Method for detecting and correcting lens distortion in an electron diffraction system

Abstract

A method for correcting distortions in a coherent electron diffraction imaging (CEDI) image induced by a projection lens utilizes a known secondary material that is imaged along with a sample of interest. Reflections produced from the secondary material are located in the image, and these observed reflections are used to approximate the beam center location. Using the known lattice structure of the secondary material, Friedel pairs are located in the image, and unit cell vectors are identified. A predicted location is then determined for each of the secondary material reflections, and the location difference between the observed and predicted reflections is used to construct a relocation function that can be applied to the overall image. The relocation function is then used to correct the location of the image components to correct for the distortions.

Description

[Technical field]

[0001] The present invention relates generally to the field of electron diffraction, and more specifically to correcting lens distortion in electron diffraction systems. [Background technology]

[0002] 2. Description of Related Art Coherent electron diffraction imaging (CEDI) is a technique for 2D or 3D reconstruction of nanoscale objects such as nanotubes, nanocrystals, and proteins. A highly coherent beam of electrons is incident on the sample, which results in a diffraction image that is recorded using an active pixel array sensor. The sample is supported by a material that is highly transparent to electrons in the desired energy range of 10-30 keV, such as a thin film of graphene or silicon nitride. In some cases, the sample can be self-supported, such as a free-standing carbon nanotube. The entire beam path from the electron gun to the detector is located in high or ultra-high vacuum.

[0003] In a typical CEDI system, the electron beam path contains electrostatic and magnetic lenses that shape the beam and determine its properties such as beam size and beam divergence, while minimizing chromatic and spherical aberrations that degrade beam coherence and prevent the formation of a diffraction image. On the secondary side (after diffraction by the sample), a magnetic projection lens is used to magnify the diffraction image on the detector. However, distortions are often introduced into the CEDI image by the projection lens and need to be corrected before reconstruction is attempted.

[0004] In conventional CEDI systems, samples are reconstructed from diffraction data using several recursive methods, such as Error Reduction (ER) and Hybrid-Input-Output (HIO). These dual-space methods alternate between reciprocal and real space, and reconstruction depends on the accuracy of the data. Simulations show that perturbations introduced by projection lens distortions can often prevent successful reconstruction. Figure 1A shows a CEDI image of a double-walled carbon nanotube (DWCNT) along the simulated tube axis. Figure 1B shows a successful reconstruction of a DWCNT from simulated data. However, Figures 1C-1F show a different failed reconstruction using simulated data after applying projection lens distortions from experimental CNT data.

[0005] In lens distortions of optical systems, a classical distinction is made between pincushion, barrel, and spiral distortions. Examples of such distortions are given in GC Capitani, et al., A practical method to detect and correct for lens distortion in the TEM, Ultramicroscopy 106, 66 (2006), which are reproduced in 2A-2E of this specification. Figure 2A represents an undistorted image of an object. Figure 2B shows how pincushion-type distortion is accompanied by a magnification that increases with the distance from the center and depends on the direction in which it is measured. Figure 2C shows barrel-type distortion, where the magnification decreases with the distance from the center. Figure 2D represents spiral distortion, where a line is imaged as a sigmoid shape. Finally, Figure 2E shows an elliptical distortion, where the magnification differs in the radial direction. For pincushion and barrel distortions, the magnitude of the distortion is proportional to the radial distance from the beam center. In its simplest form (represented by equation (1) below), the distortion scales linearly with distance, but higher-order dependencies are common as well.

[0006]

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[0007] In the aforementioned study by Capitani et al., it was determined that the most relevant distortion in transmission electron microscopy (TEM) is elliptical distortion, which is described by the following equation (2):

[0008]

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[0009] Detection and correction of projection lens distortion requires a pattern of fiducial markers from which the coefficients of radial and elliptical distortion can be derived. used diffraction patterns from 3D crystals such as vesuvianite to detect and correct lens distortion in a TEM before the actual sample was introduced. However, in modern CEDI instruments, additional measures are required to minimize the detrimental effects of such lens distortion. Summary of the Invention

[0010] In accordance with the present invention, a method is provided for correcting distortions imparted to a diffraction image of a sample of interest by a projection lens in a coherent electron diffraction imaging (CEDI) system. The method involves first acquiring a CEDI image of the sample of interest along with a secondary material having a known lattice structure. The observed positions of reflections corresponding to the secondary material are then located within the image. Based on the relative positions of the observed reflections, and by knowing the nature of the secondary material, a unit cell vector associated with the secondary material is then identified.

