Charged-particle microscope providing depth-resolved imagery

Abstract

A method of examining a sample using a charged-particle microscope, comprising mounting the sample on a sample holder; using a particle-optical column to direct at least one beam of particulate radiation onto a surface S of the sample, thereby producing an interaction that causes emitted radiation to emanate from the sample; using a detector arrangement to detect at least a portion of said emitted radiation, the method of which comprises embodying the detector arrangement to detect electrons in the emitted radiation; recording an output O.sub.n of said detector arrangement as a function of kinetic energy E.sub.n of said electrons, thus compiling a measurement set M={(O.sub.n, E.sub.n)} for a plurality of values of E.sub.n; using computer processing apparatus to automatically deconvolve the measurement set M and spatially resolve it into a result set R={(V.sub.k, L.sub.k)}, in which a spatial variable V demonstrates a value V.sub.k at an associated discrete depth level L.sub.k referenced to the surface S, whereby n and k are members of an integer sequence, and spatial variable V represents a physical property of the sample as a function of position in its bulk.

Claims

1. A method of examining a bulk sample using a charged-particle microscope, comprising: mounting the sample on a sample holder; performing an investigation of the sample using a procedure comprising the following steps: directing at least one beam of particulate radiation onto a surface S of the bulk sample, thereby producing an interaction that causes emitted radiation to emanate from the sample; using a detector arrangement to detect at least a portion of said emitted radiation, performing energy resolved detection of said emitted radiation to compile at least one measurement set corresponding to multiple energy levels present in said emitted radiation; interrupting said investigation of the sample; performing a sample modification using a procedure comprising the following steps: moving a mechanical cutting tool into a position proximal the surface S; using said tool to remove at least one layer of material from the sample, thereby exposing a new surface S on the bulk sample; withdrawing said tool to a location distal from said sample; resuming said investigation, now performed on the new surface S on the bulk sample; inputting the at least one measurement set corresponding to multiple energy levels into a computational slicing technique to deconvolve a signal from the detector to obtain depth-resolved imagery of the sample.

2. A method according to claim 1, wherein said tool is selected from the group comprising a microtome, a diamond knife, and combinations hereof.

3. A method according to claim 2, wherein said layer of material is removed non-destructively and is preserved for later use.

4. A method according to claim 2, wherein said investigation further comprises at least one of the following procedures: varying a beam parameter of said beam of particulate radiation, and recording detector output as a function of said variation; for a given beam parameter, performing angularly-resolved detection of said emitted radiation; for a given beam parameter, performing energy-resolved detection of said emitted radiation, thus compiling at least additional one measurement set M={(O.sub.n,X.sub.n)}where O.sub.n is detector output corresponding to a selected parameter value X.sub.n, with X.sub.n selected from the group comprising beam parameter P.sub.n, emission angle .sub.n and emission energy E.sub.n, respectively.

5. A method according to claim 1 wherein, during said investigation, an electric field is applied to a vicinity of the sample.

6. A method according to claim 5, wherein said location to which said tool is withdrawn is chosen to be distal from said field when applied.

7. A method according to claim 6, wherein said investigation comprises at least one of the following procedures: varying a beam parameter of said beam of particulate radiation, and recording detector output as a function of said variation; for a given beam parameter, performing angularly-resolved detection of said emitted radiation; for a given beam parameter, performing energy-resolved detection of said emitted radiation, thus compiling at least one measurement set M={(O.sub.n, X.sub.n)}, where O.sub.n is detector output corresponding to a selected parameter value X.sub.n, with X.sub.n selected from the group comprising beam parameter P.sub.n, emission angle .sub.n and emission energy E.sub.n, respectively.

8. A method according to claim 5, wherein said investigation comprises at least one of the following procedures: varying a beam parameter of said beam of particulate radiation, and recording detector output as a function of said variation; for a given beam parameter, performing angularly-resolved detection of said emitted radiation; for a given beam parameter, performing energy-resolved detection of said emitted radiation, thus compiling at least one measurement set M={(O.sub.n, X.sub.n)}, where O.sub.n is detector output corresponding to a selected parameter value X.sub.n, with X.sub.n selected from the group comprising beam parameter P.sub.n, emission angle .sub.n and emission energy E.sub.n, respectively.

9. A method according to any of claims 1, wherein said investigation further comprises at least one of the following procedures: varying a beam parameter of said beam of particulate radiation, and recording detector output as a function of said variation; for a given beam parameter, performing angularly-resolved detection of said emitted radiation; for a given beam parameter, performing energy-resolved detection of said emitted radiation, thus compiling at least one additional measurement set M={(O.sub.n, X.sub.n)}, where O.sub.n is detector output corresponding to a selected parameter value X.sub.n, with X.sub.n selected from the group comprising beam parameter P.sub.n, emission angle .sub.n and emission energy E.sub.n, respectively.

