NL2028887B1 - Method for determining a wavefront generated by means of a diffractive optical element - Google Patents
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- 230000003287 optical effect Effects 0.000 title claims abstract description 111
- 238000000034 method Methods 0.000 title claims abstract description 55
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- 238000012360 testing method Methods 0.000 claims description 88
- 238000005259 measurement Methods 0.000 claims description 18
- 238000012876 topography Methods 0.000 claims description 15
- 230000008569 process Effects 0.000 claims description 12
- 238000004422 calculation algorithm Methods 0.000 claims description 9
- 238000004364 calculation method Methods 0.000 claims description 9
- 238000012937 correction Methods 0.000 claims description 6
- 238000001393 microlithography Methods 0.000 claims description 5
- 230000003993 interaction Effects 0.000 claims description 4
- 238000001514 detection method Methods 0.000 claims description 3
- 238000004088 simulation Methods 0.000 claims description 3
- 230000006870 function Effects 0.000 description 26
- 238000001228 spectrum Methods 0.000 description 15
- 238000011156 evaluation Methods 0.000 description 11
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- 238000000691 measurement method Methods 0.000 description 1
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B9/00—Measuring instruments characterised by the use of optical techniques
- G01B9/02—Interferometers
- G01B9/02034—Interferometers characterised by particularly shaped beams or wavefronts
- G01B9/02038—Shaping the wavefront, e.g. generating a spherical wavefront
- G01B9/02039—Shaping the wavefront, e.g. generating a spherical wavefront by matching the wavefront with a particular object surface shape
Abstract
A method for determining a wavefront (54) generated by means of a diffractive optical element (24) comprises the following steps: providing an n-dimensional aperiodic pattern (58) which represents diffraction structures (34) arranged on the diffractive optical element, where n has a value of 1 or 2, and computationally determining the wavefront (54) generated by means of the diffractive optical element while taking account of the n-dimensional aperiodic pattern, wherein the n-dimensional aperiodic pattern (58) is embedded in a higher-dimensional representation during the computational determination of the wavefront, the diffraction structures being represented by a periodic pattern (64) in said higher-dimensional representation.
Description
P180779NL00
Title: Method for determining a wavefront generated by means of a diffractive optical element
This application claims priority to the German Patent Application No. 2020 209 580.9 filed on July 30, 2020. The entire disclosure of this patent application is incorporated into the present application by reference.
Background of the invention 10 The invention relates to a method for determining a wavefront generated by means of a diffractive optical element and to a method and a measuring apparatus for the interferometric measurement of a shape of a surface of a test object.
Interferometric measuring apparatuses and methods in which a diffractive optical element generates a test wave and a reference wave from an input wave are known for the highly accurate interferometric measurement of optical surfaces down to the subnanometre range. The diffractive optical element allows the wavefront of the test wave to be adapted to a target surface of the test object in such a way that said wavefront is incident on each location of the target shape in substantially perpendicular fashion and reflected back onto itself from the latter.
Deviations from the target shape can then be determined with the aid of an interferogram formed by superposing the reflected test wave on the reference wave.
US2018/0106591A1 describes an embodiment of such an interferometric measuring apparatus, in which a complex encoded computer-generated hologram (CGH) finds use as a diffractive optical element. In addition to the test wave directed at the surface to be measured and the reference wave running through a reference arm, the CGH generates a plurality of calibration waves from an input wave.
Very accurate knowledge about the wavefront of the test wave generated by the diffractive optical element is required to precisely determine the shape of the surface of the test object. To this end, FEM ("finite element method") processes for solving Maxwell's equations in the real domain are available in the prior art. FEM allow local electromagnetic fields to be computed very accurately.
However, for the interferometric approach the result must be available in the Fourier domain, i.e, the interest lies in decomposing the electromagnetic fields into plane waves which describe the direction to the test surface or to the reference surface. Therefore, the FEM calculations must be available over a very large area so that the Fourier representation of the electromagnetic fields can be computed with the required accuracy.