[0011] Using the unit cell vectors, predicted positions of each of the secondary material reflections are determined, which correspond to respective Bravais-Miller indices assigned to each of the reflections based on the unit cell vectors and the known lattice structure of the secondary material. The difference between the observed positions and the predicted positions of the secondary material reflections is then used to relocate the position of the image components within the CEDI image.

[0012] In an exemplary embodiment of the invention, the secondary material is graphene, although other materials can be used. This embodiment also involves identifying the unit cell vector of the secondary material by first finding the difference vectors of pairs of observed CEDI reflections that correspond to the secondary material. These difference vectors are then grouped according to length and direction, and the vectors of each group are averaged. The shortest averaged group vector with similar length is then selected as the unit cell vector. The predicted location of each secondary material reflection can then be determined by finding a linear combination of the unit cell vectors that correspond to locations in the image that are proximate to the observed location of that reflection.

[0013] In an exemplary embodiment, the observed and predicted locations of the secondary material reflections are found relative to the beam center location in the image. The beam center location may be determined by averaging the centroids of the reflection pairs using the observed locations of the secondary material reflections. The beam center location may then be used as a reference point in characterizing the distortion in the image. In particular, for each secondary material reflection, a ratio may be found between the distance of the observed reflection to the beam center location and the distance of the predicted reflection to the beam center location. These ratios may then be used for distortion characterization.

[0014] The lens distortion may be represented by a transformation function that indicates how the image components in the CEDI image are displaced relative to what would be their correct position in the image. The relevant coefficients of this function may be determined from the observed secondary material reflection positions and the distance ratios associated with the respective distances between the observed and predicted positions of the secondary material reflections. Such a function may then be used to adjust the image pixel by pixel, which not only corrects the information associated with the secondary material, but also the information corresponding to the sample. In an exemplary embodiment, this function is as follows:

[0015]

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[0016] In the formula, c x and c y where a, b, c, and d represent the x and y coordinates of the beam center position, and a, b, c, and d represent additional coefficients determined from the observed position and distance ratios of the secondary material reflections, as described above. The coefficients may be found by fitting the above equation to the observed position and distance ratios of the secondary materials. Such fitting may utilize, for example, a least squares refinement process. The coefficients in the equation are then replaced by the values ​​found during the fitting process to provide a transformation function that may be used in correcting lens distortion. [Brief description of the drawings]