10. A method according to claim 9, wherein said computational technique comprises the following steps: defining a Point Spread Function that, for each value of an integer sequence n, has a kernel value K.sub.n representing a behavior of said beam of particulate radiation in a bulk of the sample; defining a spatial variable V that represent a physical property of the sample as a function of position in its bulk; defining an imaging quantity that, for each value of n, has a value Q.sub.n that is a multi-dimensional convolution of K.sub.n and V, such that Q.sub.n =K.sub.n *V; for each value of n, computationally determining a minimum divergence
min D (O.sub.n K.sub.n *V) between O.sub.n and Q.sub.n, wherein one solves for V while applying constraints on the values K.sub.n.

11. A method according to claim 1, in which inputting results of said investigation into the computational technique comprises inputting results of said investigation into a computational virtual slicing technique to combine physical removal of a layer with computational slicing.

12. A method according to claim 1, in which inputting results of said investigation into the computational technique to obtain depth-resolved imagery of the sample comprises observing particles emerging from the sample to provide the signal and deconvolving the signal to provide sub-surface virtual imaging of the sample to increasing penetration depths.

13. A method according to claim 1, in which performing energy resolved detection of said emitted radiation further comprises performing energy-filtered detection of electrons emanating from the sample.

14. A charged-particle microscope comprising: a sample holder for holding a bulk sample; a particle-optical column, to direct at least one beam of particulate radiation onto a surface of the sample, thereby producing an interaction that causes emitted radiation to emanate from the sample; a detector arranged to detect at least a portion of said emitted radiation; a mechanical cutting tool for performing a surface modification to the sample; a processing apparatus programmed to execute the following steps: perform an investigation on a surface S of the bulk sample, comprising activating said particle-optical column and detector, and recording a detector output by performing energy resolved detection of said emitted radiation to compile at least one measurement set corresponding to multiple energy levels present in said emitted radiation; interrupt said investigation of the sample; perform a sample modification, comprising the following operations: moving said mechanical cutting tool into a position proximal the surface S; using said tool to remove at least one layer of material from the sample, thereby exposing a new surface S on the bulk sample; withdrawing said tool to a location distal from said sample; resume said investigation of the sample, now performed on the new surface S on the bulk sample; input the at least one measurement set corresponding to multiple energy levels into a computational slicing technique to deconvolve a signal from the detector to obtain depth-resolved imagery of the sample.

15. A charged-particle microscope according to claim 14, further comprising an electrode arrangement for applying an electric field to a vicinity of the sample, which electrode arrangement can be activated by said processing apparatus as part of said investigation.

16. A charged-particle microscope according to claim 15, wherein said location to which said tool is withdrawn is distal from said field when applied.

17. A charged-particle microscope according to claim 16, wherein said tool is selected from the group comprising a microtome, a diamond knife, and combinations hereof.

18. A charged-particle microscope according to claim 15 wherein said tool is selected from the group comprising a microtome, a diamond knife, and combinations hereof.

19. A charged-particle microscope according to claim 14, wherein said tool is selected from the group comprising a microtome, a diamond knife, and combinations hereof.

20. A method of examining a bulk sample using a charged-particle microscope, comprising: performing an investigation of the sample using a procedure comprising the following steps: directing at least one beam of particulate radiation onto a surface S of the bulk sample, thereby producing an interaction that causes emitted radiation to emanate from the sample; using a detector arrangement to detect at least a portion of said emitted radiation performing energy resolved detection of said emitted radiation to compile at least one measurement set corresponding to multiple energy levels present in said emitted radiation; interrupting said investigation of the sample; performing a sample modification using a procedure comprising the following steps: using a mechanical cutting tool to remove at least one layer of material from the sample, thereby exposing a new surface S on the bulk sample; resuming said investigation, now performed on the new surface S on the bulk sample; inputting the at least one measurement set corresponding to multiple energy levels into a computational slicing technique to deconvolve a signal from the detector to obtain depth-resolved imagery of the sample.

21. A method according to claim 20, in which performing energy resolved detection of said emitted radiation further comprises performing energy-filtered detection of electrons emanating from the sample.

22. The method of claim 20, wherein said computational technique comprises the following steps: defining a Point Spread Function that, for each value of an integer sequence n, has a kernel value Kn representing a behavior of said beam of particulate radiation in a bulk of the sample; defining a spatial variable V that represent a physical property of the sample as a function of position in its bulk; defining an imaging quantity that, for each value of n, has a value Qn that is a multi-dimensional convolution of Kn and V, such that Qn=Kn*V; for each value of n, computationally determining a minimum divergence
min D (OnKn*V) between On and Qn, wherein one solves for V while applying constraints on the values Kn.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will now be elucidated in more detail on the basis of exemplary embodiments and the accompanying schematic drawings, in which:

(2) FIGS. 1A and 1B are mutually associated flowcharts that depict a general scheme for performing the method according to the present invention.

