation corresponding to phase place.Phase term has: to the contribution ω of the object path in the interferometer (object path) _{β, k}, it comprises the film influence from object surface; And to the contribution υ of reference path _{β, k}, it comprises other optical device in reference mirror and the object lens.

Total interference signal and following equation in the pupil plane upper integral are proportional:

$I_{\mathrm{\ζ},h}=\underset{0}{\overset{\∞}{\∫}}\underset{0}{\overset{1}{\∫}}{g}_{\mathrm{\β},k,\mathrm{\ζ},,h}{U}_{\mathrm{\β}}{V}_k\mathrm{\βd\βdk}...\left(4\right)$

Wherein, U _βFor pupil plane light distributes, and V _kBe spectral distribution.Weighting factor β in the equation (4) is from the cos (ψ) that is attributable to projectional angle and be used for diameter sin (ψ) of anchor ring of width d ψ of pupil plane and draw:

cos(ψ)sin(ψ)dψ＝-βdβ (5)

Here, as shown in Figure 5, suppose object lens obedience Abb é sine condition.This weighting of simplifying relatively is possible for the spatial coherence irradiation (wherein, all light shafts are independently of one another) of random polarization.At last, represent 0≤β≤1, and represent 0≤k≤∞ based on the limit of integration of the spectral integral of all wave numbers based on the limit of integration of all incident angles.

In frequency-domain analysis (FDA), we at first calculate interference strength signal I _{ζ, h}Fourier transform.Analyze for direct amount (literal) (nonnumeric), we will use not standardized fourier integral:

$q_{K,h}=\underset{-\∞}{\overset{\∞}{\∫}}{I}_{\mathrm{\ζ},h}\exp \left(\mathrm{iK\ζ}\right)\mathrm{d\ζ}...\left(6\right)$

Wherein, K is a spatial frequency, for example, with every micron cycle be unit.Frequency domain value q _{K, h}The unit that has wave number reciprocal, for example micron.Draw power spectrum from it:

Q _{K, h}=| q _{K, h}| ²(7) and phase spectrum:

φ _K，h″＝arg(q _K，h) (8)

Be used for φ _{K, h}" double quotation marks mean: in the clause rank, have double-deck (two-fold) uncertain (uncertainty), its be individual element, and on the whole with the relevant form of starting point that scans.Traditional FDA directly advances to subsequently by by power spectrum φ _{K, h}" the phase spectrum Q of weighting _{K, h}Linear fit and the determining of the surfac topography that carries out.This fits to each pixel slope is provided:

σ _h≈dφ″/dK (9)

And intercept (intercept):

A″≈φ _K＝0，h″ (10)

" h is irrelevant with height, but has the uncertain double quotation marks of inheriting of clause rank from phase data to note intercept or " phase place gap " A.Slope σ does not have this uncertainty." and the slope σ according to intercept A _h, we can be certain sense (mean) or nominal spatial frequency K0 and define " relevant profile (profile) ":

Θ _h＝σ _hK0 (11)

And " phase outline "

θ _h″＝Θ _h+A″ (12)

For the dielectric surface of the perfect consistent and uniform of no film and dissimilar material influence and for simple, the ideal situation of the optical system of the perfect balance of chromatic dispersion, phase place and relevant profile are proportional linearly in surface elevation:

h _Θ＝Θ _h/K0 (13)

h _θ″＝θ _h″/K0 (14)

In these two high computational, based on the height value h of phase place _θ" more accurate, but it has the uncertainty in striped rank (fringe order) characteristic that monochromatic interference is measured.For high resolving power, we can use unambiguity but more coarse height value h based on the coherence _Θ, eliminating this uncertainty, and produce end value h _Θ

Traditional FDA supposition: even for relatively poor Utopian situation, interferometric phase φ _{K, h}" still almost be the linear function of spatial frequency.Yet for present embodiment, we are by comparing experimental data with the theoretical prediction of the modulation that can comprise height nonlinear phase spectrum and associated power spectrum, and the key parameter of definite surface structure.

To this, we merge into the definition of Fourier transform equation (6) and interference signal equation (4) the following equation that is used for the FDA spectrum predicted:

$q_{K,h}=\underset{-\∞}{\overset{\∞}{\∫}}\underset{0}{\overset{\∞}{\∫}}\underset{0}{\overset{1}{\∫}}{g}_{\mathrm{\β},k,\mathrm{\ζ},h}\exp \left(\mathrm{iK\ζ}\right)U_{\mathrm{\β}}{V}_k\mathrm{\βd\βdkd\ζ}...\left(15\right)$

For improving counting yield, can carry out the part of the triple integral in the equation (15) and directly measure evaluation (evaluate).

The direct component analysis of equation (15) begins along with the change of integration order, with at first to each interference signal g under fixing β and k, on all scanning position ζ _{β, k, ζ, h}Evaluation:

$q_{K,h}=\underset{0}{\overset{\∞}{\∫}}\underset{0}{\overset{1}{\∫}}{U}_{\mathrm{\β}}{V}_k\mathrm{\β}\left\{\underset{-\∞}{\overset{\∞}{\∫}}{g}_{\mathrm{\β},k,\mathrm{\ξ},h}\exp \left(\mathrm{iK\ξ}\right)\mathrm{d\ξ}\right\}\mathrm{d\βdk}...\left(16\right)$

At g _{β, k, ζ, h}In the expansion of cosine term with the usual way that uses following equation after,

2cos(u)＝exp(iu)+exp(-iu) (17)

Integrates evaluation on the ζ is:

$\underset{-\∞}{\overset{\∞}{\∫}}{g}_{\mathrm{\β},k,\mathrm{\ζ},h}g\exp \left(\mathrm{iK\ζ}\right)\mathrm{d\ζ}={\mathrm{\δ}}_K(R_{\mathrm{\β},k}+Z_{\mathrm{\β},k})...$

$+{\mathrm{\&delta;}}_{(K-2\mathrm{\&beta;k}{n}_0)}\sqrt{R_{\mathrm{\&beta;},k}{Z}_{\mathrm{\&beta;},k}}\exp [\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="A20048001232300561.png,A20048001232300571.png"\; state="\{\{state\}\}">i</figure-callout>}2\mathrm{\&beta;}{\mathrm{kn}}_{0}h+i({\mathrm{\&upsi;}}_{\mathrm{\&beta;},k}-{\mathrm{\&omega;}}_{\mathrm{\&beta;},k})]...\left(18\right)$

$+{\mathrm{\&delta;}}_{(K+2\mathrm{\&beta;k}{n}_0)}\sqrt{R_{\mathrm{\&beta;},k}{Z}_{\mathrm{\&beta;},k}}\exp [-\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="A20048001232300561.png,A20048001232300571.png"\; state="\{\{state\}\}">i</figure-callout>}2\mathrm{\&beta;}{\mathrm{kn}}_{0}h-i({\mathrm{\&upsi;}}_{\mathrm{\&beta;},k}-{\mathrm{\&omega;}}_{\mathrm{\&beta;},k})]$

Wherein, we have used

${\mathrm{\&delta;}}_K=\underset{-\&infin;}{\overset{\&infin;}{\&Integral;}}\exp \left(\mathrm{iK\&zeta;}\right)\mathrm{d\&zeta;}...\left(19\right)$

${\mathrm{\&delta;}}_{(K\&PlusMinus;2\mathrm{\&beta;}{\mathrm{kn}}_0)}=\underset{-\&infin;}{\overset{\&infin;}{\&Integral;}}\exp \left(\mathrm{iK\&zeta;}\right)\exp (\&PlusMinus;i2\mathrm{\&beta;k}{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0\mathrm{\&zeta;})d\mathrm{\&zeta;}...\left(20\right)$

The δ function is a wave number reciprocal with its physical unit reciprocal that has independent variable in this case.

These delta functions make spatial frequency K and long-pending 2 β kn ₀Between equivalence effective.Therefore, the logical changes that is used for the variable of next integration is:

$\mathrm{\&beta;}=\widehat{\mathrm{\&kappa;}}/2\mathrm{<figure-callout\; id="0"\; label="k\; n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{n}_{}{}_0...\left(21\right)$

$\mathrm{d\&beta;}=d\widehat{\mathrm{\&kappa;}}/2{\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0...\left(22\right)$

Wherein,

Have the meaning identical, but will be used as the free product variation per minute with spatial frequency K.Equation (18) can be written as:

$q_{K,h}=\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{0}{\overset{2\mathrm{<figure-callout\; id="0"\; label="k\; n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{n}_{}{}_0}{\&Integral;}}{\mathrm{\&delta;}}_K(R_{\widehat{\mathrm{\&kappa;}},k}+Z_{\widehat{\mathrm{\&kappa;}},k}){\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}\mathrm{dk}...$

$+\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{0}{\overset{2{\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0}{\&Integral;}}{\mathrm{\&delta;}}_{(K-\widehat{\mathrm{\&kappa;}})}\sqrt{R_{\widehat{\mathrm{\&kappa;}},k}{Z}_{\widehat{\mathrm{\&kappa;}},k}}\exp [i\widehat{\mathrm{\&kappa;}}h+i({\mathrm{\&upsi;}}_{\widehat{\mathrm{\&kappa;}},k}-{\mathrm{\&omega;}}_{\widehat{\mathrm{\&kappa;}},k})]{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}\mathrm{dk}...\left(23\right)$

$+\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{0}{\overset{2{\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0}{\&Integral;}}{\mathrm{\&delta;}}_{(K+\widehat{\mathrm{\&kappa;}})}\sqrt{R_{\widehat{\mathrm{\&kappa;}},k}{Z}_{\widehat{\mathrm{\&kappa;}},k}}\exp [-i\widehat{\mathrm{\&kappa;}}h-i({\mathrm{\&upsi;}}_{\widehat{\mathrm{\&kappa;}},k}-{\mathrm{\&omega;}}_{\widehat{\mathrm{\&kappa;}},k})]{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}\mathrm{dk}$

Wherein

${\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}=\frac{U_{\widehat{\mathrm{\&kappa;}},k}{V}_k\widehat{\mathrm{\&kappa;}}}{4k^2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2}...\left(24\right)$

Notice that because the change in the variable, it is right to become for the β dependence of R, Z in the equation (23), υ, ω item Dependence with k.