[0017] [Figure 1A] FIG. 1A shows a simulated CEDI image of a double-walled carbon nanotube (DWCNT) along the tube axis. [Figure 1B] FIG. 1B illustrates the successful reconstruction of the DWCNT of FIG. 1A from simulated data. [Figure 1C] FIG. 1C shows the first failed reconstruction of the DWCNT of FIG. 1A using simulated data after applying the projection lens distortion from the experimental carbon nanotube (CNT) data. [Figure 1D] FIG. 1D shows a second failed reconstruction of the DWCNT of FIG. 1A using simulated data after applying the projection lens distortion from the experimental CNT data. [Figure 1E] FIG. 1E shows a third failed reconstruction of the DWCNT of FIG. 1A using simulated data after applying the projection lens distortion from the experimental CNT data. [Figure 1F] FIG. 1F shows a fourth failed reconstruction of the DWCNT of FIG. 1A using simulated data after applying the projection lens distortion from the experimental CNT data. [Figure 2A] FIG. 2A is a schematic diagram of an undistorted image of an object according to the prior art. [Figure 2B]FIG. 2B is a schematic diagram of the object shown in FIG. 2A subjected to pincushion-type distortion. [Figure 2C] FIG. 2C is a schematic diagram of the object shown in FIG. 2A subjected to a barrel distortion. [Figure 2D] FIG. 2D is a schematic diagram of the object shown in FIG. 2A subjected to a spiral type distortion. [Figure 2E] FIG. 2E is a schematic diagram of the object shown in FIG. 2A, illustrating an elliptical distortion. [Diagram 3] FIG. 3 is a schematic diagram of a hexagonal lattice of graphene with a rhombic unit cell ABCD. [Figure 4] FIG. 4 shows an exemplary CEDI image of graphene. [Figure 5A] FIG. 5A is a perspective view of flat graphene in real space. [Figure 5B] FIG. 5B is a perspective view of corrugated graphene in real space. [Figure 5C] FIG. 5C is a schematic diagram of the interspace of flat graphene sheets. [Figure 5D] FIG. 5D is a schematic diagram of the interspace of corrugated graphene sheets. [Figure 5E] FIG. 5E is a schematic diagram showing how the diffraction spots become blurred at large angles for a corrugated graphene sheet. [Figure 6A] FIG. 6A shows a first section of a flow diagram of a method according to the present invention. [Figure 6B] FIG. 6B illustrates a second section of the flow diagram of FIG. 6A. [Figure 7A] FIG. 7A shows a CEDI image of graphene similar to that of FIG. 4, but showing the material lattice structure. [Figure 7B] FIG. 7B is a table of measured reflectance data for the image of FIG. 7A. [Figure 7C] FIG. 7C is a table of predicted reflectances for the image of FIG. 7A. [Figure 7D] FIG. 7D is a table of distance ratios of reflections in the image of FIG. 7A. [Figure 7E] FIG. 7E illustrates a coefficient fit of the distortion characterization equation associated with the image of FIG. 7A. [Figure 7F] FIG. 7F is a table showing matched distance ratios for the reflections in the image of FIG. 7A. [Figure 8] FIG. 8 shows the CEDI image of FIG. 7A with arrows superimposed on it indicating the direction and distance each reflection shown in the image would need to be displaced to be in its correct position if there was no lens distortion. [Figure 9] FIG. 9 shows a CEDI image of gold nanocrystals supported by two layers of graphene. [Figure 10] FIG. 10 shows a CEDI image of a free-standing carbon nanotube. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0018] Detailed Description of the Invention The present invention utilizes Bragg diffraction peaks from secondary materials such as graphene to identify and correct CEDI lens distortion errors. In many CEDI experiments, graphene is used as a support for the sample under test due to its high transparency to low energy electrons. Graphene is an allotrope of carbon in the form of a single atomic layer in a 2D hexagonal lattice with one atom forming each vertex. An example of a hexagonal lattice of graphene is shown in FIG. 3, with a rhomboid unit cell ABCD and four adjacent cells indicated with circled numbers 1, 2, 3, and 4. Graphene is also the basic structural element of other allotropes, including carbon nanotubes. The rhomboid unit cell of graphene contains two chemically identical carbon atoms, labeled P1 and P2 in FIG. 3, and has unit cell dimensions b1=b2=2.46 Å. Graphene has a lattice symmetry of p6m. An example of a CEDI image of graphene is shown in FIG. 4.

[0019] Because of the hexagonal symmetry, the relative distances of reflections from the beam center can be derived from their Bravais-Miller index, as shown below in equation (3). The ratio between the observed distances and those predicted from the symmetry provides the input for determining the distortion parameters.

[0020]

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[0021] As a 2D crystal, instead of a spheroid like in 3D crystals, the Bragg reflection of graphene in reciprocal space is a cone (a rod for idealized flat graphene). Thus, the Bragg condition is satisfied for all reflections at any angle of incidence of the electron beam on graphene, eliminating the need to carefully align the graphene with the beam, as is the case for 3D crystals.

[0022] For 3D reconstructions where the angle of incidence is continuously changed during a tomographic scan, the graphene reflections move away from the scan axis as the sharpness of the angle of incidence increases. This spreads out the reflections on the detector surface, allowing for the sampling of different regions of the distortion function and a more accurate determination of the distortion parameters. This is shown in Figures 5A-5E.

[0023] Figures 5A-5E show the properties of flat and corrugated graphene sheets, as described in C. Meyer et al., The Structure of Suspended Graphene Sheets, Nature 446 (2007) 60-63. Figure 5A is a perspective view of flat graphene in real space, and Figure 5B is a similar view of corrugated graphene. Figure 5C shows the interspace of a flat graphene sheet, as shown, a set of rods oriented perpendicular to the hexagonal interlattice of graphene. Figure 5D shows how, in the case of a corrugated graphene sheet, the superposition of diffracted beams from a microscopic flat area effectively turns the vertical rods of Figure 5C into a cone-shaped volume. As a result, the diffraction spots from such a surface become smeared at large angles, as shown by the dotted line in Figure 5E. Moreover, this effect becomes more pronounced away from the tilt axis.

[0024] The present invention uses measured CEDI images of graphene as part of a method for identifying and compensating for such lens distortions. An exemplary embodiment of this method is outlined in the steps shown in Figures 6A and 6B. This embodiment may be applied, for example, as part of a CEDI measurement in which one or more samples of interest are attached to a graphene support surface.