(3) FIG. 2 illustrates a hybrid technique involving the alternate use of computational slicing and physical slicing in accordance with an embodiment of the current invention.

(4) FIG. 3 shows simulation results pertaining to emission of BS electrons from a sample in a SEM, illustrating numbers of BS electrons emanating from different depths within the sample for different BS electron energy values.

(5) FIG. 4 renders a longitudinal cross-sectional view of aspects of a particle-optical microscope (in this case a SEM) with which the method according to the current invention can be implemented.

(6) FIG. 5 illustrates the operating principle of a particular detector arrangement that can be used to produce energy-filtered outputs O.sub.n for a detected beam of electrons of mixed energies E.sub.n, thus serving to compile a measurement set M={(O.sub.n, E.sub.n)} as set forth in the current invention.

(7) In the Figures, where pertinent, corresponding parts are indicated using corresponding reference symbols.

(8) Embodiment 1

(9) FIGS. 1A and 1B are mutually associated flowcharts that depict a general scheme for performing the method according to the present invention. With reference to the nomenclature introduced in the discussion above, it is noted that: FIG. 1A depicts an algorithm for a given PSF kernel K.sub.n at iteration 1. Multiple iteration cycles for a given K.sub.n are applied sequentially. The iterative scheme in FIG. 1A can be sequentially applied to each PSF and to the spatial variable V. For any pair K.sub.n,V, one can have one or more iterations at each cycle.

(10) In the depicted flowcharts, the indicated steps will now be elucidated in more detail. Starting with FIG. 1A: 201: This step represents the value of K.sub.n at iteration l (i.e. K.sub.n.sup.l). In the special case l=1, a preceding initialization procedure will have been performed, so as to kick start the iteration procedure. 203: Similarly, this step represents the value of V at iteration l (i.e. V.sup.l). Once again, in the special case l=1, a preceding kick start initialization procedure will have been performed. 205: The convolution K.sub.n.sup.l*V.sup.l is calculated using the output of steps 201 and 203. One now introduces a quantity I.sub.n that is a dimensionless/scaled version of the quantity O.sub.n. For example, if O.sub.n is measured in volts, its numerical value in volts is dimensionless, and can, if desired, be scaled by the value of the fundamental electron charge (e) so as to effect a conversion to a numerical value in electron-volts (eV), for example. This is purely a matter of choice in any given situation, as will be readily grasped by the skilled artisan. The quantity will be referred to hereinafter as an image. In step 205, a divergence between image and the convolution K.sub.n.sup.l*V.sup.l is determined, i.e. D (I.sub.nK.sub.n.sup.l*V.sup.l) is calculated. 207: Here, it is determined if the divergence calculated in step 205 is minimal, i.e. if convergence has been attained. If it is (Yes), then one has distilled the sought values K.sub.n and V;
if it is not (No), then one returns to the top of the flowchart for the next iteration (l+1).

(11) Turning now to FIG. 1B, this figure represents a generalization of FIG. 1A. Instead of just showing the procedure for only one element n of the measurement sequence [1, . . . , N], it now depicts all the elements 1 . . . N in this sequence: 211, 213, 215: Each of these steps corresponds to the cumulative steps 201, 203 and 205 of FIG. 1A, but now shown for the individual cases n=1 (211), n=2 (213) and n=N (215). 217: This step corresponds to step 207 of FIG. 1A.

(12) For a specific example as to how the minimum divergence problem set forth above can be formulated and solved, reference is made to the next Embodiment below.

(13) Embodiment 2

(14) One intuitive way to consider the variable-kernel deconvolution task at hand is to formulate it using so-called Bayesian statistics.

(15) One first defines a number of probabilities that will be used throughout the elucidation below: Pr(V|I.sub.n) is the probability of distilling the spatial variable V, given the acquired input values (see the above discussion of step 205 in the flowchart of FIG. 1A for an explanation of the concept of image value I.sub.n). Similarly, Pr(I.sub.n|V) is the probability of observing the image values I.sub.n given a sample structure described by V. Pr(V) is the so-called prior probability associated with V, representing one's knowledge about the structure to be reconstructed. Pr(I.sub.n) is the probability associated with the acquired images; however, this is essentially a constant, given that the images are actually observed/measured values.