For following step, we at first notice:

$\underset{0}{\overset{2\mathrm{<figure-callout\; id="0"\; label="k\; n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{n}_{}{}_0}{\&Integral;}}{\mathrm{\&delta;}}_K{f}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}={\mathrm{\&delta;}}_K\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{(2k{n}_0-\widehat{\mathrm{\&kappa;}})}{f}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}...\left(25\right)$

$\underset{0}{\overset{2{\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0}{\&Integral;}}{\mathrm{\&delta;}}_{(K-\widehat{\mathrm{\&kappa;}})}{f}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}=f_{K,k}{H}_K{H}_{(2k{n}_0-K)}...\left(26\right)$

$\underset{0}{\overset{2{\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0}{\&Integral;}}{\mathrm{\&delta;}}_{(K+\widehat{\mathrm{\&kappa;}})}{f}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}=f_{-K,k}{H}_{-K}{H}_{(2\mathrm{<figure-callout\; id="0"\; label="k\; n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{n}_{}{}_0+K)}...\left(27\right)$

Wherein, H is the Heaviside step function (unitless Heavisidestep function) by the no unit of following equation definition.

And f is the arbitrary function of K and k.Use equation (25) to (27), equation (23) becomes:

$q_{K,h}={\mathrm{\&delta;}}_K\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{({2\mathrm{kn}}_0-\widehat{\mathrm{\&kappa;}})}(R_{\widehat{\mathrm{\&kappa;}},k}+Z_{\widehat{\mathrm{\&kappa;}},k}){\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}\mathrm{dk}...$

$+\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_K{H}_{({2\mathrm{kn}}_0-K)}\sqrt{R_{K,k}{Z}_{K,k}}\exp [\mathrm{iKh}+i({\mathrm{\&upsi;}}_{K,k}-{\mathrm{\&omega;}}_{K,k})]{\mathrm{\&Gamma;}}_{K,k}\mathrm{dk}...\left(29\right)$

$+\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{-K}{H}_{({2\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0+K)}\sqrt{R_{-K,k}{Z}_{-K,k}}\exp [\mathrm{iKh}-i({\mathrm{\&upsi;}}_{K,k}-{\mathrm{\&omega;}}_{K,k})]{\mathrm{\&Gamma;}}_{-K,k}\mathrm{dk}$

Use now

$\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{(2{\mathrm{kn}}_0-\widehat{\mathrm{\&kappa;}})}{f}_{\widehat{\mathrm{\&kappa;}},k}d\widehat{\mathrm{\&kappa;}}\mathrm{dk}=\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{(2k{n}_0-\widehat{\mathrm{\&kappa;}})}{f}_{\widehat{\mathrm{\&kappa;}},k}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}...\left(30\right)$

$\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_K{H}_{({2\mathrm{kn}}_0-K)}{f}_{K,k}\mathrm{dk}=H_K\underset{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{f}_{K,k}\mathrm{dk}...\left(31\right)$

$\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{-K}{H}_{({2\mathrm{<figure-callout\; id="0"\; label="kn"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">kn</figure-callout>}}_0+K)}{f}_{-K,k}\mathrm{dk}=H_{-K}\underset{-K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{f}_{-K,k}\mathrm{dk}...\left(32\right)$

We have had last result

(33)

Because there is less integration, so equation (33) is more efficient significantly on calculating than the triple integral of original (15).

For find the solution analytically, some limited case are interesting.For example, if phase place contribution (υ _{K, k}-ω _{K, k})=0 and reflectivity R, Z and incident angle and Wavelength-independent, so, equation (33) is reduced to:

$q_{K,h}={\mathrm{\&delta;}}_K(R+Z)\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{\widehat{\mathrm{\&kappa;}}/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}$

$+H_K\exp \left(\mathrm{iKh}\right)\sqrt{\mathrm{RZ}}\underset{k/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{K,k}\mathrm{dk}...\left(34\right)$

$+H_{-K}\exp \left(\mathrm{iKh}\right)\sqrt{\mathrm{RZ}}\underset{-k/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{-K,k}\mathrm{dk}$

And we only must handle the weighting factor Γ that relates to definition in equation (24) _{K, k}Integration.This Utopian situation has been simplified the evaluation to two other limited case of equation (34), described two other limited case promptly: near monochromatic irradiation by high NA object lens and the broadband irradiation by low NA.

For having narrow spectral bandwidth k _ΔThe situation of nearly monochromatic source, we have the standardization frequency spectrum:

$V_k=\frac{1}{k_{\mathrm{\&Delta;}}}{H}_{(k-k0)}{H}_{(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0+k_{\mathrm{\&Delta;}}-k0)}...\left(35\right)$

Wherein, k0 is a standardization source wave number.Now, the integration in the equation (34) has following form:

$\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{K/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}=\frac{1}{{4\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2{k}_{\mathrm{\&Delta;}}}\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{(k0-\widehat{\mathrm{\&kappa;}}/2n_0)}\widehat{\mathrm{\&kappa;}}\underset{k0}{\overset{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0+k_{\mathrm{\&Delta;}}}{\&Integral;}}\frac{U_{\widehat{\mathrm{\&kappa;}},\mathrm{<figure-callout\; id="2"\; label="k\; k"\; filenames="A20048001232300561.png,A20048001232300571.png"\; state="\{\{state\}\}">k</figure-callout>}}}{k^{}}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}...\left(36\right)$

$\underset{K/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{K,k}\mathrm{dk}=\frac{1}{4{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2{k}_{\mathrm{\&Delta;}}}{H}_{(k0-K/{2n}_0)}K\underset{k0}{\overset{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0+k_{\mathrm{\&Delta;}}}{\&Integral;}}\frac{U_{K,\mathrm{<figure-callout\; id="2"\; label="k\; k"\; filenames="A20048001232300561.png,A20048001232300571.png"\; state="\{\{state\}\}">k</figure-callout>}}}{k^{}}\mathrm{dk}...\left(37\right)$

Suppose U _{K, k}At little bandwidth k _ΔLast substantial constant, we have:

$\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}=\underset{0}{\overset{\&infin;}{\&Integral;}}{H}_{(k0-\widehat{\mathrm{\&kappa;}}/2n_0)}{U}_{\widehat{\mathrm{\&kappa;}},\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0}\frac{\widehat{\mathrm{\&kappa;}}}{{4\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{0}^2}d\widehat{\mathrm{\&kappa;}}...\left(38\right)$

$\underset{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{K,k}\mathrm{dk}=H_{(k0-K/{2n}_0)}{U}_{K,k0}\frac{K}{4{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{0}^2}....\left(39\right)$

Wherein, in the evaluation of integration, we have used:

$\frac{-1}{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0+k_{\mathrm{\&Delta;}}}+\frac{1}{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0}\&ap;\frac{k_{\mathrm{\&Delta;}}}{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}0}...\left(40\right)$

It is for narrow bandwidth k _Δ＜＜k0 is effective.Especially, frequency spectrum just, non-0 part abbreviation is:

$q_{K>0,h}=\frac{H_k{H}_{(k0-K/2n_0)}{U}_{K,{2n}_{0}k0}K\sqrt{\mathrm{RZ}}}{{4\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">k</figure-callout>}{0}^2}\exp \left(\mathrm{iKh}\right)...\left(41\right)$

Thus, for narrow spectral bandwidth light source, constant reflectivity R, Z and there is not phase place contribution

These special circumstances,

φ _K，h″＝Kh (42)

In these special circumstances, consistent with traditional FDA, phase place and surface elevation are proportional linearly.Spatial frequency also has direct corresponding with direction cosine:

K＝β2n ₀k0 (43)

Thus, between the spatial frequency coordinate of FDA frequency spectrum and incident angle, there is man-to-man relation.Be also noted that, calculate fourier modulus according to equation (41) In the K weighting.This clearly, the figure shows the theoretical prediction to the perfect consistent filling that begins the pupil plane on the scope till the direction cosine restriction that is applied by object lens NA from vertical incidence in example frequency spectrum Fig. 6 (a):

${\mathrm{\&beta;}}_{\mathrm{NA}}=\sqrt{1-{\mathrm{NA}}^2}---\left(44\right)$

As second example, consider to have the situation of shining that is restricted near the consistent broadband of shine of the close limit β Δ of the direction cosine of vertical incidence.So, standardized pupil plane is distributed as:

$U_{\mathrm{\&beta;}}=\frac{1}{{\mathrm{\&beta;}}_{\mathrm{\&Delta;}}}{H}_{1-\mathrm{\&beta;}}{H}_{\mathrm{\&beta;}-(1-{\mathrm{\&beta;}}_{\mathrm{\&Delta;}})}---\left(45\right)$

After changing variable,

$U_{K,k}=\frac{1}{{\mathrm{\&beta;}}_{\mathrm{\&Delta;}}}{H}_{({2\mathrm{kn}}_0-K)}{H}_{[K-2{\mathrm{kn}}_0(1-{\mathrm{\&beta;}}_{\mathrm{\&Delta;}})]}...\left(46\right)$

In this case, the form of the definite integral in the equation (34) is:

$\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}=\frac{1}{{\mathrm{\&beta;}}_{\mathrm{\&Delta;}}}\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{\widehat{\mathrm{\&kappa;}}/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\widehat{\mathrm{\&kappa;}}/(1-{\mathrm{\&beta;}}_{\mathrm{\&Delta;}})2n_0}{\&Integral;}}\frac{V_k\widehat{\mathrm{\&kappa;}}}{4k^2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}...\left(47\right)$

$\underset{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{K,k}\mathrm{dk}=\frac{1}{{\mathrm{\&beta;}}_{\mathrm{\&Delta;}}}\underset{K/{2n}_0}{\overset{K/(1-{\mathrm{\&beta;}}_{\mathrm{\&Delta;}}){2N}_0}{\&Integral;}}\frac{V_{k}K}{4k^2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2}\mathrm{dk}...\left(48\right)$

It is evaluated as:

$\underset{0}{\overset{\&infin;}{\&Integral;}}\underset{K/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{\widehat{\mathrm{\&kappa;}},k}\mathrm{dkd}\widehat{\mathrm{\&kappa;}}=\underset{0}{\overset{\&infin;}{\&Integral;}}\frac{V_{\widehat{\mathrm{\&kappa;}}/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}}{{2n}_0}d\widehat{\mathrm{\&kappa;}}...\left(49\right)$

$\underset{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\overset{\&infin;}{\&Integral;}}{\mathrm{\&Gamma;}}_{K,k}\mathrm{dk}=\frac{V_{K/2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}}{{2n}_0}---\left(50\right)$

Wherein, we have used

$\frac{(1-{\mathrm{\&beta;}}_{\mathrm{\&Delta;}})2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\widehat{\mathrm{\&kappa;}}}-\frac{{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}{\widehat{\mathrm{\&kappa;}}}=-\frac{2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0{\mathrm{\&beta;}}_{\mathrm{\&Delta;}}}{\widehat{\mathrm{\&kappa;}}}...\left(51\right)$

Frequency spectrum just, non-0 part is used for the irradiation of this broad band source, and therefore, nearly vertical incidence is:

$q_{K>0,h}=\frac{V_{K/{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}\sqrt{\mathrm{RZ}}}{{2\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0}\exp \left(\mathrm{iKh}\right)...\left(52\right)$

This is approximate corresponding to such result that is familiar with, that is: for example, as Fig. 6 (b) that to be used for nominal or middle wavelength k0 be the Gaussian spectrum at center shown in, fourier modulus

With source spectrum distribution V _K/2n0Proportional.Notice that consistent with traditional FDA, equation (25) is also followed the supposition that linear phase develops (evolution), that is:

${\mathrm{\&phi;}}_{K,h}^{\&prime;\&prime;}=\mathrm{Kh}...\left(53\right)$

Because from interference strength I _{ζ, h}Fourier transform derive fourier modulus $\sqrt{Q_{K,h}}=\left|q_{K,h}\right|$ And phase place

${\mathrm{\&phi;}}_{K,h}^{\&prime;\&prime;}=\arg \left(q_{K,h}\right),$ So inverse Fourier transform is got back to us and is done solid work in the territory that relates to signal:

$I_{\mathrm{\&zeta;},h}=\underset{-\&infin;}{\overset{\&infin;}{\&Integral;}}{q}_{\widehat{\mathrm{\&kappa;}},h}\exp (-i\widehat{\mathrm{\&kappa;}}\mathrm{\&zeta;})d\widehat{\mathrm{\&kappa;}}...\left(54\right)$

Wherein, we have reused Being used for spatial frequency, is the free variable of the integration in the equation (54) to emphasize it.Thus, a kind of mode of calculating strength signal is: generate Fourier component q by equation (33) _{K, h}, and use equation (54) and be transformed to I _{ζ, h}

We suppose the random polarization of the light source in the current model.Yet this does not mean that we should ignore the polarization influence.On the contrary, in above calculating, the coherent superposition (superposition) of the equal weighting of quilt that our supposition obtains from two orthogonal polarization state s and p by the plane of incidence definition of irradiation.By polarization being used the subscript note,

$q_{\mathrm{\&beta;},k}=q_{\mathrm{\&beta;},k}^s+q_{\mathrm{\&beta;},k}^p...\left(55\right)$

Therefore, the average phase angle of the nonpolarized light under this β, k will for:

$<{\mathrm{\&phi;}}_{\mathrm{\&beta;},k}^{\&prime;\&prime;}>=\arg (q_{\mathrm{\&beta;},k}^s+q_{\mathrm{\&beta;},k}^p)...\left(56\right)$

Note, unless amplitude is identical for two polarization contributions, in most cases:

$<{\mathrm{\&phi;}}_{\mathrm{\&beta;},k}^{\&prime;\&prime;}>\&NotEqual;({\mathrm{\&phi;}}_{\mathrm{\&beta;},k}^{\&prime;\&prime;s}+{\mathrm{\&phi;}}_{\mathrm{\&beta;},k}^{\&prime;\&prime;p})/2...\left(57\right)$

And, unless q _{β, k} ^sAnd q _{β, k} ^pPerfect parallel in complex plane, otherwise

$<Q_{\mathrm{\&beta;},k}>\&NotEqual;(Q_{\mathrm{\&beta;},k}^s+Q_{\mathrm{\&beta;},k}^p)/2...\left(58\right)$

Same discussion (observation) is applied to system and object reflectivity R respectively _{β, k} ^s, R _{β, k} ^pAnd Z _{β, k} ^s, Z _{β, k} ^pUnless they have same phase place, otherwise, can not be directly to they summations.

Suppose that we suitably influence with respect to the polarization in the calculating object surface reflectivity, still suitable simple and clear (straightforward) of modeling then, and enough flexibly, to handle the more interesting situation of (further down theline) polarized light.

Consider software development, following step is: be converted to discrete numerical value equation.We use following discrete Fourier transform (DFT), and redefine interference signal I _{ζ, h}With fourier spectrum q _{K, h}Between relation:

$q_{K,h}=\frac{1}{\sqrt{N}}\underset{\mathrm{\&zeta;}}{\mathrm{\&Sigma;}}{I}_{\mathrm{\&zeta;},h}\exp \left(\mathrm{iK\&zeta;}\right)...\left(59\right)$

$I_{\mathrm{\&zeta;},h}=\frac{1}{\sqrt{N}}[{\mathrm{<figure-callout\; id="0"\; label="q"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">q</figure-callout>}}_0+\underset{K>0}{\mathrm{\&Sigma;}}{q}_{K,h}\exp (-\mathrm{iK\&zeta;})+\underset{K>0}{\mathrm{\&Sigma;}}{\stackrel{\&OverBar;}{q}}_{K,h}\exp \left(\mathrm{iK\&zeta;}\right)]...\left(60\right)$

Wherein, For

Complex conjugate, and, at interference signal I _{ζ, h}In have N discrete sample.Equation (60) and below equation in, we have cancelled (set aside) free variable

Use, this is important in differential (derivation), but not its substituting as spatial frequency K of needs.So, the positive frequency FDA complex frequency spectrum of prediction is:

q K≥0，h＝ρ K≥0exp(iKh)

...(61)

Wherein, the coefficient of standardized height independent (height-independent) is:

...(62)

...(63)

Wherein, being standardized as limit of integration:

Heaviside step function H in the equation (62) has prevented the unnecessary contribution to summation.Weighting factor Г K, k is as defined in the equation (24).

Compare for testing with theory, we use equation (61) to generate experiment FDA frequency spectrum, and use equation (62) that its conversion is back to I _{ζ, h}The spatial domain of theoretical prediction.It is carried out in full blast ground by fast Fourier transform (FFT).The attribute of FFT is determined the scope of K value.If I _{ζ, h}N discrete sample with increment ζ _StepAnd at interval, then will have since 0 and rise to N/2+1 the positive space frequency in every data tracking (datatrace) N/2 cycle, described data tracking is the interval with following increment:

$K_{\mathrm{step}}=\frac{2\mathrm{\&pi;}}{{\mathrm{N\&zeta;}}_{\mathrm{step}}}...\left(65\right)$

For helping the phase unwrapping in the frequency domain, we attempt adjusting 0 position of scanning, make its approach signal peak value, reduce the phase slope in the frequency domain thus.Because first data point that FFT is rendered as in scanning is in 0 always, so, should suitably make this signal bias.

Now, our concentrated discussion is to having the sample surface modeling of film.

Fig. 7 shows two kinds of surface types that have and do not have film.For described two kinds of situations, we define effective amplitude reflectivity z according to following equation _{β, k}:

$z_{\mathrm{\&beta;},k}=\sqrt{z_{\mathrm{\&beta;},k}}\exp \left(i{\mathrm{\&omega;}}_{\mathrm{\&beta;},k}\right)...\left(66\right)$

Wherein, Z _{β, k}Be intensity reflectivity, and ω _{β, k}Be phase change to reflection.Subscript β, k have emphasized the dependence to the direction cosine of irradiation

β ₀＝cos(ψ ₀) (67)

Wherein, ψ ₀Be incident angle, and, on wave number

k＝(2π/λ) (68)

Wherein, λ is the wavelength of light source.Subscript β will be understood that to represent the first incident direction cosine β ₀

Partly the refractive index by the surface characterizes the surface.This index (index) of surrounding medium (being generally air) is n ₀For simple surfaces Fig. 7 (a), a refractive index n is only arranged ₁For the film among Fig. 7 (b), two surface refractive indexs, that is: n are arranged ₁, be used for the film of transparent or partially transparent; n ₂, be used for substrate.More generally, these refractive indexes are the plural number with feature of real part and imaginary part.For example, the typical index that for example is used for chromium under λ=550nm is n ₁=3.18+4.41i, wherein, we have adopted and wherein imaginary part have been defined as positive agreement.

The refractive index of material depends on wavelength.Refractive index n ₀In chromatic dispersion be not very remarkable for air, but be important for multisample surface (particularly metal).Based near wavelength shift little the nominal value k0, most materials have the approximately linear of the wave number of depending on, and make us can write following equation:

...(69)

Wherein, v ₁ ⁽⁰⁾, v ₁ ⁽¹⁾Be respectively under nominal wave number k0, be used for refractive index n ₁Intercept and slope.

The modal use of refractive index is snell law.With reference to Fig. 7 (b), the deflecting light beams angle of film inside is:

...(70)

Wherein, Ψ ₀For at refractive index n ₁The top surface of medium on incident, at refractive index n ₀Medium in angle, and, Ψ _{L, β, k}Be the refraction angle.If described refractive index is partly to indicate the plural number that (evanescentpropagation) propagated in decay, then complex values might be got in these angles.

The complex amplitude reflectivity on the border between the two media depends on polarization, wavelength, incident angle and refractive index.Provide the s and the p polarized reflectance on the item surface of the film among Fig. 7 (b) by Fresnel equation:

...(71)

...(72)

Dependence to β, k derives from angle Ψ ₀, Ψ _{L, β, k}, emergence angle Ψ _{L, β, k}Pass through refractive index n _{L, k}And introducing k dependence.Similarly, substrate-membrane interface reflectivity is:

Notice that in the Fresnel equation, if incident is identical with the refraction angle, the reflectivity of then described two polarizations becomes 0.

For simple surface (no film), the sample surface reflectivity is identical with the top surface reflectivity:

Thus, the phase change (PCOR) of the reflection that is caused by surface reflection is

Notice that for satisfying boundary condition, s polarization in a single day reflection just can " reverse " (=dielectric π phase shift), and the p polarization can be not like this.It is meaningless fully that difference between the described polarization state becomes under the situation of vertical incidence, and in any case, this causes being removed by 0 in the Fresnel equation, and different equations is handled this limited case.

When using plural number part to use plus sige approximately regularly, absorb big more (plural part), then PCOR ω as refractive index _{β, k}Big more.In other words, bigger absorption coefficient is equivalent to reduce (decrease) of effective surface height.This produces visual sense---and the imagination is in light beam absorption during penetrable material before reflection, rather than in boundary (clean) reflection completely with transmit light.Agreement commonly used (wherein, the increase of surface elevation is corresponding to the just change of the phase differential between reference and the surface measurements) according to us deducts front surface PCOR from the interferometer phase place.

Film is the special circumstances of reflected in parallel.Light is by the top surface (referring to Fig. 7) of partial reflection, and continues to advance to and have the substrate surface that has with respect to second reflection of the phase delay of first reflection on it.Yet this is not the end of process.When returning,, thereby produce once more (south) to the south additional reflecting bundle towards substrate from the light quilt partial reflection once more of substrate reflection by top surface.This can continue in theory always, and wherein each additional reflection slightly weakens than last one.Suppose that all these multiple reflections exist,, so, obtain infinite sequence and be so that final surface reflectivity is made contributions:

β _1，β，k＝cos(Ψ，1 _β，k) (78)

As remarks clearly, recall β again _{1, β, k}The β dependence refer to refractive index n ₀The incident direction cosine β of surrounding medium ₀Dependence.Same equation (77) is applicable to described two polarization states with corresponding single surface reflectivity.

