Method and device for beam analysis

Abstract

A method and an apparatus for beam analysis in an optical system are disclosed, wherein a plurality of beam parameters of a beam propagating along an optical axis are ascertained. The method includes: splitting the beam into a plurality of partial beams which have a focus offset in the longitudinal direction in relation to the optical axis; recording a measurement image produced by these partial beams; carrying out a forward simulation of the beam in the optical system on the basis of estimated initial values for the beam parameters in order to obtain a simulated image; and calculating a set of values for the beam parameters on the basis of the comparison between the simulated image and the measurement image.

Claims

1. A method, comprising: a) splitting a beam in an optical system into a plurality of partial beams which have a focus offset in a longitudinal direction in relation to an optical axis; b) recording a measurement image produced by the partial beams; c) performing a forward simulation of the beam in the optical system based on estimated initial values for beam parameters to obtain a simulated image; and d) calculating a set of values for the beam parameters based on a comparison between the simulated image and the measurement image; e) iteratively performing c) and d), wherein the respectively calculated values for the beam parameters form the basis of a forward simulation that follows in each case; and f) outputting output values for the beam parameters, ascertained by the iteration ine), wherein, during the iterative performance of c) and d), at least one of the following holds: the number of varied beam parameters is varied; and varying an algorithm used in the iteration.

2. The method of claim 1, further comprising recording a near-field image produced by the beam.

3. The method of claim 2, further comprising, simultaneously with recording the near-field image, recording a far-field image that corresponds to the measurement image produced by the partial beams.

4. The method of claim 1, wherein the plurality of beam parameters comprises at least one member selected from the group consisting of beam size, beam decentration, beam inclination, beam diver-gence, astigmatism, coma and spherical aberration.

5. The method of claim 1, further comprising, based the output values for the beam parameters, manipulating the beam while adapting at least one of the beam parameters.

6. The method of claim 5, further comprising, in real time during operation of the optical system, outputting the output values and manipulating the beam while adapting at least one of the beam parameters.

7. The method of claim 1, wherein a) comprising using a beam-splitting optical arrangement which brings about spherical wave-front deformations of the beam.

8. The method as claimed in claim 7, wherein the beam-splitting optical arrangement comprises a diffractive structure.

9. The method of claim 7, wherein the beam-splitting optical arrangement splits a beam incident on the arrangement into partial beams, and points of incidence of the partial beams form a two-dimensional, grid-like distribution on a plane extending transversely to the light propagation direction of the beam.

10. The method of claim 8, wherein the beam-splitting optical arrangement comprises two diffractive structures extending in mutually different directions.

11. The method of claim 10, wherein the diffractive structures differ by at least a factor of three in terms of their focal length related to the first positive order of diffraction in each case.

12. The method of claim 1, wherein the beam is a laser beam.

13. The method of claim 1, wherein the optical system comprises a laser plasma source.

14. The method of claim 1, wherein the method comprises using an apparatus which comprises: a beam-splitting optical arrangement configured to split a beam incident on the beam-splitting optical arrangement along an optical axis into a plurality of partial beams having a focus offset in a longitudinal direction relative the optical axis; and a sensor arrangement configured to capture the partial beams.

15. The method of claim 1, wherein the method comprises using an arrangement configured to split a beam incident on the arrangement along an optical axis into a plurality of partial beams having a focus offset in a longitudinal direction relative to the optical axis, wherein points of incidence of the partial beams form a two-dimensional, grid-like distribution on a plane extending transversely to a propagation direction of the beam.

16. The method of claim 15, wherein the arrangement comprises two diffractive structures extending in mutually different directions.

17. The method of claim 16, wherein the diffractive structures differ by at least a factor of three in terms of their focal length related to the first positive order of diffraction in each case.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The disclosure is explained in greater detail below on the basis of exemplary embodiments illustrated in the accompanying figures, in which:

(2) FIG. 1 shows a schematic illustration of a measurement arrangement used in an exemplary manner in a method according to the disclosure;

(3) FIGS. 2A and 2B show a schematic illustration (FIG. 2A) and a diagram (FIG. 2B) for explaining construction and functionality of an exemplary embodiment of a beam-splitting optical arrangement;

(4) FIG. 3 shows a schematic illustration of a further embodiment of a measurement arrangement used in an exemplary manner in a method according to the disclosure;

(5) FIGS. 4 and 5 show schematic illustrations for explaining the possible sequence of a method according to the disclosure;

(6) FIGS. 6A and 6B show schematic illustrations for explaining a problem underlying the disclosure.