The structures on the CGH surface must be scanned very accurately. The resultant grid and hence the resultant memory requirements are huge.
Therefore, such an FEM calculation requires a supercomputer and would take several days. Hence, the FEM processes are not practicable in the real domain.
Document DE 10 2012 217800 Al describes a method for determining a wavefront generated by means of a diffractive optical element. A spherical output shaft (58) passes through optical elements to an interferometer camera (45). An interference pattern forms on a detection surface (47), indicating deviations from its ideal wavefront. These deviations, determined using a calibration sphere (62), are recorded as calibration discrepancies.
Object addressed
It is an object of the invention to provide methods and a measuring apparatus of the type set forth at the outset, which solve the aforementioned problems, and, in particular, to ascertain a diffractive optical element- generated wavefront with great accuracy and, at the same time, reasonable computational complexity and hence within a comparatively short time frame.
Solution according to the invention
According to the invention, the aforementioned object can be achieved, for example, using a method for determining a wavefront generated by means of a diffractive optical element, the method comprising the following steps: providing an n-dimensional aperiodic pattern which represents diffraction structures arranged on the diffractive optical element, where n has a value of 1 or 2, and computationally determining the wavefront generated by means of the diffractive optical element while taking account of the n-dimensional aperiodic pattern, wherein the n- dimensional aperiodic pattern is embedded in a higher-dimensional representation during the computational determination of the wavefront, the diffraction structures being represented by a periodic pattern in said higher-dimensional representation.
For n = 2, the aperiodic pattern is a 2-dimensional aperiodic pattern, i.e. the pattern is aperiodic in both dimensions. The higher-dimensional representation should be understood to mean an at least (n+1)-dimensional representation. One-time encoded diffraction patterns are periodic, and so no aperiodic n-dimensional representation can be specified therefor. By contrast, two-times encoded diffraction patterns are also periodic as a rule.
However, for the case in which the diffraction pattern only diffracts in one direction, 1.e., the diffraction only occurs in a plane, the diffraction pattern can also be aperiodic and hence can be specified in the aforementioned aperiodic one-dimensional representation. Three- and more- times encoded diffraction patterns are aperiodic as a matter of principle and could therefore be specified in the aforementioned aperiodic two- dimensional representation.
Embedding the n-dimensional aperiodic pattern into a higher- dimensional representation in which the diffraction structures are represented by a periodic pattern, in accordance with the invention, facilitates the computation of the wavefront generated by means of the diffractive optical element with a high accuracy and, at the same time, greatly reduced computational complexity, and so the time frame for ascertaining the wavefront can be kept comparatively short.
According to one embodiment, during the computational determination of the wavefront, the pattern emerging from the higher- dimensional periodic representation is represented by means of a Fourier series and the n-dimensional aperiodic pattern is converted into a Fourier representation by means of expansion coefficients contained in the Fourier series of the periodic pattern. The expansion coefficients are also referred to as Fourier coefficients. The expansion coefficients of the Fourier series can be ascertained by means of a one-dimensional FFT algorithm (in particular by means of the "rank-1 lattice" algorithm known to a person skilled in the art).
According to a further embodiment, the wavefront generated by means of the diffractive optical element is determined by means of an algorithm on the basis of the Fourier representation of the n-dimensional aperiodic pattern and a topography of the diffraction structures arranged on the diffractive optical element. The algorithm can be based on Fourier modal methods, in particular the RCWA (rigorous coupled wave analysis) method known to a person skilled in the art.
According to a further embodiment, the aperiodic pattern is two- dimensional and represents an at least 3-times encoded diffraction pattern.
A multiple-times encoded diffraction pattern or multiple-times encoded
CGH is understood to mean a complex encoded phase grating or CGH, which has a plurality of different phase functions. Expressed differently, a multiple-times encoded diffraction pattern comprises a plurality of diffractive structure patterns arranged overlaid on one another; three diffractive structure patterns arranged overlaid on one another are present in the case of a 3-times encoded diffraction pattern. Each of the different phase functions serves to generate a separate wave when an input wave is radiated onto the diffractive optical element. As a rule, the individual generated waves have different directions of propagation and can be configured with different wavefronts.