Inspection to these equatioies shows: what traditional FDA is treated to can be false when having film.Traditional FDA uses with broadband (in vain) light that generates the fourier space frequency expansion, by the linear fit of being composed by the Fourier phase of Fourier power spectrum weighting, and determines surface elevation.This thought is: phase evolution is from the linear phase dependence to the expection of surface elevation.By characterized systematically or by ignoring not those phase places contributions that change with the position, field simply, and eliminate any other systematic offset or the linear coefficient (for example, " chromatic dispersion ") that is associated with character of surface.

This plays perfect effect for simple surface.By nonpolarized light, also more likely by circularly polarized light, the wavelength dependency of PCOR almost is linear with wave number, and, be constant for given material.Yet in having the situation of film, traditional analysis is false.Phase place becomes non-linear, and phase slope becomes the film thickness sensitivity that changes crossing over visual field (field of view).Therefore, present analysis is passed through our knowledge (for example, how film is modulated the reflectivity on surface) of use, experimental data is compared with theoretical prediction, and determines the key parameter of surface structure, as film thickness.

Now, more how the storehouse (library) that we discuss experimental data and theoretical prediction provides the surface structure parameter, as film thickness and for the phase change (PCOR) that reflects.Under the situation of the film of unknown thickness, a lot of possible film thicknesses can be contained in the storehouse that is used for single surface type (for example, silicon dioxide on the silicon).In frequency domain embodiment, this thought is: to the coupling of this library searching to those characteristics of the FDA spectrum that the is independent of surfac topography difference structure of the amplitude spectrum that produces from the film interference effect (for example, with).Subsequently, computing machine uses this storehouse to compose and compensates the FDA data, thereby allows more accurate surface configuration mapping.

In one embodiment, this storehouse comprises the example FDA spectrum of surface structure, and each frequency spectrum provides as a series of complex coefficient ρ function of spatial frequency K, the expression fourier coefficient _KThese frequency spectrums are the intensity data I that obtains during the scanning ζ of interferometric optical path length _{ζ, h}Fourier transform.The angular wave number k=2 π/λ of the fragment of spatial frequency K and source spectrum, the refractive index n of surrounding medium ₀, and direction cosine β=cos (ψ) proportional, wherein, ψ is the incident angles of light shafts towards object surface:

K＝2βkn ₀ (79)

The ρ in prediction storehouse _KCoefficient comprises the optical properties on the surface of profile except surface elevation, that can influence the FDA spectrum.

Prediction FDA spectrum relate to expression light shafts irrelevant on the scope of the incident angle ψ of source light and angular wave number k and.As mentioned above, but the numerical integration abbreviation be: by factor Г _{K, k}In calculating weighting, on the N angular wave number k effectively (computationally efficient) single and:

Weighting factor is:

${\mathrm{\&Gamma;}}_{K,k}=\frac{{\mathrm{KU}}_{K,k}{V}_k}{4k^2{\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">n</figure-callout>}}_0^2}...\left(82\right)$

Wherein, V _kBe the source spectrum, and U _{K, k}For pupil plane light distributes.Corresponding standardization Υ be weighting factor on all spatial frequencys and.

Wherein, the standardization of Υ for will briefly defining, and H is the Heaviside step function.

As above described in detail, the distinctive characteristics of object surface structure (especially film) is by object path phase place ω _{K, k}With reflectivity Z _{K, k}Compose ρ and enter into _KIn.That of equal importance is the reference path phase place υ that depends on scanning interferometer meter self _{K, k}With reflectivity R _{K, k}As described further below, can determine such factor by the scanning interferometer meter being carried out theoretical modeling or calibrating it originally by test specimens with known attribute.

The typical prediction storehouse that is used for film is by a series of spectrum ρs of film thickness L as index _KThe spectrum of being stored only covers interested narrow spatial frequency zone (ROI), and usually, obtaining for 256 frame intensity datas is 15 or 16 values, and its residual value outside this ROI is 0.The definition of spatial frequency is followed in the restriction of ROI:

K ^min＝2β ^mink ^minn ₀ (84)

K ^min＝2β ^maxk ^maxn ₀ (85)

Typical range based on the spatial frequency of the scanning interferometer meter of 100X Mirau object lens and narrow bandwidth, 500-nm light source is 2.7 μ m ^-1To 4.0 μ m ^-1For counting yield, can use by 0.5 to 5nm tight look-up table between the sample frequency spectrum (dense look up table) as index, do not use equation (80)-(83) and each pixel is recomputated several times analysis search routine and do not use to relate to.

Library searching relates to following steps: (1) from the corresponding storehouse of particular surface type select prediction FDA spectrum, (2) the using character function calculates the tightness degree of this spectrum and experimental data coupling, repeat by concentrated some or all of database data (3) subsequently, provides optimum matching to determine which theoretical spectrum.We sought is " signature (signature) " that relates to uniquely such as in the frequency domain of the interactional character of surface of film, dissimilar material, step structure, roughness and they and interferometric optical system.Therefore, this relatively explicitly filtering: as a characteristic of the FDA spectrum that directly changes along with surfac topography also thereby, phase place incoherent with library searching with respect to the linear velocity of the change of spatial frequency.

When comparing frequency spectrum, separate phase place and amplitude contribution that quality is calculated and be good.Thus, in theory, we have:

P _K＝|ρ _K| (86)

φ _K＝connect _K[(ρ _K)] (87)

Wherein, connect _KFor eliminating φ _{K, h}The spatial frequency dependence in the function of 2-π step (step).For experimental data, we have:

$P_K^{\mathrm{ex}}=\left|q_{K,h}^{\mathrm{ex}}\right|...\left(88\right)$

${\mathrm{\&phi;}}_{K,h}^{\&prime;\&prime;\mathrm{ex}}=\mathrm{connec}{t}_K\left[\arg \left(q_{K,h}^{\mathrm{ex}}\right)\right]...\left(89\right)$

Be used for φ _{K, h} ^{" ex}Double quotation marks indication individual element and be directed to uncertainty in the striped rank of the starting point in the scanning generally.Experimental data must comprise the slope term that relates to the local surfaces height; This is to use the q symbol to substitute the reason of ρ symbol.

For test (trial) surface parameter of a particular group, we can calculate phase differential:

The presumptive test parameter is correct, then phase differential

(compensated) FDA phase place for compensation.Theoretical matched well with experiment produces phase place

, it is the simple linear function of the spatial frequency K under the situation in 0 intercept (that is 0 phase place gap) in principle.Thus, can reckon with that what we the dirtiest (downstream) offered traditional FDA analysis is the phase place that successfully compensates

, it is directly proportional that described traditional FDA analyzes slope and the surface elevation of supposition phase place in the frequency space.

There is compensation of phase in discussion based on first previous paragraphs Two interested features, it allows we and surface elevation irrespectively to obtain coupling theoretical and experiment.First feature is phase place gap A " or the K=0 values of intercept that obtains by linear fit

, and second feature is non-linear with respect to the remainder of wave number after linear fit.For example, Dui Ying quality function is:

${\mathrm{\&chi;}}_{\mathrm{\&phi;}}={[\frac{A^{\&prime;\&prime;}}{2\mathrm{\&pi;}}-\mathrm{round}\left(\frac{A^{\&prime;\&prime;}}{2\mathrm{\&pi;}}\right)]}^2---\left(91\right)$

${\mathrm{\&chi;}}_{\mathrm{\&phi;non}}=\frac{\underset{K>0}{\mathrm{\&Sigma;}}{({\mathrm{\&zeta;}}_{K,h}^{\&prime;\&prime;}-{\mathrm{\&sigma;}}_{h}K-A^{\&prime;\&prime;})}^2{P}_{K,h}^{\mathrm{ex}}}{\underset{K>0}{\mathrm{\&Sigma;}}{P}_{K,h}^{\mathrm{ex}}}---\left(92\right)$

Wherein, σ _hFor with the phase place that is compensated The slope of linear fit.Round () function in the equation (91) " is restricted to scope ± π with phase place gap A.

Although phase information can be only used in library searching, promptly by making quality functional value χ _φIn and/or χ _{φ non}In one or Minimize All and carrying out, but we also have signature important and useful in fourier modulus.The interested especially aspect of amplitude is: it is independent of surface elevation in itself.Thus, for example, we can be similar to the phase place quality and define following amplitude quality function similarly:

${\mathrm{\&chi;}}_P={\left[\frac{\underset{K>0}{\mathrm{\&Sigma;}}(P_{K,h}^{\mathrm{ex}}-P_{K,h})}{\underset{K>0}{\mathrm{\&Sigma;}}(P_{K,h}^{\mathrm{ex}}+P_{K,h})}\right]}^2---\left(93\right)$

${\mathrm{\&chi;}}_{\mathrm{Pnon}}=\frac{\underset{K>0}{\mathrm{\&Sigma;}}{({\mathrm{\&Omega;}}^{-1}{P}_{K,h}^{\mathrm{ex}}-P_{K,h})}^2}{\underset{K>0}{\mathrm{\&Sigma;}}{({\mathrm{\&Omega;}}^{-1}{P}_{K,h}^{\mathrm{ex}}+P_{K,h})}^2}---\left(94\right)$

Wherein, Ω is experiment scale factor (scaling factor):

$\mathrm{\&Omega;}=\underset{K>0}{\mathrm{\&Sigma;}}{P}_{K,h}^{\mathrm{ex}}/\underset{K>0}{\mathrm{\&Sigma;}}{P}_{K,h}---\left(95\right)$

Wherein, quality χ _PThe most relevant with the total reflectivity of object surface, irrelevant with the spatial frequency dependence, and χ _PnonRepresentation theory and experiment amplitude curve are in shape coupling good degree.

Amplitude quality function) χ _PWith/χ _PnonBe phase place quality χ _φAnd/or χ _{φ non}Replenish or even substitute.Therefore, common library searching quality function is:

χ＝w φ+χ φ+w φnonχ φnon+w Pχ P+w Pnonχ Pnon

...(96)

Wherein, w is a weighting factor.In principle, can under the situation of the standard deviation of knowing various parameters, determine weight in the equation (96).More experimental method is: the data to true and emulation are attempted various weights, and watch them and play effect how.For following Example, we select the equal weight w for all quality contributions _φ=w _{φ non}=w _P=W _Pnon=1.