(16) Using Bayes' rule one now obtains:

(17) $\begin{array}{cc}\mathrm{Pr}\left(V.Math.I_n\right)=\frac{\mathrm{Pr}\left(I_n.Math.V\right)\mathrm{Pr}\left(V\right)}{\mathrm{Pr}\left(I_n\right)}& \left(1\right)\end{array}$

(18) In the Bayesian framework, the current problem can be expressed as the following maximization task:
{circumflex over (V)}=argmax.sub.V0 {Pr(V|I.sub.n)},(2)
in which one needs to enforce the positivity of the reconstructed variable V. This is necessary in order to obtain a physically meaningful solution. More commonly, one will use the so called log-likelihood function to simplify the calculations:
{circumflex over (V)}=argmin.sub.V0{log(Pr(V|I.sub.n))}(3)
Concretely, the current imaging process is well represented by a Poisson process. Given the nature of charged-particle and X-ray detectors, one can assume that, at each voxel x in a 3D grid , the image is formed by the realization of independent Poisson processes. This leads to:

(19) $\begin{array}{cc}\mathrm{Pr}\left(I_n.Math.V\right)=\underset{x}{.Math.}\frac{{\left(\left(K_n*V\right)\left(x\right)\right)}^{I_n\left(x\right)}\exp \left(-\left(K_n*V\right)\left(x\right)\right)}{I_n\left(x\right)!},& \left(4\right)\end{array}$
wherein it should be noted that x is not the linear Cartesian coordinate x, but is instead an algebraic denotation of a three-dimensional position.

(20) To recover the volume V, one needs to minimize the criterion:

(21) $\begin{array}{cc}\begin{array}{c}J\left(\left(I_n.Math.V\right)\right)=-\log \left(\mathrm{Pr}\left(I_n.Math.V\right)\right)\\ =\underset{x}{\overset{}{.Math.}}\left(\left(K_n*V\right)\left(x\right)\right)-I_n\left(x\right).Math.\log \left(\left(K_n*V\right)\left(x\right)\right)+\log \left(I_n\left(x\right)!\right)\end{array}& \left(5\right)\end{array}$

(22) Given that the .sub.x log(I.sub.n(X)!) term does not contain any variables, the criterion can be redefined as:
J((I.sub.n|V))=.sub.x((K.sub.n*V)(x))I.sub.n(x).Math.log((K.sub.n*V)(x))(6)

(23) It is important to note that this criterion is related to Kullback-Leibler generalized I-divergence IDIV(I.sub.nV). This can be seen from the definition of I-divergence:

(24) $\begin{array}{cc}\mathrm{IDIV}\left(I_n.Math..Math.V\right)\stackrel{\mathrm{def}}{=}\underset{x}{\overset{}{.Math.}}{I}_n\left(x\right)\log \left(\frac{I_n\left(x\right)}{\left(K_n*V\right)\left(x\right)}\right)-\underset{x}{.Math.}I{(}_n\left(x\right)-\left(K_n*V\right)\left(x\right))& \left(7\right)\end{array}$
from which one can obtain:
IDIV(I.sub.nV)=J((I.sub.n|V)).sub.xI.sub.n(x).Math.log(I.sub.n(x))(8)

(25) The second term in (8) is a constant with regard to minimization and, hence, minimizing J((I.sub.n|V)) is equivalent to minimizing IDIV(I.sub.nV).

(26) Reference is now made to the following journal article: [1] H. Lantri, M. Roche, C. Aime, Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms, Inverse Problems, vol. 18, pp. 1397-1419, 2002,
in which it was shown that a positivity-constrained minimization problem of the type (2) above can be solved using the following iterative scheme:

(27) $\begin{array}{cc}{V}^{l+1}\left(x\right)=V^l\left(x\right).Math.\left(\frac{I_n\left(x\right)}{\left(K_n*V^l\right)\left(x\right)}*K_n\left(-x\right)\right)& \left(9\right)\end{array}$

(28) This algorithm is also known as the Maximum-Likelihood Expectation Maximization algorithm, which is further described, for example, in the following references: [2] L. Shepp, Y. Vardi, Maximum-Likelihood reconstruction for emission tomography, IEEE Transactions on Medical Imaging, MI-5, pp. 16-22, 1982. [3] Richardson, William Hadley. Bayesian-Based Iterative Method of Image Restoration, JOSA 62 (1), pp 55-59, 1972.

(29) Convergence in expression (9) can be accelerated by using the exponent q as follows:

(30) $\begin{array}{cc}{V}^{l+1}\left(x\right)=V^l\left(x\right).Math.{\left(\frac{I_n\left(x\right)}{\left(K_n*V^l\right)\left(x\right)}*K_n\left(-x\right)\right)}^q& \left(10\right)\end{array}$

(31) Typically, q[1, 1.5] and, in addition to acceleration, it can act as a regularizing factor. In the current case, the iterative algorithm needs to be sequentially used for all kernels K.sub.n associated with the different PSFs. Convergence can be assessed empirically or based on other criteria, such as the relative change in the variables.