[0025] The method begins with the acquisition of a CEDI image in step 600. In step 602, graphene Bragg reflection locations are located in the CEDI image. This may be done manually by a user visually identifying the reflections and noting their pixel locations in the overall image. Alternatively, automated procedures such as blob detection methods (e.g., labeling connected components) may be applied. In general, manual approaches work best for images that also contain diffraction data from the specimen, or when multiple graphene domains are present.

[0026] In step 604, a correction for the incidence tilt angle of the electron beam with respect to the graphene sheet is applied. This can be done by modifying the image so that the reflection position approaches the projection of the scan axis on the image by a factor of 1 / cosφ, where φ=0° for normal incidence. The invention may utilize only one image, but it is also possible to use a set of images taken using different angles of incidence. As discussed above with respect to Figures 5A-5E, different angles of incidence allow the sampling of different regions of the distortion function, which can improve the determination of the distortion parameters. Thus, the method shown in Figures 6A and 6B includes the possibility of repeating steps 602 and 604 for each of the images that may be used, as indicated by a decision branch (step 606) that accounts for all of the different images that may be desired.

[0027] Once the graphene reflection positions of each image have been located and corrected, the difference vectors of all reflection pairs in each image are calculated (step 608). The difference vectors are grouped according to their length and orientation, using specified threshold criteria to limit the maximum fractional deviation of vector length for grouping and the maximum angular deviation from collinearity of vectors within a group. The average difference vector of each group is then calculated.

[0028] In step 610, the three shortest group difference vectors are found and these vectors form a hexagonal lattice. The vectors should be similar in length and oriented at a 60° angle with respect to each other. The two vectors with the closest length are selected and these vectors become the unit cell vectors in reciprocal space. The beam center position (c x , c y ) is estimated by calculating the centroid of all reflection locations (step 612).

[0029] A Bravais-Miller index is assigned to each reflection (step 614) by finding the best linear combination of the two unit cell vectors from the beam center to the observed reflection position. In a preferred embodiment of the invention, the curvature of the Ewald sphere is ignored and approximated as a plane. This is a reasonable approximation given that the de Broglie wavelength of the electron is very small and the diffraction angles of the observed reflections are very small. An improved beam center position is then calculated by averaging over the centroids of all Friedel pairs of observed reflections (step 616).

[0030] For each reflection, the ratio of the observed and calculated distances to the beam center is determined (step 618). Using equation (3) above, the distance between the observed position of each reflection and the beam center, and the distance between the calculated position of each reflection for that state and the beam center are determined. For each reflection, the ratio between these two values ​​is found and recorded.

[0031] In step 620, equation (4) shown below is fitted to the calculated ratios and observed reflectance pixel locations (x,y) to obtain coefficients a, b, c, d, c x and c y Determine the set of

[0032]

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[0033] The coefficients define the type and magnitude of distortion exhibited by the error in the distance of the reflection from the center of the beam. x and C ywhere a represents the coordinate of the beam center, and the values ​​a, b, c, d provide different respective information regarding the nature of the distortion itself. These are defined herein as follows: 1) "a" represents the average magnification; 2) "b" represents the radial; distortion coefficient; 3) "c" represents the elliptical distortion coefficient; and 4) "d" represents the tilt angle of the ellipse. Thus, by finding the magnitude of these coefficients, an indication of the type of distortion produced by the projection lens can be determined. Furthermore, knowing these values ​​allows corrections to be applied to the image data of a sample that have been shown to significantly improve data quality.

[0034] The fitting process using equation (4) may follow known fitting techniques, and in this embodiment is the modified least squares method. x and C y The values ​​of a, b, c, and d are used and starting values ​​for the coefficients a, b, c, and d are chosen empirically. In this embodiment, starting values ​​of a=1, b=0, and d=0 are chosen. The starting value of c is the maximum distance ratio found in step 618. Those skilled in the art will recognize that the beam center coordinate c x , c y It will be appreciated that although values ​​of a, b, c, d were previously obtained, these values ​​are refined during the fitting process along with the coefficients a, b, c, d to improve the overall result.

[0035] Once the coefficients are found, the equation can be used as a general tool to correct distortions in CEDI images that affect both image components related to secondary materials, e.g., graphene, and image components related to the specimen under test. In this context, equation (4) can be used as a transformation function to reconstruct the image pixel by pixel. In the application of the equation, a, b, c, d, c x and c y are replaced with the actual values ​​determined during the fitting process, and the corrected x and y values ​​can be calculated for any pixel location as follows:

[0036]

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[0037] In the formula, (x rev ,y rev ) is the modified pixel location, (x,y) is the original pixel location, and the distortion coefficients f(x,y) are the evaluation of equation (4) at the original pixel location using the coefficients determined in the fitting process.