Example among Fig. 8-13 illustrates the quality function search procedure for silicon dioxide film thickness on 6 silicon (that is, 0,50,100,300,600 and 1200nm) respectively.The single library that is used for all examples contain with 2-nm be at interval, the scope from 0 to 1500nm.These data are muting emulation.As in all examples of here describing, scanning step is 40nm, and source wavelength is 498nm, and source Gauss FWHM is 30nm (accurate monochromatic).

The most interested aspect of these emulation search is the state (behavior) of 4 quality functions.Usually, we observe: comprise that these 4 functions have helped to reduce the ambiguity (ambiguity) of final quality value, thereby have the strong periodicity for each quality value as the function of film thickness.What another was general is viewed as: the most effective based on nonlinear quality in phase place and amplitude at 300nm and above place, and peace equal amplitude in phase place gap is preponderated below the 300nm film thickness.This shows: χ _φ, χ _PThe quality function is particularly useful for real film, and it is for phase place gap and the direct-coupled characterized systematically of amplitude result and occupy consequence.

In case we have determined film thickness (or having discerned material and other use that is used for this algorithm), FDA handles and just carries out with usual way, and it still uses the FDA phase place of correction

, rather than original experiment phase data.In principle, if modeling is successful, then

Should not have non-linearly, and the phase place gap should be 0.Therefore, following step is to phase spectrum Linear fit.Use amplitude spectrum P _kSubstitute squared magnitude and for high NA FDA, seem more effective.This fits to each pixel slope is provided:

And intercept (phase place gap)

" the striped rank uncertainty that has from phase data is inherited and next double quotation marks to note phase place gap A.Slope σ _hThere is not this uncertainty." and the slope σ according to intercept A _h, we define " relevant profile (profile) " for certain sense or nominal spatial frequency K0

Θ _h＝σ _hK0 (99)

And " phase outline ":

θ " _h=Θ _h" (100) subsequently, we eliminate in phase theta+A _h" in by the striped rank uncertainty of pixel:

${\mathrm{\&theta;}}^{\&prime;}={\mathrm{\&theta;}}^{\&prime;\&prime;}-2\mathrm{\&pi;round}\left[\frac{A^{\&prime;\&prime;}-{\mathrm{\&alpha;}}^{\&prime;}}{2\mathrm{\&pi;}}\right]...\left(101\right)$

Wherein, α ' is original phase gap A " approach 2 π step-lengths of its no individual element.

At last, draw height profile from following equation:

h′＝θ′/K0 (102)

Note, there is no need to deduct phase deviation γ, this be because: it is generating compensation of phase

In time, finished.

First example (Figure 14) that surface configuration is measured is pure emulation.Surfac topography is 0 throughout, but to exist with 10nm be increment, from the 0 basic rete that develops into 1500nm.Use the prediction storehouse identical with Fig. 8-13, this test specification run through non-ambiguous scope, film thickness of predicting the storehouse and determine, although it is used for perfect noise free data.

Next example (Figure 15) also is emulation, but it has additional noise.Additive noise at random is the Gaussian noise with 2 standard deviations among average 128 intensity positions, and it appears as typical True Data.Although the reflectivity between silicon dioxide and the silicon has significant difference (4% pair 45%), described result obviously is gratifying.

Now, we are directed to characterized systematically and describe.

We use the data of collecting during the characterized systematically process to define phase deviation) γ _SysWith linear deviation (dispersion) τ _SysFor comprising the characterized systematically data, we use following equation, proofread and correct the experimental data q of Fourier transform before any other FDA processing that before the library searching and at individual element is the basis _{K, h} ^Ex:

$q_{K>0}^{\mathrm{ex}}=M^{-1}\exp [-i{\mathrm{\&gamma;}}_{\mathrm{sys}}-i(K-K0){\mathrm{\&tau;}}_{\mathrm{sys}}]q_{K>0}^{\mathrm{ex}}---\left(103\right)$

Wherein, K0 is the nominal spatial frequency, the nominal spatial frequency of the FDA data set that its expression is for example discerned by the mid point of surveying ROI.Notice that theoretical library remains unchanged.Scale-up factor M (Greek capitalization " M ") is new characterized systematically, and it makes and uses the object surface reflectivity to become possibility as the parameter in the library searching.

Can be with phase deviation γ as the function of position, field _SysWith system phase gap A _SysTame and docile the function that is stored as (field) position, and calculate real systems deviation according to following equation:

τ _Sys=(γ _Sys-A _Sys)/K0 (104) range coefficient M also depends on the field.

With with carry out the establishment of characterized systematically data for the similar mode of the mode of object samples as mentioned above.We transfer to the pseudomorphism with known features, measure it, and for perfect system is desired how different are arranged with us by watching the result, and determine characterized systematically.Especially, use it is pre-determined the known sample of correct storehouse clauses and subclauses, we generate as at the phase place gap A in the equation (98) " and as the final height h ' in the equation (102).Subsequently, suppose it is perfect dull and stereotyped pseudomorphism, our computing system phase deviation:

γ _sys＝K0h′ (105)

And system phase gap:

A _sys＝connect _xy(A″) (106)

Wherein, connect _Xy() is the phase unwrapping of individual element.Amplitude is mapped as:

$M_{\mathrm{sys}}=\underset{K>0}{\mathrm{\&Sigma;}}{P}_{K,h}^{\mathrm{ex}}/\underset{K>0}{\mathrm{\&Sigma;}}{P}_{K,h}...\left(107\right)$

In certain embodiments, may use to have on the scope of simple types and use (for example, silicon dioxide on the silicon) similarly surface structure, and some characterized systematicallies are averaged with final.

In above a lot of descriptions and emulation, we have been directed to the film surface structure and have described, yet, also can be with the complex surface structures of analytical applications in other type.Below, we show: analysis scan interferometry data how, and to consider surface structure less than the microscopical optical resolution of scanning interferometer.Optical resolution is finally collected the NA of optical device by the wavelength of light source and light and is limited.

Figure 16 a shows the light source that uses the 500-nm nominal wavelength, and according to the actual scanning interferometry data of every millimeter 2400 line (1pmm) grating and definite height profile, wherein, described grating has the peak of 120nm to paddy (PV) depth of modulation.Top profile among Figure 16 a shows the height profile that uses traditional FDA to analyze and determine.Traditional analysis indicates the only PV depth of modulation of about 10nm, and this has greatly underestimated actual depth of modulation.Because grating has the feature in the restriction of the optical resolution of 500-nm instrument, so this out of true occurs.Even the pixel resolution of the camera in the instrument also can be like this than enough accurate resolution grating is bigger.

A kind of mode of considering this influence is: when the adjacently situated surfaces position have surface characteristics with respect to enough sharp keen (sharp) of optical wavelength, with optical diffraction during to first pixel, also comprise contribution with the corresponding scanning interferometer measuring-signal that is used for first camera pixel in first surface position usually from those adjacently situated surfaces positions that add.From the surface elevation feature of those adjacently situated surfaces positions destroyed to the traditional analysis of the corresponding scanning interferometer measuring-signal in first surface position.

Yet, simultaneously, this means: comprise near the information of relevant complex surface feature with the corresponding scanning interferometer measuring-signal in first surface position.Figure 17 by illustrate from the scanning interferometer measuring-signal of the corresponding pixel of all places of relevant step height feature, and illustrate this situation.For the signal in (a), step height is on the right of pixel and higher, and for the signal in (b), step is directly by this pixel, and for the signal in (c), and step height is on the left side of pixel and lower.A signature that manifests immediately in described signal is: striped contrast is with respect to (a) and reducing (c) (b).For example, if 1/4th and location of pixels that step height equals wavelength definitely corresponding to the position of step height, so because disappear mutually each other definitely from the interference meeting of the both sides of this step, so, (b) in the striped contrast should wholely disappear.At (a) with in the signal (c) a lot of information are arranged also.For example, Figure 18 show respectively produce by near step height, the signal (a) of Figure 17 and the nonlinear distortion of frequency domain phase spectrum (c).In Figure 18, these spectrums are designated as (a) and (b) respectively.Under the situation that does not have step height, what the frequency domain phase spectrum will be for linearity.Thus, with the frequency domain phase spectrum of the surface location corresponding pixel adjacent with step height in nonlinear characteristic still comprise the information of relevant step height.

For there being such surface profile of measuring test surfaces under the situation of the surface characteristics of (under-resolved) below the resolution more accurately, we can use the above-mentioned library searching technology that is used for film.For example, for situation about having, generate a series of model FDA for the different value of PV depth of modulation and deviation post and compose at the test surfaces of differentiating following grating.As in the film example, the surface elevation of model spectra is maintained fixed.Subsequently, except not being comes parametrization representation model spectrum with film thickness but come parametrization to represent them with depth of modulation and deviation post, analysis is as in the upper film example and continue.Can use between the signature of the FDA spectrum of actual test surfaces and different model spectra relatively come to determine mate.Based on this coupling, eliminated existence by grating cause, for the distortion in the actual FDA spectrum of each pixel, so that can use traditional processing to determine the surface elevation of each pixel.The result who uses with such analysis of the above-mentioned identical quality function that is used for film has been shown among Figure 16 b and the 19b.

The library searching analysis of the grating of describing by reference Figure 16 a above Figure 16 b shows and uses that is used for every millimeter 2400 line and definite height profile.In Figure 16 a and Figure 16 b, used identical data, yet the PV depth of modulation that the library searching analysis will be used for grating is defined as 100nm, than handling the 10-nm result that determines more near the 120-nm depth of modulation of reality by traditional FDA among Figure 16 a.Figure 19 a and 19b show the similar analysis that is used for by discrete step height emulation and supposes the 500-nm light source of nominal.Figure 19 a shows with the true altitude profile (dotted lines) that is used for emulation and compares, uses traditional FDA to handle and definite height profile (solid line).Figure 19 b shows with the true altitude profile (dotted lines) that is used for emulation and compares, uses the library searching method and definite height profile (solid line).The parameter that is used for model spectra in library searching is position and step height amplitude.As illustrated, the library searching analysis makes lateral resolution improve about 0.5 micron to about 0.3 micron.

In above-mentioned labor, in frequency domain, taken place in the real data information and with different model information corresponding between comparison.In other embodiments, can in the scanning coordinate territory, carry out described comparison.For example, although the change in the absolute position of striped contrast envelope indicates the change in the surface elevation with the corresponding first surface of described signal position usually, but the shape of this signal (irrelevant with its absolute position) comprises the information of complex surfaces structure, as the surface structure of locational basal layer of first surface and/or adjacent position.

A simple situation is: the amplitude of considering striped contrast envelope self.For example, when film thickness with respect to the scope of the wavelength that produces by light source very hour, the interference effect that is produced by film becomes and Wavelength-independent, in this case, the amplitude of the direct modulation stripe contrast of film thickness envelope.So usually, the amplitude that striped can be contrasted is compared with the described amplitude that is used for the different corresponding models of film thickness, with the coupling (consider system contribution from interferometer self) of identification for specific film thickness.