(32) If one needs to recover or adjust the values of the PSF kernels K.sub.n, one can use alternate minimization of the spatial variable V and the K.sub.n variables. One then obtains the following algorithm:

(33) 0$\begin{array}{cc}{V}^{l+1}\left(x\right)=V^l\left(x\right).Math.{\left(\frac{I_n\left(x\right)}{\left(K_n^l*V^l\right)\left(x\right)}*K_n^l\left(-x\right)\right)}^q{K}_n^{l+1}\left(x\right)=K_n^l\left(x\right).Math.{\left(\frac{I_n\left(x\right)}{\left(K_n^l*V^{l+1}\right)\left(x\right)}*V^{l+1}\left(-x\right)\right)}^q& \left(11\right)\end{array}$
One can choose to have more iterations for the kernels K.sub.n or for the spatial variable V at each cycle; such a choice can be determined based on experience/experimentation. For example, it is generally noticed that V tends to converge faster, and hence more iterations can be spent searching for the different values K.sub.n.

(34) If prior knowledge about the PSF or V is available, it can be incorporated into the Bayesian formulation using a combination of conditional Pr(.|.) and joint probabilities Pr(.,.) as follows:

(35) $\begin{array}{cc}\mathrm{Pr}\left(V,K_n.Math.I_n\right)=\frac{\mathrm{Pr}\left(I_n.Math.V,K_n\right)\mathrm{Pr}\left(V\right)\mathrm{Pr}\left(K_n\right)}{\mathrm{Pr}\left(I_n\right)}& \left(12\right)\end{array}$

(36) It follows that the minimization problem (2) is then modified as follows:
{circumflex over (V)}=argmax.sub.V0{Pr(V,K.sub.n|I.sub.n)}(13)
and the log-likelihood criterion to be minimized then becomes

(37) $\begin{array}{cc}\begin{array}{c}J\left(V,K_n.Math.I_n\right)=-\log \left(\mathrm{Pr}\left(I_n.Math.V,K_n\right)\right)-\\ \log \left(\mathrm{Pr}\left(V\right)\right)-\log \left(\mathrm{Pr}\left(K_n\right)\right)\\ =J\left(I_n.Math.V,K_n\right)+J\left(V\right)+J\left(K_n\right)\end{array}& \left(14\right)\end{array}$
While the first term is the data term that ensures that one fits the observations, the second and third terms are known as regularization terms that use one's knowledge and assumptions about the variables to limit the space of solutions and reduce the effects of noise. The criterion J(V,K.sub.n|I.sub.n) can be minimized using the Maximum Likelihood Expectation Maximization approach. Optimization can be also carried out using a variety of other convex and non-convex methods, as set forth, for example, in the following reference: [4] William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, Second Edition (1992).

(38) For completeness, it is noted that the approach set out in the current Embodiment can be regarded as a hybrid/variant of the so-called Richardson-Lucey Algorithm (RLA). The RLA is a known mathematical technique that can be applied to solve a variety of problems. For example, it was used by NASA scientists in an attempt to computationally improve blurred imagery from the original (i.e. uncorrected) Hubble Space Telescope.

(39) Embodiment 3

(40) The prior distributions of the sample structure [Pr(V)] and the PSF [Pr(K.sub.n)] can play an important role of regularization in the process of maximizing the so-called posterior probability Pr(V,K.sub.n|I.sub.n). Several well-known regularization methods are set forth in the following publications. [5] A. N. Tikhonov, On the Stability of Inverse Problems, Proceedings of Doklady Akademii Nauk SSSR, Russian Academy of Sciences, 1943, pp. 195-198. [6] D. Strong, T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 2003, 19: S165-S187. [7] P. O. Hoyer, Non-negative Matrix Factorization with Sparseness Constraints, Journal of Machine Learning Research 5, 2004, pp. 1457-1469. [8] WD. Dong, H J. Feng, Z. H. Xu, Q. Li, A piecewise local regularized Richardson-Lucy algorithm for remote sensing image deconvolution, Optics and Laser Technology 43, 2011, pp. 926-933.
Known regularization methods include Tikhonov regularization [5], Total Variation (TV) regularization [6], sparse prior regularization [7], piecewise local regularization [8], etc.