[0038] FIG. 7A shows a CEDI image of graphene similar to FIG. 4, without the presence of a beam stop. The positions of many or all of the different reflections shown in the figure may have position errors due to distortions of the projection lens. The image was obtained with a normal incidence angle, so no correction for the incidence angle is necessary. The vector from the measured distances of all pairs of reflections in the image, the shortest of which, turns out to be associated with two unit cell vectors that define a hexagonal lattice and are also among the most frequently occurring. The unit cell vectors are at an angle of about 60° with respect to each other. Of the reflections shown in the figure by dashed lines, the distance vectors from the beam center to any two adjacent reflections form a valid set of unit cell vectors.

[0039] The measured reflection data of the image shown in FIG. 7A is shown in the table of FIG. 7B. The table shows a list of all the measured difference vectors, and at the bottom the lengths (in pixels) of the three shortest vectors are shown along with their relative angles. As shown, the three vectors are similar in length and have angles of about 60° with respect to each other. Two of these have very similar lengths (283.0 and 283.3 pixels respectively), which would represent unit vectors of the graphene material. Also shown in the table are the coordinates of the beam center (1030, 1020), which is determined by finding the centroid of all observed reflection positions.

[0040] Once the unit vectors are determined, Bravais-Miller indices can be assigned to the observed reflections, as shown in the table of FIG. 7C. An improved value of the beam center is found by averaging over the centroids of all Friedel pairs of observed reflections found from the indexing step. For each reflection, a predicted position can then be found from the updated value of the beam center using equation (3), which relies on the symmetry properties of the graphene crystal to make an accurate calculation that is less sensitive to distortion than the observed reflection positions. A ratio is then determined between the calculated distance of the observed reflection to the beam center and the distance of the predicted reflection to the beam center, as shown in the table of FIG. 7D.

[0041] The fit of equation (4) to the calculated ratios from FIG. 7D and the observed pixel locations (x, y) is shown in FIG. 7E. As shown, the resulting values ​​of the coefficients are a=1.167, b=-0.00009, c=-0.017, and d=-0.36. The resulting value of the beam center coefficient is c x =1021 and c y = 1025. Using these values, one can then determine a fit ratio for each of the reflections, as shown in FIG. 7F. Thus, FIG. 7D shows the observed ratios, and FIG. 7F shows the fit ratios. The Fit column in FIG. 7F shows the observed ratios divided by the fit ratios, indicating the quality of the fit. These fit ratios indicate, for each reflection, the magnitude of the displacement of that reflection away from or towards the beam center due to lens distortion. This is visually shown in the version of the CEDI image of FIG. 7A shown in FIG. 8, on which are overlaid arrows indicating the direction and distance each reflection shown in the image would need to be displaced to be in its correct position if there was no lens distortion. That is, the corrected positions of the graphene reflections are shown as circles, with arrows pointing out from their distorted (observed) positions. Because the coefficient b is close to zero, this particular example may exhibit a predominantly elliptical distortion.

[0042] Once the distortion correction is known, it can be applied to the entire image, thus providing correction for image components related to other materials inspected together with the graphene, such as a sample of interest mounted on a graphene substrate. Although both the sample and secondary material are typically imaged together, it may also be possible to collect an image of only the secondary material by moving the electron beam to an area where there is no sample material, and then use this image to calibrate the distortion function used for the image of the sample and secondary material collected together. Once the correction for lens distortion is obtained, it can be applied to the unknown material based on the distortion characteristics determined by inspection of the secondary material, in this case graphene.

[0043] While graphene was the material chosen for this embodiment, one skilled in the art would understand that other materials with known properties could also be used to characterize lens distortion when considered along with the sample of interest. One alternative material is, for example, silicon nitride. Although it is a three-dimensional crystal rather than a two-dimensional one like graphene, a similar process of finding predicted reflection positions and fitting equation (4) to determine coefficients indicative of the quality of distortion could be used. However, such a 3D crystal needs to be aligned with its axis of symmetry parallel to the electron beam.

[0044] In another embodiment of the invention, more than one layer of secondary material may be used. Shown in FIG. 9 is a CEDI image of gold nanocrystals supported by two layers of graphene. As shown, this produces two sets of graphene reflections, each associated with a different graphene layer. The reflection regions of these two graphene domains are distinguished in the diagram by a circle and a square, respectively, and adjacent to them are their Bravais-Miller indices.