Another simple situation is: the relative spacing of watching the zero crossing of striped by means of striped contrast envelope.For the graphic simple surface structure by the frequency distribution of symmetry, the relative spacing between the different zero crossings should be identical in nominal.Therefore, the variation of relative spacing indicates complex surfaces structure (when the system contribution considered from interferometer self), and it can be compared with the model that is used for different complex surface structures, with the coupling of identification and specific surface structure.

Another situation is: carry out this scanning field signal and with the corresponding scanning field signal of different test surfaces models between relevant.Usually, coupling is corresponding to having the relevant of peak-peak, and it indicates its scanning field signal has and the similar shapes of the shape of actual signal.Notice that usually, such analysis and surface elevation are irrelevant, this is because the surface elevation of actual sample and the difference between each mold surface height only make the offset of the peak value in the related function, and generally can not influence peak value self.On the other hand, in case discerned correct model,, and do not need further analysis (as traditional FDA) just the peak in the related function of correct model produces the surface elevation of test sample book.

As the analysis in spatial frequency domain, the analysis in the scanning coordinate territory can be used for a lot of dissimilar complex surfaces, and it not only comprises film, also comprises other complex surface, as above-mentioned surface elevation feature below differentiating.

Now, we describe the scanning coordinate library searching in detail and analyze, and it relates to the signal that is used for test sample book and is used for relevant between the respective signal of various test sample book models.

This method is cancelled the arbitrary assumption of relevant interference figure, rather than says: comprise for each pixel the identical basis that only is offset (also may being regulated in proportion again) on the position, local interference figure with all pixels of the corresponding data centralization of surface location with identical complex surface characteristic.The actual appearance that seems of signal, it is for Gaussian envelope or have the linear phase state or be not problem in any case in frequency domain.This thought is: the simple signal or the template that generate this local interference figure of expression for the different complex surface structures models that are used for tested object, and, for finding its local interference figure, each pixel mates the model of the shape of actual local interference figure best subsequently, and, find the interior scanning position of data set that the optimum matching between interference figure template and the observation signal (it provides surface elevation) is provided for model.Some technology can be used for pattern match.A kind of method is: it is relevant that each template and data are carried out mathematics.By being that each model uses multiple (that is, adding void in fact) stencil function, we recover two profiles, and one closely is associated with signal envelope, and another phase place with the underlying carrier signal is associated.

For example, in one embodiment, will comprise for the analysis of each pixel: (1) selects test template from the template base that the particular value for customized parameter (as film thickness) calculates or writes down; (2) use selected test template and correlation technique to find local surfaces height (its example is described below); (3), write down peak value quality functional value for selected test template based on correlation technique; (4) for the whole or subclass of the template in the storehouse and repeating step 1-3; (5) determine which test template provides optimum matching (=the highest peak value quality functional value); (6) write down the value (for example, film thickness) of customized parameter for the template of optimum matching; And (7) are invoked at the height value that the peak value matched position is provided in the data tracking again.

Now, we describe the correlation technique that is fit to based on multiple correlation.We generate the template interference figure according to each test surfaces model

Wherein, index j indication is used for the particular model of die plate pattern.Function m _Temp ^j(ζ) and _ _Temp ^j(ζ) characterize complex surface structures, but with irrelevant with the corresponding locational surface elevation of signal (it is set as 0).In a preferred embodiment, function m _Temp ^j(ζ) and _ _Temp ^j(ζ) also consider contribution from interferometric system.Subsequently, we use the complex representation of die plate pattern:

We further use window function to select the specific part of multiple stencil function:

${\tilde{I}}_{\mathrm{pat}}^k\left(\mathrm{\&zeta;}\right)=w\left(\mathrm{\&zeta;}\right){\stackrel{\&OverBar;}{I}}_{\mathrm{temp}}^k\left(\mathrm{\&zeta;}\right)...\left(111\right)$

For example, suitable window can be:

${\mathrm{\&zeta;}}_{\mathrm{start}}=-\frac{\mathrm{\&Delta;\&zeta;}}{2}$

${\mathrm{\&zeta;}}_{\mathrm{stop}}=+\frac{\mathrm{\&Delta;\&zeta;}}{2}$ (112)

Wherein, window width Δ ζ can manually be provided with.

Now, we have had the interference figure template

We prepare to use it to come to compare with the real data collection.When this is prepared, generate the complex signal that begins from real experimental data collection

Will be easily:

I _ex(ζ，x)＝D _ex(x)+...

..AC _ex(x)m _ex[ζ-h _ex(x)]cos{-[ζ-he _x(x)]K ₀+_ _ex[ζ-h _ex(x)]}

(113)

The Fourier transform of this signal is

q _ex(K，x)＝FT{I _ex(ζ，x)} (114)

$q_{\mathrm{ex}}(K,x)=\mathrm{\&delta;}\left(K\right)D{C}_{\mathrm{ex}}\left(x\right)+\frac{1}{2}A{C}_{\mathrm{ex}}\left(x\right)[G_{\mathrm{ex}}^{*}(-K-{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">K</figure-callout>}}_0,x)+G_{\mathrm{ex}}(K-{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">K</figure-callout>}}_0,x)]...\left(115\right)$

Wherein,

G _ex(K)＝FT{m _ex(ζ)exp[i_ _ex(ζ)]}exp[iKh _ex(x)] (116)

Subsequently, we are from the positive frequency section construction partial frequency spectrum of this frequency spectrum:

${\tilde{q}}_{\mathrm{ex}}\left(K\right)={\mathrm{AC}}_{\mathrm{ex}}\left(x\right)G_{\mathrm{ex}}(K-{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="A20048001232300611.png,A20048001232300621.png"\; state="\{\{state\}\}">K</figure-callout>}}_0,x)...\left(117\right)$

Subsequently, inverse Fourier transform is:

${\tilde{I}}_{\mathrm{ex}}\left(\mathrm{\&zeta;}\right)={\mathrm{FT}}^{-1}\left\{{\tilde{q}}_{\mathrm{ex}}\left(K\right)\right\}...\left(118\right)$

Here, this complex function

Real part be original experimental data I _ExIn addition, can be by shirtsleeve operation discrete phases and envelope, for example, we can use complex function

Amplitude, obtain signal intensity AC _Ex(x) and envelope m _ExProduct:

${\mathrm{AC}}_{\mathrm{ex}}\left(x\right)m_{\mathrm{ex}}[\mathrm{\&zeta;}-h_{\mathrm{ex}}\left(x\right)]=\left|{\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;},x)\right|...\left(120\right)$

According to the basic theory of this technology, we estimate m _ExA significant at least part have and be used for the m of correct model _Temp ^jIdentical general shape, only difference are linear deflection h _ExWith scale factor AC _Ex(x).We also estimate: for correct model, and experiment and the phase deviation of interference figure template _ _Ex, _ _Pat ^jBetween difference not with height h _ExProportional linearly.

Task on the horizon is: at the experimental data collection The interior detection by the interference figure template

And the particular signal pattern of expression, and determine there is the coupling of which kind of degree for each different model j.Hereinafter, we will cancel index j, and note, continue The matching analysis for each model.

First step is: the shape m that finds envelope _Ex, m _Pat, and _ _Ex, _ _PatScanning position ζ with its optimum matching _BestA kind of feasible method is: based on the quality function of the standardization relevant (correlation) of the signal in interference figure template and the scanning fragment that defines by following window w:

$\mathrm{\&Pi;}(\mathrm{\&zeta;},x)=\frac{{\left|\tilde{I}(\mathrm{\&zeta;},x)\right|}^2}{<m_{\mathrm{pat}}^2><{\left|{\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;},x)\right|}^2>}...\left(121\right)$

Wherein,

$\tilde{I}(\mathrm{\&zeta;},x)=\frac{1}{N}\underset{-\&infin;}{\overset{\&infin;}{\&Integral;}}{\tilde{I}}_{\mathrm{pat}}^{*}\left(\widehat{\mathrm{\&zeta;}}\right){\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;}+\widehat{\mathrm{\&zeta;}},x)d\widehat{\mathrm{\&zeta;}}...\left(122\right)$

Be compound correlative function, and,

$<{\mathrm{<figure-callout\; id="2"\; label="m\; pat"\; filenames="A20048001232300561.png,A20048001232300571.png"\; state="\{\{state\}\}">m</figure-callout>}}_{\mathrm{pat}}^{}>=\frac{1}{N}\underset{-\&infin;}{\overset{\&infin;}{\&Integral;}}{\left|{\tilde{I}}_{\mathrm{pat}}\left(\widehat{\mathrm{\&zeta;}}\right)\right|}^{2}d\widehat{\mathrm{\&zeta;}}...\left(123\right)$

$<{\left|{\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;},x)\right|}^2>=\frac{1}{N}\underset{-\&infin;}{\overset{\&infin;}{\&Integral;}}{\left|{\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;}+\widehat{\mathrm{\&zeta;}},x)\right|}^{2}w\left(\widehat{\mathrm{\&zeta;}}\right)d\widehat{\mathrm{\&zeta;}}...\left(124\right)$

For producing the standardization of the quality function П that has nothing to do with signal intensity.For _ _Ex, _ _PatThe situation of coupling, the complex conjugate of template

The use cancellation synchronous linear phase term K ₀ζ, and make the П maximization.The absolute value that should be correlated with || eliminated any remaining complex phase position.

Run into singular point for the high value that prevents П (ζ) generation error or at the low-signal levels place, it is careful adding minimum value to denominator, as

$<{\left|{\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;},x)\right|}^2>\&LeftArrow;<{{\tilde{I}}_{\mathrm{ex}}(\mathrm{\&zeta;},x)|}^2>+\mathrm{MinDenom}\&CenterDot;\max (<{\left|{\tilde{I}}_{\mathrm{ex}}\right|}^2>)...\left(125\right)$

Wherein, max () function return signal intensity

Maximal value on full scan length ζ, and MinDenom thinks effective minimum relative signal intensity in quality function search for us.Can be 5% or certain other little value with the value hard coded (hard code) of MinDenom, or it is left adjustable parameter.