(7) FIGS. 7, 8A-8E, 9, 10, 11A, 11B and 11C show schematic illustrations for explaining further embodiments of the disclosure; and

(8) FIG. 12 shows a schematic illustration of the design of an EUV light source in accordance with the prior art.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

(9) According to FIG. 1, a beam to be analyzed, which is produced by a laser light source (not depicted here) and comes from a telescope unit 101 (which has, inter alia, an attenuator 102 in FIG. 1), is initially incident on an optical beam splitter 103 in an exemplary measurement arrangement, part of the beam being decoupled immediately by the optical beam splitter and being steered onto a sensor arrangement 120 (e.g. In the form of a CMOS arrangement or a CCD arrangement). The portion transmitted through the beam splitter 103 reaches a beam-splitting optical arrangement 110 via deflection mirrors 104, 106 (between which a further attenuator 105 is arranged) and the sensor arrangement 120 from the beam-splitting optical arrangement via a further deflection mirror 107. Consequently, according to FIG. 1, a near-field image that is produced by the immediately decoupled part of the beam is also recorded in addition to the far-field image that is produced by the part of the beam that is steered via the beam-splitting optical arrangement 110. As already explained, this is advantageous in that the absolute value of the amplitude is already present in an immediate form and therefore it is only still involved to substantially determine the wavefront or the phase of the complex amplitude by way of a retrieval scheme yet to be explained below.

(10) In the exemplary embodiment, as indicated in FIG. 2A, the beam-splitting optical arrangement 110 has a diffractive structure 111 and a refractive optical element (refractive lens element) 112, which have a monolithic embodiment here and together form a multi-focal optical element. In a specific exemplary embodiment, the refractive optical element 112 can be a plano-convex lens element, wherein the diffractive structure 111 can be formed on the plane surface of this plano-convex lens element. The diffractive structure and refractive optical element or lens element can also have a separate configuration and a (preferably small) distance from one another in further embodiments. In principle, in accordance with the occurring orders of diffraction, a diffractive lens element has positive and negative focal lengths in accordance with

(11) $\begin{array}{cc}{f}_{\mathrm{diff}}=\frac{f_1}{k},k=0,1,2,\mathrm{.Math.}& \left(1\right)\end{array}$

(12) Here, f.sub.1 denotes the focal length of the first positive order of diffraction and k denotes the beam index or the order of diffraction. Here, the intensity of the respective focus depends directly on the embodiment and approximation form of the underlying (approximately parabolic) phase profile. In combination with a refractive lens element with a focal length of f.sub.0, a multi-focal optical system emerges with a plurality of used focal lengths f.sub.k, k=0, 1, . . . , k.sub.max, wherein the following applies approximately if the distance between the diffractive structure and the refractive lens element is neglected:

(13) $\begin{array}{cc}{f}_k\frac{f_0{f}_1}{f_1+{\mathrm{kf}}_0}& \left(2\right)\end{array}$

(14) This relation is elucidated in FIG. 2B for f.sub.1>>f.sub.0.

(15) The disclosure is not restricted to the configuration of the beam-splitting optical arrangement 110 with such a diffractive structure. Rather, what is important in the configuration of the beam-splitting optical arrangement is that it causes, where possible, spherical wavefront deformations of the beam that is incident on the beam-splitting optical arrangement. In further embodiments, use may also be made of a different beam-spitting optical arrangement suitable to this end, for example in the form of an etalon.

(16) The partial beams emanating from the beam splitting optical arrangement are thereupon incident onreference once again being made to FIG. 1the sensor arrangement 120, on which different spot images corresponding to the focus offset are produced, the size of the spot images being smallest in the middle or in the perfect focus in the shown example and increasing toward the edge. The recording produced by the sensor arrangement 120 is denoted 121.