[0045] When multiple layers of secondary material are used, as in Figure 9, the method according to the invention proceeds in the same manner, with the additional layers providing additional sets of reflections for characterizing the distortion imparted by the projection lens. Those skilled in the art will appreciate that, although the two layers must be evaluated separately with respect to establishing the expected locations of the reflections, both sets of reflections are evaluated to find the beam center and to estimate the coefficients a, b, c, d, c x and c y It will be appreciated that this can be used to adapt equation (4) to find

[0046] Another embodiment In another embodiment, characterization of lens distortion can be performed using known material reflections that may be unique to the sample of interest. Figure 10 shows a CEDI image of a free-standing carbon nanotube that inherently produces a reflection that corresponds to the graphene contained therein. In the figure, the graphene reflections are shown as circles with their respective Bravais-Miller indices adjacent to them. As in the previous embodiment, the predicted reflection positions can be used to determine the displacement of the reflections in the image due to lens distortion, and a fit of equation (4) provides coefficients to characterize the distortion. An overall image correction can then be performed accordingly.

Claims

1. 1. A method for correcting distortion imparted to a diffraction image of a sample of interest by a projection lens in a coherent electron diffraction imaging (CEDI) system, comprising: The method comprises: a) acquiring a CEDI image of the sample of interest together with a secondary material having a known lattice structure; b) locating within the CEDI image an observed location of a reflection that corresponds to the secondary material; c) identifying a unit cell vector for the secondary material based on observed positions of reflections that correspond to the secondary material; and d) determining a predicted position of each secondary material reflection corresponding to a respective Bravais-Miller index assigned to the reflection based on the unit cell vector and the known lattice structure of the secondary material; e) relocating image components within the CEDI image using a difference between an observed position of a secondary material reflection and a predicted position of a secondary material reflection.

2. The method of claim 1 , wherein the secondary material comprises graphene.

3. 3. The method of claim 1 or 2, wherein identifying a unit cell vector comprises finding difference vectors of pairs of reflections of the CEDI image corresponding to the secondary material, grouping the difference vectors according to length and direction, averaging the difference vectors of each group, and selecting the shortest averaged group vector having a similar length as the unit cell vector.

4. 4. The method of claim 1, wherein determining the predicted location of each secondary material reflection comprises finding a linear combination of the unit cell vectors that correspond to locations in the CEDI image that are proximate to the observed location of the reflection.

5. 5. The method of claim 1, wherein the difference between observed and predicted positions of secondary material reflections is found relative to a beam center position in the CEDI image.

6. 6. The method of claim 5, wherein the beam center location is found by averaging the centroids of Friedel pairs using the observed locations of reflections corresponding to the secondary material.

7. 6. The method of claim 5, wherein for each secondary material reflection, a ratio between a distance of the observed reflection relative to the beam center position and a distance of the predicted reflection relative to the beam center position is found, and the ratio is used to characterize the distortion in the CEDI image relative to the beam center position.

8. The ratio and the observed reflection position of the secondary material are determined by the coefficients a, b, c, d, c that best satisfy the following equation: x and c y The method of claim 7, which is used to determine For each of said ratios and observed secondary material reflection locations, [0010] Here, c x and c y represent the x and y coordinates of the beam center position, and the equations with the determined coefficients a, b, c, d, cx and cy are used for the repositioning of image components.

9. The method of claim 8 , wherein a modified least squares method is used to fit the equation to the ratios and observed secondary material reflection positions.

10. 9. The method of claim 8, wherein the corrected x and y values ​​of a pixel location in the CEDI image are determined as follows: [0025] In the formula, x rev and y rev are the modified x and y pixel values, respectively, where x and y are the original pixel values, respectively, and f(x,y) is a calculated version of equation (4) using the determined coefficients.

11. 11. The method of any one of claims 1 to 10, wherein the CEDI image is a first CEDI image, the method further comprising acquiring a plurality of CEDI images of the sample of interest together with the secondary material at a plurality of angles of incidence, the reflections from each of the CEDI images being corrected for positional changes with angle of incidence before being used in the method.

12. 12. The method of claim 1, wherein the secondary material comprises multiple layers, each layer producing a distinct reflection for use in the method.

13. The method of claim 1 , wherein the secondary material comprises a substrate on which the sample is located.

14. 14. The method of claim 1, wherein the secondary material is part of the sample.

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