Also can use correlation theorem, in frequency domain, carry out correlation integral

$\tilde{I}\left(\mathrm{\&zeta;}\right)={\mathrm{FT}}^{-1}\left\{{\tilde{q}}_{\mathrm{pat}}^{*}\left(K\right){\tilde{q}}_{\mathrm{ex}}\left(K\right)\right\}...\left(126\right)$

Wherein, I has utilized

$\mathrm{FT}\left\{{\tilde{I}}_{\mathrm{pat}}^{*}(\mathrm{\&zeta;},x)\right\}={\tilde{q}}_{\mathrm{pat}}^{*}(-K,x)...\left(127\right)$

Wherein,

${\tilde{q}}_{\mathrm{pat}}(K,x)=\mathrm{FT}\left\{{\tilde{I}}_{\mathrm{pat}}(\mathrm{\&zeta;},x)\right\}...\left(128\right)$

Find the search of peak value to produce best match position ζ by П _Best, and the value of П is the measurement of the quality of match of scope from 0 to 1, wherein 1 corresponding to Perfect Matchings.Being the peak value of each different Model Calculation quality function, is optimum matching with definite which model, and, subsequently, be used for the best match position ζ of this model _BestProvide surface elevation.

Figure 20-24 illustrates the example of this technology.Figure 20 shows the actual scanning interferometry signal of the basic silicon substrate of no film.Figure 21 and 22 shows the interference die plate pattern of membrane structure that is used for the bare silicon substrate and has 1 micron silicon dioxide on silicon respectively.Figure 23 and 24 shows the quality function as the function of the scanning position of the stencil function in Figure 21 and 22 respectively.The quality function shows: the interference die plate pattern (peak value is 0.92) that is used for exposed substrate is mated much betterly than the interference die plate pattern that is used for the thin-film template pattern (peak value is 0.76), and therefore indicated: test sample book is exposed substrate.In addition, the peak that is used for the quality function of correct die plate pattern has provided the apparent surface's height and position that is used for test sample book.

Said method and system are particularly useful in semiconductor application.Additional embodiment of the present invention comprises: use above-mentioned any measuring technique and be directed to following any semiconductor application; And the system that is used to carry out measuring technique and semiconductor application.

At present very interested in semicon industry is the measures of quantization that carries out surfac topography.Because the small size of typical chip feature, typically, the instrument that is used to carry out these measurements must have at the high spatial resolution that is parallel and perpendicular on both directions of chip surface.Slip-stick artist and scientist use the surface configuration measuring system to carry out process control, and detect the defective that occurs in manufacture process, particularly by the defective that causes such as etching, polishing, cleaning and patterning.

For being useful especially process control and defects detection, the surfac topography measuring system should have the lateral resolution that can compare with the lateral dimension of typical surface characteristics and the longitudinal frame that can compare with the minimal surface step height that is allowed.Typically, this need be less than 1 micron lateral resolution and less than the longitudinal frame of 1 nanometer.And, preferably, for such system, on contact chip surface not or apply thereon under the situation of potential destructive power and carry out the measurement of system, so that avoid changing the surface or introduce defective.In addition, as is known, the effect of a lot of processes of using in chip manufacturing depends on local factor consumingly, as the pattern density and the edge degree of approach, for the surface configuration measuring system, it also is important having the ability of carrying out intensive sampling on high measurement handling capacity and the large tracts of land in the zone that may comprise one or more interested surface characteristics.

In chip maker, the electrical interconnection of using so-called " dual damascene copper (dual damascene copper) " process to make between the different piece of chip is just becoming usual.This is an example that can use the process that suitable surfac topography system effectively characterizes.The dual damascene process can be considered to have 5 parts: (1) interlayer dielectric (ILD) deposition (deposition) wherein, is deposited to the wafer surface of (comprising a plurality of independently chips) with dielectric substance (as polymkeric substance or glass); (2) chemically mechanical polishing (CMP) wherein, is polished dielectric layer, is suitable for the typographic level and smooth surface of precise optical so that create; The combination of (3) photoetching (lithographic) patterning and retroaction ion etching (reactive ion etch) step, wherein, create complicated network, it comprises the narrow raceway groove (trench) that is parallel to wafer surface and extends, and the cat walk that extends to lower (previous definition) electrically-conductive layer from the bottom of raceway groove, (4) produce the combination of being crossed the metal deposition step of the raceway groove that is full of (over-filled) and path by copper, and (5) last chemically mechanical polishing (CMP) step, wherein, remove too much copper, stay the network of the raceway groove that is filled up by copper (and possible path) round dielectric substance.

Typically, in the scope that the thickness of the copper in the channel region (that is channel depth) and dielectric on every side thickness drop on 0.2 to 0.5 micron.The width of resulting raceway groove can be in the scope of 100 to 100000 nanometers, and, the copper zone in each chip can be in some zones the pattern (as the array of parallel lines) of formation rule, and in other zone, they can not have the pattern that manifests.Equally, in some zones, can be to covering copper zone thick and fast, surface, and in other zone, the copper zone may be very sparse.Understand polishing rate and after polishing thereby remaining copper (and dielectric) thickness is strong and depend on polishing condition (as stuffing pressure (pad pressure) and brilliant polish (polishing slurry) composition) and copper and the partial detailed arrangement of dielectric area on every side (that is, towards, the degree of approach and shape) in the mode of complexity is important.

This " depends on the polishing rate of position " and has been considered to cause the variable surface configuration on a lot of lateral length yardsticks.For example, it may mean: be positioned near the chip at the edge of the wafer on the condensate and more promptly polished with being positioned to compare near those chips at center, thereby created than near thin and more desired than the center thick copper the zone desired edge.This is the example of " wafer scale (wafer scale) " process unevenness, that is, its can with length dimension that wafer diameter is compared on occur.Zone with high-copper gully density also is known to polish with higher speed than the near zone with low copper cash density.This causes being called as the phenomenon of " corrosion that CMP lures into " in the high-copper density area.This is the example of " chip dimension (chip scale) " process unevenness, that is, it occurs can comparing with the linear dimension of single chip on the length dimension of (and, sometimes much smaller than this linear dimension).(it causes comparing with dielectric substance on every side, with higher speed polishing) appears in the chip dimension unevenness that is called as another type of " depression (dishing) " in the channel region that single copper is filled.For greater than several microns raceway groove, depression may become seriously, and it makes and demonstrate too much resistance after the affected line, and causes failure of chip.

Wafer that CMP lures into and chip processes unevenness are difficult to prediction in itself, and they submit to the temporal change as the condition in the development of CMP disposal system.For for guaranteeing that any unevenness remains on that purpose in the acceptable limit is monitored effectively and adjustment process condition suitably, for the process engineer, it is important that a large amount of and extensive multiple position on chip produces frequent noncontact surfac topography measurement.This utilizes the embodiment of above-mentioned interference measuring technique and becomes possibility.

Can hardware or software or described both combination realize above-mentioned any computer analysis method.Can follow describing method and figure here, use the computer program of standard program technology to realize described method.Program code is applied to import data, carrying out function described herein, and generates output information.Output information is applied to one or more output devices, as display monitor.Can level process or Object-Oriented Programming Language realize each program, to communicate by letter with computer system.Yet, if expectation can be collected or machine language realizes described program.In any case, this language can be compiling or interpreted language.In addition, this program can be moved on the special IC of programming in advance for this purpose.

Preferably, the computer program that each is such (for example is stored in the storage medium that can be read by the programmable calculator of universal or special purpose or equipment, ROM or disk) on, so that when reading this storage medium or equipment by computing machine the configuration and operate this computing machine, to carry out process described herein.Computer program also can reside in during program run in cache memory or the primary memory.Also analytical approach can be embodied as the computer-readable recording medium that disposes by computer program, wherein, be configured to make computing machine to operate this storage medium, to carry out function described herein in specific and predefined mode.

A large amount of embodiment of the present invention has been described.Yet, will understand, can under the situation that does not deviate from the spirit and scope of the present invention, make various modifications.

Claims (77)

1, a kind of method, it comprises:

The information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics that are used for tested object with described a plurality of model parameterizations,

Wherein, comprise with a plurality of model information corresponding: with the information of at least one amplitude components of the conversion of corresponding, the relevant scanning interferometer measuring-signal of the model of each tested object.

2, the method for claim 1 also comprises: the precise characteristics that is identified for tested object based on described comparison.

3, the method for claim 1 also comprises: determine the apparent surface's height for the first surface position based on described comparison.

4, method as claimed in claim 3, wherein, the determining of apparent surface height comprises: determine which model corresponding to the precise characteristics in the characteristic of tested object based on described comparison, and use is calculated the apparent surface highly with the corresponding model of precise characteristics.

5, method as claimed in claim 4 wherein, comprises with the use of the corresponding model of precise characteristics: compensation is from the data of scanning interferometer measuring-signal, with the contribution that reduces to produce from this precise characteristics.

6, method as claimed in claim 5, wherein, compensation to data comprises: eliminate the phase place contribution that produces from this precise characteristics from the phase component for the conversion of the scanning interferometer measuring-signal of tested object, and, wherein, also comprise with the use of the corresponding model of precise characteristics: after eliminating the phase place contribution that produces from this precise characteristics, calculate apparent surface's height according to the phase component of this conversion.

7, the method for claim 1 also comprises: the information that can derive from the scanning interferometer measuring-signal that is used for the additional surfaces position and compare with a plurality of model information corresponding.

8, method as claimed in claim 7 also comprises: the surface elevation profile that is identified for tested object based on described comparison.

9, the method for claim 1, wherein describedly relatively comprise: calculate indicate the information that can derive from the scanning interferometer measuring-signal and and each model information corresponding between one or more quality functions of similarity.

10, the method for claim 1, wherein describedly relatively comprise: information that can derive from the scanning interferometer measuring-signal and expression match mutually corresponding to the information of model.

11, the method for claim 1, wherein can comprise: the information of at least one amplitude components of the conversion of relevant scanning interferometer measuring-signal for tested object from the information that the scanning interferometer measuring-signal is derived.

12, method as claimed in claim 11 wherein, describedly relatively comprises: the relative intensity of at least one amplitude components of tested object is compared with the relative intensity of at least one amplitude components of each model.

13, be the function that is used for the coordinate of conversion the method for claim 1, wherein with a plurality of model information corresponding.

14, method as claimed in claim 13 wherein, comprises with a plurality of model information corresponding: the amplitude profile that is used for the conversion of each model.

15, method as claimed in claim 14 wherein, describedly relatively comprises: will for the amplitude profile of the conversion of the scanning interferometer measuring-signal of tested object with for each amplitude profile phase of model relatively.

16, method as claimed in claim 14 wherein, describedly more also comprises: will for the information in the phase outline of the conversion of the scanning interferometer measuring-signal of tested object with compare for the information in the phase outline of the conversion of each model.

17, method as claimed in claim 16, wherein, the information in the phase outline comprises: phase outline is non-linear with respect to coordinate transforming.

18, method as claimed in claim 16, wherein, the information in the phase outline comprises: relate to the phase place gap width with tested object and each model.