(17) FIG. 3 shows a further embodiment of a measurement arrangement, wherein components analogous or substantially functionally identical to FIG. 1 are designated by reference numerals increased by 200. The embodiment of FIG. 3 differs from that of FIG. 1 in that provision is made here of a diffractive structure 310 in the form of a reflective element, with provision further being made of a spherical deflection mirror 307.

(18) In principle, the recording of these individual spot images assigned to different focus positions in each case by the application of known, so-called phase retrieval methods (e.g. Gerchberg-Saxton algorithm) would allow a back calculation to the phase of the wave-front if the individual spot images were independent of one another (i.e. if there were no mutual influencing by way of interference). However, unavoidable interferences between the individual spot images are present in this case, the interferences leading to a pronounced mutual disturbance as indicated in FIG. 6 (with FIG. 6A showing ideal spot images without taking account of the interference and FIG. 6B showing real spot images with taking account of the interference).

(19) Mathematically, these circumstances mean that no unique back transformation is possible for directly calculating the beam parameters. In order to take account of this problem, iterative comparisons between respectively calculated or simulated images and the recorded measurement image are performed according to the disclosure in a model-based approach, as described below with reference to FIG. 4 and FIG. 5:

(20) As indicated in the schematic diagram of FIG. 4, this is initially based on estimated values for the sought-after beam parameters (step S410), with the corresponding parameters set being denoted by a.sub.1, a.sub.2, a.sub.3, . . . in this case.

(21) These parameters for describing the beam may be, for example, the beam size, beam decentration in the x-direction, beam decentration the y-direction, beam inclination in the x-direction, beam inclination in the y-direction, beam divergence, astigmatism in the x-direction, astigmatism in the y-direction, coma in the x-direction, coma in the y-direction and spherical aberration. Here, a Zernike parameterization may also be effectuated when desired in order to describe and ascertain corresponding wavefront aberrations of higher order too.

(22) Thereupon there is a forward simulation (step S420) for ascertaining a calculated image. According to FIG. 5, this forward simulation includes, in particular, a free space propagation P.sub.1 in the form of a Fourier transform upstream of the beam-splitting optical arrangement 110 or 310 (with respect to the light propagation direction) and a free space propagation P.sub.2, likewise in the form of a Fourier transform, downstream of the beam-splitting optical arrangement 110 or 310 (with respect to the light propagation direction), which each act on the complex amplitude function u={square root over (I.Math.e.sup.i)}.

(23) If the assumption of a beam propagation in the positive z-direction is made, the beam amplitude to be determined z.sub.0 in the range of scalar diffraction) is denoted by u(x,y|z.sub.0) at the location z.sub.0 in the reference plane (ideally near-field plane). After passing over the free space path between the reference plane and the plane of the effective optical element (beam-splitting optical arrangement 110 or diffractive optical structure), the amplitude present at the entrance of the optical element forming the beam-splitting optical arrangement 110 or 310 at the position z.sub.1 is given by
u(x,y|z.sub.1)={circumflex over (P)}.sub.1u(x,y|z.sub.0)=IFT.sub.xy[(z.sub.1z.sub.0).Math.FT.sub.xy[u(x,y|z.sub.0)]](3).

(24) The optical element forming the beam-splitting optical arrangement 110 or 310, in the approximation of the infinitely thin element, multiplicatively impresses the amplitude function T(x,y)=t(x,y).Math.exp(i(x,y))u(x,y|z.sub.1) according to
u(x,y|z.sub.1+)=T(x,y)u(x,y|z.sub.1)(4).