(41) For the popular TV regularization, a regularization term J(V) is defined by the integration of absolute gradients of the sample structure V, as follows:

(42) $J\left(V\right)=-\log \left(\mathrm{Pr}\left(V\right)\right)=.Math.\underset{x}{\overset{}{.Math.}}.Math.V\left(x\right).Math.$

(43) The total function to be minimized is then:

(44) $\begin{array}{c}J\left(V.Math.I_n\right)=J\left(I_n.Math.V\right)+J\left(V\right)\\ =\underset{x}{\overset{}{.Math.}}\left(\left(K_n*V\right)\left(x\right)\right)-I_n\left(x\right).Math.\log \left(\left(K_n*V\right)\left(x\right)\right)+.Math.\underset{x}{\overset{}{.Math.}}.Math.V\left(x\right).Math.\end{array}$
The derivative of J(V) with respect to V is

(45) $\frac{}{V}J\left(V\left(x\right)\right)=-.Math.\mathrm{div}\left(\frac{V\left(x\right)}{.Math.V.Math.\left(x\right)}\right),$
where div stands for divergence in the context of vector calculus (as opposed to divergence in the context of statistical distance). One minimizes J(V|I.sub.n) by setting the derivative of J(V|I.sub.N) to be zero, with .sub.xK.sub.n(x)=1, which results in the following iterative scheme:

(46) $V^{l+1}\left(x\right)=\frac{V^l\left(x\right)}{1-.Math.\mathrm{div}\left(\frac{V}{.Math.V.Math.}\right)}.Math.\left(\frac{I_n\left(x\right)}{\left(K_n*V^l\right)\left(x\right)}*K_n\left(-x\right)\right)$
where effectively controls the weight of TV regularization during the optimization. The main advantage of TV as a regularization method is that it preserves the edges in the resulting image while reducing noise in homogeneous regions.
Embodiment 4

(47) As an alternative to the mathematics presented above, the following deconvolution methods also deserve mention in the context of the present invention.

(48) (I) Maximum Entropy Methods

(49) The Maximum Entropy (ME) method has been widely used with success for many years and is, for example, one of the practical choices in radio astronomy for image restoration applications. In contrast to the Maximum Likelihood (ML) approach, which aims to maximize a probability function, the general approach of ME reconstruction is to maximize an entropy function subject to constraints on the image estimation:
{circumflex over (V)}=argmax {Ent(V)}
such that I.sub.nK.sub.n*V
where Ent represents the entropy function. The idea behind the ME method is to find the solution that is not only compatible with the image formation process but that also has the highest information content.

(50) The most popular entropy function in image restoration is the Shannon entropy, defined directly on the gray-levels of the image as:

(51) $\mathrm{Ent}\left(V\left(x\right)\right)=\underset{x}{\overset{}{.Math.}}V\left(x\right).Math.\log V\left(x\right)$
which has its origin in information theory. Another entropy function found in the literature is the Burg entropy:

(52) $\mathrm{Ent}\left(V\left(x\right)\right)=\underset{x}{\overset{}{.Math.}}\log V\left(x\right)$
The ME problem can, for example, be solved as a deterministic constrained convex optimization problem using the Multiplicative Algebraic Reconstruction Technique (MART), which minimizes the negative Shannon entropy function via an iterative scheme. The MART technique involves a multiplicative correction to the voxel intensity based on the ratio of the recorded pixel intensity I.sub.n(j) and the projection of voxel intensities (K.sub.n*V.sup.I)(j) from the previous iteration:

(53) $V^{l+1}\left(j\right)=V^l\left(j\right){\left(\frac{I_n\left(i\right)}{\left(K_n*V^l\right)\left(i\right)}\right)}^{{\mathrm{wK}}_n\left(i,j\right)}$
where w is a relaxation parameter that controls the step size, V.sup.I(j) is the j.sup.th element of the i.sup.th iteration on V, and K.sub.n(i,j) is the i,j.sup.th element of K.sub.n. Each voxel's intensity is corrected by one projection at a time, which means a single iteration is completed only after every projection has been considered.

(54) For more information on MART, reference is to made to: [9] R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques for three-dimensional electron microscopy and x-ray photography, J. Theoretical Biology 29, 1970, pp 471-481.
(II) Linear Methods

(55) For an image formation model applied without further assumptions on noise statistics, the estimate for the sample structure is given by:

(56) 0$V=F^{-1}\left(\frac{F\left(I_n\right)}{F\left(K_n\right)}\right)$
where F and F.sup.1 denote Fourier and inverse Fourier transforms, respectively. Because K.sub.n is band-limited, the denominator of this expression is close to zero at many frequencies, and the direct deconvolution tends to suffer from severe noise amplification. One way to tackle this problem is by using a truncated inverse filter (see reference [10] below):

(57) $V=\{\begin{array}{cc}{F}^{-1}\left(\frac{F\left(I_n\right)}{F\left(K_n\right)}\right)& \mathrm{if}.Math.F\left(K_n\right).Math.\mathrm{.Math.}\\ 0& \mathrm{else}\end{array}$
where is a small positive constant. The solution is generally ill-posed, and a regularization term can be introduced to find a stable solution. For example, Tikhonov regularization consists of minimizing the term:
J(V)=I.sub.nK.sub.n*V+H*V
where H denotes a high-pass filter. The solution is obtained in the Fourier space (see reference [11] below):

(58) $V=F^{-1}\left(\frac{F\left(K_n\right)F\left(I_n\right)}{{.Math.F\left(K_n\right).Math.}^2+.Math.{.Math.F\left(H\right).Math.}^2}\right)$
where is the regularization parameter, representing the trade-off between fidelity to the data and smoothness of the restored image.