19, the method for claim 1, wherein described a plurality of models are corresponding to the fixed surface height of the tested object at primary importance place.

20, the method for claim 1, wherein described series of characteristics comprises: a series of values of at least one physical parameter of tested object.

21, method as claimed in claim 20, wherein, described tested object comprises the thin layer with thickness, and described physical parameter is the film thickness at primary importance place.

22, the method for claim 1, wherein described series of characteristics comprises the series of characteristics of tested object in the second surface position that is different from the first surface position.

23, method as claimed in claim 22, wherein, tested object comprises: the structure in the second surface position of making contributions with optical diffraction and to the scanning interferometer measuring-signal that is used for the first surface position.

24, method as claimed in claim 22, wherein, the series of characteristics of second surface position comprises: in the arrangement of the amplitude of the step height at second place place and the location of the second place.

25, method as claimed in claim 22, wherein, the series of characteristics of second surface position comprises: be used for the arrangement of depth of modulation of grating and the deviation post of grating, wherein, grating extends on the second place.

26, the method for claim 1, wherein described series of characteristics is a series of surfacings that are used for tested object.

27, the method for claim 1, wherein described series of characteristics is a series of superficial layer configurations that are used for tested object.

28, the method for claim 1, wherein described Fourier transform that is transformed to.

29, the method for claim 1, wherein produce the scanning interferometer measuring-signal by the scanning interferometer measuring system, and, wherein, describedly relatively comprise: consider to produce by the scanning interferometer measuring system, to system's contribution of scanning interferometer measuring-signal.

30, method as claimed in claim 29 also comprises: use another tested object with known attribute, and the contribution of the system of calibration scan interferometer measuration system.

31, the method for claim 1, wherein, by to sending from tested object with on the detecting device and the reference light test light of interfering is carried out imaging and the optical path length from the common source to the detecting device that changes between the interference portion of test and reference light is poor, produce the scanning interferometer measuring-signal, wherein, derive test and reference light from common source, and, wherein, the scanning interferometer measuring-signal corresponding to: when the optical path length difference changes, by the interference strength of detectors measure.

32, method as claimed in claim 31 also comprises: produce the scanning interferometer measuring-signal.

33, method as claimed in claim 31, wherein, test and reference light have greater than test and 5% bands of a spectrum of the centre frequency of reference light wide.

34, method as claimed in claim 31, wherein, common source has the spectrum coherent length, and the optical path length difference changes on the scope greater than the spectrum coherent length, to produce the scanning interferometer measuring-signal.

35, method as claimed in claim 31, wherein, be used for test light is directed on the tested object and with its optical device definition that is imaged onto detecting device greater than numerical aperture 0.8, that be used for test light.

36, method as claimed in claim 32, wherein, described common source is the source that extend in the space.

37, a kind of equipment, it comprises:

Computer-readable medium, it has information that the processor that makes in the computing machine can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and the program of comparing with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations

Wherein, comprise with a plurality of model information corresponding: with the information of at least one amplitude components of the conversion of one of a plurality of models of tested object corresponding, relevant scanning interferometer measuring-signal.

38, a kind of equipment, it comprises:

The scanning interferometer measuring system, it is configured to produce the scanning interferometer measuring-signal; And

Electronic processors, it is coupled to the scanning interferometer measuring system, to receive the scanning interferometer measuring-signal, and be programmed to the information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations

Wherein, comprise with a plurality of model information corresponding: with the information of at least one amplitude components of the conversion of one of a plurality of models of tested object corresponding, relevant scanning interferometer measuring-signal.

39, a kind of method, it comprises:

The information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object,

Wherein, described a plurality of models are corresponding with the fixed surface height of the tested object that is used for the first surface position, and a series of characteristics that are used for tested object by being different from the fixed surface height and by parametrization.

40, method as claimed in claim 39 also comprises: the precise characteristics that is identified for tested object based on described comparison.

41, method as claimed in claim 39 also comprises: determine the apparent surface's height for the first surface position based on described comparison.

42, method as claimed in claim 41, wherein, the determining of apparent surface height comprises: determine which model corresponding to the precise characteristics in the characteristic of tested object based on described comparison, and use is calculated the apparent surface highly with the corresponding model of precise characteristics.

43, method as claimed in claim 42 wherein, comprises with the use of the corresponding model of precise characteristics: compensation is from the data of scanning interferometer measuring-signal, with the contribution that reduces to produce from this precise characteristics.

44, method as claimed in claim 43, wherein, compensation to data comprises: eliminate the phase place contribution that produces from this precise characteristics from the phase component for the conversion of the scanning interferometer measuring-signal of tested object, and, wherein, also comprise with the use of the corresponding model of precise characteristics: after eliminating the phase place contribution that produces from this precise characteristics, calculate apparent surface's height according to the phase component of this conversion.

45, method as claimed in claim 42, wherein, use and to calculate the apparent surface with the corresponding model of precise characteristics and highly comprise: determine to be used for and to be used for the information of tested object and the position of the peak value of the related function that is used for comparing with the information of the corresponding model of precise characteristics.

46, method as claimed in claim 39 also comprises: the information that can derive from the scanning interferometer measuring-signal that is used for the additional surfaces position and compare with a plurality of model information corresponding.

47, method as claimed in claim 46 also comprises: the surface elevation profile that is identified for tested object based on described comparison.

48, method as claimed in claim 39 wherein, describedly relatively comprises: calculate indicate the information that can derive from the scanning interferometer measuring-signal and and each model information corresponding between one or more quality functions of similarity.

49, method as claimed in claim 39 wherein, describedly relatively comprises: information that can derive from the scanning interferometer measuring-signal and expression match mutually corresponding to the information of model.

50, method as claimed in claim 39, wherein, the information that can derive and just be compared from the scanning interferometer measuring-signal is a number.

51, method as claimed in claim 39, wherein, the information that can derive and just be compared from the scanning interferometer measuring-signal is function.

52, method as claimed in claim 51, wherein, described function is the function of spatial frequency.

53, method as claimed in claim 51, wherein, described function is the function of scanning position.

54, method as claimed in claim 39 wherein, can be transformed to the conversion of spatial frequency domain according to the scanning interferometer measuring-signal that will be used for tested object, and derives the information that is used for tested object.

55, method as claimed in claim 54, wherein, the described Fourier transform that is transformed to.

56, method as claimed in claim 54, wherein, the information that is used for tested object comprises the information of the amplitude profile of relevant conversion.

57, method as claimed in claim 54, wherein, the information that is used for tested object comprises the information of the phase outline of relevant conversion.

58, method as claimed in claim 39, wherein, the information that is used for tested object is relevant with shape at the scanning interferometer measuring-signal that is used for tested object at primary importance place.

59, method as claimed in claim 58, wherein, the information that is used for tested object is relevant with the striped contrast amplitude of the shape of scanning interferometer measuring-signal.

60, method as claimed in claim 58, wherein, the information that is used for tested object is relevant with relative spacing between the zero crossing of the shape of scanning interferometer measuring-signal.

61, method as claimed in claim 58, wherein, the information representation that will be used for tested object is the function of scanning position, wherein, derives described function from the shape of scanning interferometer measuring-signal.

62, method as claimed in claim 39 wherein, describedly relatively comprises: calculate and be used for the information of tested object and be used for related function between the information of each model.

63, method as claimed in claim 62, wherein, described related function is a compound correlative function.

64, method as claimed in claim 62 wherein, describedly more also comprises: determine the one or more peak values in each related function.

65, as the described method of claim 64, also comprise: based on the precise characteristics of determining tested object with the parametrization of the corresponding model of peak-peak.

66, as the described method of claim 64, also comprise: apparent surface's height of determining the tested object of first surface position based on the coordinate of at least one peak value in the related function.

67, method as claimed in claim 39 wherein, produces the scanning interferometer measuring-signal by the scanning interferometer measuring system, and, wherein, describedly relatively comprise: consider to produce by the scanning interferometer measuring system, to system's contribution of scanning interferometer measuring-signal.

68, as the described method of claim 67, also comprise: use another tested object with known attribute, and the contribution of the system of calibration scan interferometer measuration system.

69, a kind of equipment, it comprises:

Computer-readable medium, it has information that the processor that makes in the computing machine can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and the program of comparing with a plurality of model information corresponding of tested object,

Wherein, described a plurality of models are corresponding with the fixed surface height of the tested object that is used for the first surface position, and a series of characteristics that are used for tested object by being different from the fixed surface height and by parametrization.

70, a kind of equipment, it comprises:

The scanning interferometer measuring system, it is configured to produce the scanning interferometer measuring-signal; And

Electronic processors, it is coupled to the scanning interferometer measuring system, to receive the scanning interferometer measuring-signal, and be programmed to the information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object

Wherein, described a plurality of models are corresponding with the fixed surface height of the tested object that is used for the first surface position, and a series of characteristics that are used for tested object by being different from the fixed surface height and by parametrization.

71, a kind of method, it comprises:

The information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics that are used for tested object with described a plurality of model parameterizations,

Wherein, describedly relatively comprise: consider to produce by the scanning interferometer measuring system that is used for producing the scanning interferometer measuring-signal, to system's contribution of scanning interferometer measuring-signal.

72, as the described method of claim 71, wherein, the contribution of described system is: relevant information for the deviation from the phase change of the reflection of the assembly of scanning interferometer measuring system.

73, as the described method of claim 71, also comprise: the information that can derive from the scanning interferometer measuring-signal that is used for the additional surface position and compare with a plurality of model information corresponding.

74, as the described method of claim 73, wherein, can be for a plurality of in the surface location and the contribution of resolution system.

75, as the described method of claim 71, also comprise: use another tested object with known attribute, and the contribution of the system of calibration scan interferometer measuration system.

76, a kind of equipment, it comprises:

Computer-readable medium, it has information that the processor that makes in the computing machine can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and the program of comparing with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations

Wherein, described program also make that processor was considered to be produced by the scanning interferometer measuring system that is used for producing the scanning interferometer measuring-signal between the described comparable period, to system's contribution of scanning interferometer measuring-signal.

77, a kind of equipment, it comprises:

The scanning interferometer measuring system, it is configured to produce the scanning interferometer measuring-signal; And

Electronic processors, it is coupled to the scanning interferometer measuring system, to receive the scanning interferometer measuring-signal, and be programmed to the information that can derive from the scanning interferometer measuring-signal of the first surface position that is used for tested object and compare with a plurality of model information corresponding of tested object, wherein, by a series of characteristics of tested object that are used for described a plurality of model parameterizations

Wherein, described electronic processors also be programmed with between the described comparable period, consider to produce by the scanning interferometer measuring system that is used for producing the scanning interferometer measuring-signal, to system's contribution of scanning interferometer measuring-signal.

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