(25) By way of a further free space propagation from the optical element forming the beam-spitting optical arrangement 110 or 310 to the sensor arrangement 120 or 320 (the plane of which lies perpendicular to the z-axis at the position z.sub.2), the amplitude on the plane of the sensor arrangement 120 or 320 is finally arrived at according to
u(x,y|z.sub.2)={circumflex over (P)}.sub.2u(x,y|z.sub.1+)=IFT.sub.xy[(z.sub.2z.sub.1).Math.FT.sub.xy[u(x,y|z.sub.1+)]](5)

(26) The intensity profile detected at the spatially-resolving sensor arrangement 120 or 320 is obtained by forming the square of the absolute value according to
I.sub.Sensor(x,y)=|u(x,y|z.sub.2)|.sup.2(6)

(27) The propagator of the free space propagation is known from the formalism of Fourier optics. During the propagation from a plane perpendicular to the z-axis at the position z to a parallel plane at the position z, the amplitude is initially transformed into the frequency space according to
(f.sub.x,f.sub.y|z)=FT.sub.xy[u(x,y|z)]=dxdyu(x,y|z)exp(2i(f.sub.xx+f.sub.yy))(7)
by way of the 2D Fourier transform and there it is multiplied by the free space propagation function

(28) $\begin{array}{cc}\left(d|f_x,f_y\right)=\exp \left(2i\frac{d}{}\left(f_x,f_y\right)\right)& \left(8\right)\end{array}$
over the distance d=zz. Here, the phase in the propagation function is given by
(f.sub.x,f.sub.y)={square root over (1.sup.2(f.sub.x.sup.2+f.sub.y.sup.2))}(9),
where f.sub.x,f.sub.y denote the spatial frequencies and denotes the wavelength of the radiation. The amplitude in the plane at z in the spatial domain is finally obtained by a back transformation by way of the inverse Fourier transform according to

(29) $\begin{array}{cc}u\left(x,y|z\right)=\underset{A}{}\mathrm{dxdy}\left(z-z^{}|f_x,f_y\right)\tilde{u}\left(f_x,f_y|z^{}\right)\exp \left(+2i\left(f_{x}x+f_{y}y\right)\right)& \left(10\right)\end{array}$

(30) The correspondingly calculated image (containing the calculated intensity values I.sub.calc) is subtracted from the recorded measurement image (containing the measured intensity values I.sub.meas), whereupon appropriately modified model parameters for describing the beam are ascertained and these form the basis of a new forward simulation (step S460 in FIG. 4). Here, an optimization is performed, for example by applying a Levenberg-Marquardt algorithm. Thereupon, according to FIG. 5, the calculated intensity values I.sub.calc are ascertained anew by a forward simulation, with a new calculated image being obtained thereby, wherein the difference between the calculated image and the recorded measurement image is ascertained again. This is carried out iteratively until the difference between the calculated image and the recorded measurement image is sufficiently small (or falls below a predetermined threshold), whereupon the appropriate parameters for describing the beam are output.

(31) As already explained above, it is particularly advantageous if the absolute value of the amplitude is available from the near-field measurement and if it need not be described and fitted by a model. As a result, firstly, the number of parameters to be described is reduced, possibly significantly reduced, and, secondly, the quality of the information obtained about the beam to be measured is improved.

(32) According to the disclosure, the fact that enabling a large number of parameters leads to high numerical complexity is preferably further taken into account. Consequently, there preferably is initially a start with a comparatively small scope of the parameters set, which is then successively expanded with respect to the simultaneously varied parameters of the parameters set, i.e. adaptive fitting of the model is undertaken. Thus, for instance, if twenty parameters are sought after in principle, only ten dominating parameters may initially be enabled.

(33) Furthermore, it is also possible to adapt the respective evaluation method or the algorithm in order to obtain the fastest possible speed for the beam analysis, for example after reaching a quasi-stationary operation of the respective system (e.g. a largely stably operated plasma light source), in which, typically, only small changes in the beam parameters still occur. Here, in particular, use can be made of the already collected information in order then to be able to determine and correct, in real time, the small changes in the beam parameters that still occur. In this phase, the originally nonlinear optimization problem may also be approximable in linear form. As a result, what may be achieved thus is that, for instance in a plasma light source, the laser beam can be guided, accurately and quickly at the same time, with respect to the beam parameters.