(59) For more information on the linear methods discussed here, reference is made to the following publications: [10] J. G. McNally, T. Karpova, J. Cooper, J. A. Conchello, Three-dimensional imaging by deconvolution microscopy, Methods, vol. 19, no. 3, pp. 373-385 (1999). [11] J. L. Starck, E. Pantin, Deconvolution in Astronomy: A Review, Astronomical Society of the Pacific, 114: 1051-1069 (2002).

(60) It should be noted that linear methods as set forth here do not restore the sample structure's frequency components beyond the PSF bandwidth. In addition, these methods can give negative intensity in the estimated image, and tend to be very sensitive to errors in the PSF used for the estimation, resulting in artifacts.

(61) One may, if desired, combine different types of the methods listed here for the complete deconvolution problem involved. For example, one could first use a Maximum Posteriori (MAP) or ML method to estimate the PSF, and then use the linear or ME approach to deconvolve the sample structure.

(62) Embodiment 5

(63) FIG. 2 illustrates (in a stylized manner) an embodiment of the current invention whereby computational slicing is combined with physical slicing, so as to allow charged-particle-microscopy-based 3D volume imaging of a sample to relatively increased depths.

(64) FIG. 2A (left) depicts a computational slicing step, whereby a sample is observed at varying emergent electron energies (E.sub.1, E.sub.2, E.sub.3) and a 3D deconvolution algorithm is applied, as set forth above. This allows sub-surface virtual imaging of the sample to increasing penetration depths, here schematically labeled as (L.sub.1, L.sub.2, L.sub.3).

(65) In FIG. 2B (center), subsequent use is made of a physical slicing step, whereby a mechanical cutting device (e.g. a diamond knife) or a non-mechanical approach (e.g. involving a focused/broad beam of ions, or a focused electromagnetic beam) is used to physically skim off a certain depth of material from the sample, thus producing a newly exposed surface.

(66) In FIG. 2C (right), one executes a subsequent computational slicing operation on said newly exposed surface. This allows sub-surface virtual imaging of the sample to new penetration depths, here schematically labeled as (L.sub.4, L.sub.5, L.sub.6).

(67) Embodiment 6

(68) FIG. 3 shows simulation results pertaining to emission of BS electrons from a sample in a SEM, illustrating numbers of BS electrons emanating from different depths within the sample for different BS electron energy values. The results come from Monte Carlo simulations, and pertain to a Si target irradiated with a 5 kV incident electron beam and a fictive BS electron detector observing along an altitude angle of 90-180 degrees with respect to the incoming beam (thus relating to any BS electron with a velocity component anti-parallel to the incident beam). The graph depicts BS electron counts as a function of production depth (in nm) for non-overlapping energy bands (of width 500 eV) centered on three different BS energy values, viz. (from right to left) 4 keV, 3 keV and 2 keV.

(69) The Figure demonstrates that BS electrons of a given energy E.sub.n can be produced from a whole range of different depths, but that peak numbers are produced from a certain preferential depth for each value of E.sub.n. More specifically, in the depicted graph: 4 keV electrons show a peak in numbers corresponding to a depth of about 40 nm; 3 keV electrons show a peak in numbers corresponding to a depth of about 65 nm; 2 keV electrons show a peak in numbers corresponding to a depth of about 75 nm.

(70) It should be noted that these energy values are the kinetic energies of the electrons as they emerge from the sample surface, rather than their intrinsic kinetic energy upon production; it is thus logical that electrons from deeper layers shouldon averagehave less energy when they emerge from the sample, since they will generally have undergone greater losses in trying to escape from the sample.

(71) Embodiment 7

(72) FIG. 4 is a highly schematic depiction of a charged-particle microscope 400, which, in this case, is a SEM. The microscope 400 comprises a particle-optical column 402, which produces a charged-particle beam 404 (in this case, an electron beam). The particle-optical column 402 is mounted on a vacuum chamber 406, which comprising a sample holder/stage 408 for holding a sample 410. The vacuum chamber 406 is evacuated using vacuum pumps (not depicted). With the aid of voltage source 422, the sample holder 408, or at least the sample 410, may be biased (floated) to an electrical potential with respect to ground.

(73) The particle-optical column 402 comprises an electron source 412, lenses 414, 416 to focus the electron beam 404 onto the sample 410, and a deflection unit 418. The apparatus further comprises a computer processing apparatus (controller) 424 for controlling inter alia the deflection unit 418, lenses 414, and detectors 100, 420, and displaying information gathered from the detectors 100, 420 on a display unit 426.