(34) FIGS. 7-11 show schematic illustrations for explaining further embodiments of the disclosure. What is common to these embodiments is that, in this case, the beam-splitting optical arrangement according to the disclosure is configured in such a way that the beam splitting is effectuated in a two-dimensional manner to the extent that the beams with a longitudinal focus offset relative to one another that were produced during this split form, with respect to the points of incidence on the plane, a two-dimensional grid-like arrangement in, in each case, a plane transverse to the light propagation direction and henceaccording to an aspect of this configurationthey are suitable, in particular, for effectively filling a two-dimensional sensor or detector area. According to a further aspect of this configuration, a significant measurement range increase can also be obtained hereby, as will be described in more detail below.

(35) In order to obtain this two-dimensional beam split, it is possiblewithout the disclosure being restricted heretoto provide, for example in the embodiment schematically illustrated in FIG. 7, two diffractive optical elements 711, 712, of which, in turn, the one diffractive element 711 is arranged in an exemplary manner upstream of a refractive optical element (refractive lens element) 713, which is present in a manner analogous to the embodiment of FIG. 2A, in relation to the light propagation direction (extending in the z-direction in the plotted coordinate system) and the other diffractive optical element 712 is arranged downstream of this refractive optical element 713. In further embodiments (some of which are schematically illustrated in FIGS. 8A-8E), the diffractive structure involved for the two-dimensional beam split may also be realized in any other suitable manner.

(36) The mode of operation of the beam-splitting optical arrangement according to FIG. 7 formed by the diffractive optical elements 711, 712 and the refractive optical element 913 is elucidated in FIG. 9 (where analogous or functionally equivalent components are denoted by reference signs that have been increased by 200). Accordingly, according to FIG. 9, the first diffractive optical element 911 produces a split of the incident beam with fanning into partial beams which diverge in the xz-plane and the second diffractive optical element 911 produces a split with fanning into partial beams which diverge in the yz-plane. The resultant two-dimensional beam distribution obtained in the sensor or detector plane is denoted by 950.

(37) Proceeding from the basic construction of the beam-splitting optical arrangement according to FIG. 7 or FIG. 9, FIG. 10 shows a schematic illustration for explaining an exemplary geometric design. The two aforementioned diffractive optical elements are only indicated here by the respective planes 1011 and 1012, wherein, at the same time, a decentration by offsetting the respective diffractive structure in a plane perpendicular to the optical axis, respectively by distance d.sub.x or d.sub.y is indicated for the purposes of obtaining a lateral offset of the partial beams (break of symmetry). In FIG. 10, 1001 denotes the entrance plane of the beam and 1015 denotes the sensor or detector plane.

(38) Analogous to the embodiment of FIG. 2A, the multifocal optical system formed by the beam-splitting optical arrangement according to FIG. 7 or 9 has a plurality of used focal lengths, wherein the following approximately applies if the distance between the diffractive optical elements and the refractive lens element is neglected here:

(39) $\begin{array}{cc}{f}_{m,n}\frac{f_0{f}_1^{*}{f}_2^{*}}{f_1^{*}n+f_2^{*}m+f_0\mathrm{mn}}& \left(11\right)\end{array}$

(40) Here, f.sub.1* and f.sub.2* denote (in relation to the respective first positive order of diffraction) the respective focal lengths of the first diffractive optical element 911 and the second diffractive optical element 912, respectively, and f.sub.0 denotes the focal length of the refractive optical element 913, while m and n denote the orders of diffraction of the respective diffraction at the first optical element 911 and second diffractive optical element 912, respectively.

(41) The focal lengths of the first diffractive optical element 911 and the second diffractive optical element 912 are selected to be different from one another, with the consequence that the element with the relatively shorter focal length produces the relatively greater longitudinal focus offset, and vice versa. In a specific exemplary embodiment, it is possible, for instance, for the focal length of the first diffractive optical element 911 to be greater by a factor of five than the focal length of the second diffractive optical element 912.

(42) In the case of a suitable selection of the aforementioned parameters (i.e. the focal lengths f.sub.1*, f.sub.2* and f.sub.0) and of the used value ranges of the orders of diffraction (n, m), it is now possible to obtain and use a measurement range increase in different ways, as will be explained below with reference to FIGS. 11A-11C.