(74) The detectors 420, 100 are chosen from a variety of possible detector types that can be used to examine different types of radiation in different manners. In the apparatus depicted here, the following detector choices have been made: Detector 100 is a segmented electron detector. Such a detector can, for example, be used to investigate the angular dependence of electrons emerging from the sample 410. A detector of this type is, for example, elucidated in more detail in the aforementioned European Patent Application EP12163262. Detector 420 is used in the context of the current invention to perform energy-filtered detection of electrons emanating from the sample 410. In the present instance, the detector 420 can, for example, be a multi-channel solid-state detector of the type (C) alluded to above. Alternatively, it may be of the type (A) referred to above, and employ a deflecting field in a cavity in order to fan out an incoming beam of electrons into energy-sorted sub-beams than then land on an array of detection modules. Regardless of its internal workings, signals from this detector 420 serve as the basis for compiling a measurement set M={(O.sub.n, E.sub.n)} as discussed above, since the detector 420 provides output values O.sub.n that are associated with discrete electron energy values E.sub.n emanating from the sample 410.

(75) As here rendered, both detectors 100 and 420 are used to examine electrons; however, this is purely a design/implementation choice and, if desired, one could also elect to detect other types of stimulated radiation (e.g. X-rays) in addition to electrons.

(76) By scanning the beam 404 over the sample 410, stimulated radiationcomprising, for example, X-rays, infrared/visible/ultraviolet light, secondary electrons and backscatter (BS) electronsemanates from the sample 410. As the emitted radiation is position-sensitive (due to said scanning motion), the information obtained from the detectors 100, 420, will also be position-dependent.

(77) The signals from the detectors 100, 420 are processed by the processing apparatus 424, and displayed on display unit 426. Such processing may include operations such as combining, integrating, subtracting, false colouring, edge enhancing, and other processing known to the skilled artisan. In addition, automated recognition processes (e.g. as used for particle analysis) may be included in such processing. In the context of the current invention, the processing apparatus 424and/or a dedicated separate processing unit (not shown)can be used to perform the prescribed mathematical manipulations on said measurement set M so as to deconvolve it and spatially resolve it into the result set R discussed above.

(78) It should be noted that many refinements and alternatives of such a set-up will be known to the skilled artisan, including, but not limited to: The use of dual beamsfor example an electron beam 404 for imaging and an ion beam for machining (or, in some cases, imaging) the sample 410; The use of a controlled environment at the sample 410for example, maintaining a pressure of several mbar (as used in a so-called Environmental SEM) or by admitting gases, such as etching or precursor gases;
etc.
Embodiment 8

(79) FIG. 5 illustrates the operating principle of a particular detector arrangement that can be used to produce energy-filtered outputs O.sub.n for a detected beam of electrons of mixed energies E.sub.n, thus serving to compile a measurement set M={(O.sub.n, E.sub.n)} as set forth in the current invention. More particularly, the depicted detector arrangement is of the type (A) set forth above.

(80) In the Figure, a beam 51 of electrons (e.g. BS electrons) enters a measurement cavity 53 through an aperture 55. The beam 51 comprises a portion of the emitted radiation emanating from a sample (such as item 410 in FIG. 4) when it is irradiated by a beam of charged-particle radiation (such as item 404 in FIG. 4), and can, for example, be produced by causing said emitted radiation to pass through an aperture plate (not depicted). In FIG. 5, the beam 51 is depicted as being vertical but, in general, it can also have another orientation.

(81) The interior 57 of the cavity 53 is permeated by a suitable deflecting field (not depicted), e.g. a (uniform) magnetic field with field lines perpendicular to the plane of the Figure. When they encounter this field, electrons in the beam 51 undergo a deflection, whose magnitude will depend on the kinetic energy of the electrons in question. As a result, what enters the cavity 53 as a well-defined beam 51 is converted into a fanned-out array of sub-beamsfour of which (51a, 51b, 51c, 51d) are illustrated herewhereby relatively low-energy electrons in the beam 51 undergo relatively large deflections, and vice versa. These sub-beams 51a, 51b, 51c, 51d impinge upon respective detection modules 59a, 59b, 59c, 59d of detector arrangement 59, each of which modules 59a, 59b, 59c, 59d may be a separate energy detector (such as an SSPM) or an individual segment of a segmented detector, for example. Since the sub-beams 51a, 51b, 51c, 51d will each be characterized by a different electron energy E.sub.n (in practice, a relatively narrow band of energies), the detection modules 59a, 59b, 59c, 59d of the detector arrangement 59 will produce an energy-resolved output, allowing an output value O.sub.n to be assigned to each energy value E.sub.n.