(43) FIG. 11A initially shows a possible distribution of focal lengths obtained over the individual fanned beams using a beam-splitting optical arrangement according to the disclosure in accordance with FIG. 7 or 9. Here, a value group A, B, C, . . . of in each case seven values or points in the diagram (corresponding to the number of orders of diffraction in the value range of 3 to +3 selected in an exemplary manner) in each case corresponds to a line in the two-dimensional beam distribution obtained in the sensor or detector plane. Firstly, it is clear that, as a consequence of the two-dimensional beam fanning, a measurement value increase is obtained in relation to the one-dimensional beam fanning that took place in the exemplary embodiment of FIG. 2A (which would have only resulted in a single one of the value groups A, B, C, . . . ), with the consequence that a comparatively large measurement range of focal lengths is covered with, at the same time, a high resolution. Further, it is clear that, according to FIG. 11A, this measurement range increase can also be used for calibration purposes by virtue of redundancies namely being created to the extent that the value groups A, B, C, . . . partly overlap with respect to the respective focal length values in the diagram of FIG. 11A. In further embodiments, it is also possible, as illustrated in FIG. 11B and FIG. 11C, to dispense with such redundancies for the benefit of a further increase of the measurement range of focal lengths covered overall, wherein the individual value groups A, B, C, . . . of, in each case, seven values or points may be produced continuously (according to FIG. 11B) or else with a certain gaps or distances between the value groups A, B, C, . . . (according to FIG. 11C).

(44) The above-described measurement range increase can be used to take account of the large focus variations of the respective beam to be characterized, as occur, for example in applications of material processing, in particular at high laser powers, as a consequence of heating and deformation of the individual optical components, namely by virtue of the capture region of the respective focus values being significantly increased (for example, by approximately a factor of seven according to FIGS. 11A-11C) with an unchanged high resolution. Further particularly, this measurement range increase can be used to realize an overall ISO-compliant beam characterization to the extent that measurement points are obtained in, respectively, a sufficient number or a number that is prescribed by the respective ISO standard, both in the immediate vicinity of the focus and also at a sufficient distance from the focus (e.g. at a distance of two Raleigh lengths) of the beam.

(45) In further embodiments (some of which are schematically illustrated in FIGS. 8A-8E), the diffractive structure involved for the two-dimensional beam split may also be obtained in any other suitable manner. Here, the sensor arrangement (e.g. CCD camera) is respectively denoted by 815 in FIGS. 8A-8E.

(46) According to FIGS. 8A-8B, use can also be made of a single diffractive optical element 811 or 821, which is already inherently two-dimensional (i.e. it has periodic diffractive structures in mutually different, in particular perpendicular directions), instead of two diffractive optical elements, and the single diffractive optical element may be arranged, in relation to the light propagation direction, either upstream (FIG. 8A) or downstream (FIG. 8B) of the refractive optical element 813 or 823 in a beam-splitting optical arrangement 810 or 820. According to FIG. 8C, the diffractive structures (which, in turn, extend in mutually different, in particular perpendicular directions) may also be designed on plano-convex lens elements 831, 832 in a beam-splitting optical arrangement 830. According to FIG. 8D, a beam-splitting optical arrangement 840 may also have a diffractive optical element 841 with a complex diffractive structure (e.g. as a complex-encoded CGH), which brings about a diffraction in mutually different, in particular perpendicular directions, in combination with a refractive lens element 843, wherein, in this configuration too, the diffractive optical element 841 may be alternatively arranged downstream, in relation to the light propagation direction, of the refractive lens element 843 as well. According to FIG. 8E, a beam-splitting optical arrangement 850 may also be configured as a refractive lens element 851, which has diffractive structures on both its light entry face and its light exit face. In all of the above-described embodiments, the diffractive optical elements or diffraction gratings may be realized as amplitude gratings, phase gratings or hybrid gratings.

(47) Even though the disclosure has been described on the basis of specific embodiments, numerous variations and alternative embodiments are apparent to a person skilled in the art, for example by combination and/or exchange of features of individual embodiments. Accordingly, it goes without saying for a person skilled in the art that such variations and alternative embodiments are concomitantly encompassed by the present disclosure, and the scope of the disclosure is restricted only within the meaning of the accompanying patent claims and the equivalents thereof.