According to a further embodiment, the diffractive optical element is 5 configured for use in a measuring apparatus for the interferometric measurement of the shape of a surface of a test object. According to one embodiment, the test object is an optical element of a microlithographic projection exposure apparatus, in particular of a projection lens of a microlithographic projection exposure apparatus.
According to a further embodiment, a wavefront correction is determined by comparing a wavefront simulated on the basis of the layout of the diffraction structures to the wavefront determined taking into account the topography of the diffraction structures. The layout of the diffraction structures should be understood to mean the plan view of the pattern generated by the diffraction structures, i.e., the topography of the diffraction structures is not considered here.
In this case, the pattern can have a binary design, i.e., certain elements of the pattern bring about a specified phase step while the regions between these elements cause a different phase step or no phase step. The layout itself can have a two-dimensional design or else a one-dimensional design. The aforementioned wavefront correction takes account of the three- dimensional or rigorous effects of the diffractive structures. According to one embodiment, the wavefront simulated under two-dimensional consideration of the diffraction structures is a target wavefront which corresponds to the target shape of the surface of the test object.
Furthermore, the invention provides a method for the interferometric measurement of a shape of a surface of a test object, which comprises the following steps: radiating a test wave, which is generated by means of a diffractive optical element, onto the surface of the test object, computationally determining a wavefront of the test wave by means of the method according to any one of the above-described embodiments or embodiment variants, and evaluating an interferogram which was generated by means of the test wave after interaction with the surface of the test object, with the wavefront of the test wave determined by computation being taken into account.
According to one embodiment of the method for the interferometric measurement, the test wave and a reference wave are generated simultaneously by radiating an input wave onto the diffractive optical element, said reference wave being superposed on the test wave when generating the interferogram. During the generation of the interferogram, the reference wave is superposed on the test wave after the interaction of the latter with the surface of the test object. Alternatively, the reference wave can be generated at a Fizeau element. In particular, the surface of the test object is an optical surface, for example a mirror surface or the surface of a microlithographic lens.
According to a further embodiment, at least one calibration wave is additionally generated when the input wave is radiated onto the diffractive optical element.
According to a further embodiment, the test object 1s an optical element of a microlithographic projection exposure apparatus, in particular an optical element of a projection lens of a microlithographic projection exposure apparatus.
According to a further embodiment, the test object 1s an optical element for EUV microlithography. EUV microlithography is understood to mean microlithography with an operating wavelength of less than 100 nm, in particular an operating wavelength of approximately 13.5 nm or approximately 6.8 nm.
Furthermore, the invention provides a method for producing a diffractive optical element, which comprising the following steps: determining a structure pattern for diffraction structures of a diffractive optical element for generating a specified target wavefront on the basis of a simulation calculation that only takes the layout of the diffraction structures into account and computationally determining the wavefront generated by means of the optical element according to one embodiment or embodiment variant of the above-described method for determining a wavefront generated by means of a diffractive optical element, wherein a topography of the diffraction structures arranged on the diffractive optical element is taken into account during the computational determination of the wavefront. Furthermore, the method according to the invention comprises determining a correction for the structure pattern on the basis of a wavefront difference between the actual wavefront and the target wavefront for the purposes of minimizing the wavefront difference, and producing the diffractive optical element on the basis of the corrected structure pattern.
According to one embodiment of the aforementioned production method, the target wavefront corresponds to a target shape of an optical surface of a test object, the diffractive optical element being configured for the interferometric measurement of the latter.
Furthermore, the invention provides a measuring apparatus for the interferometric measurement of a shape of a surface of a test object, the measuring apparatus comprising: a diffractive optical element for generating a test wave that is radiated onto the surface of the test object and a wavefront determining device which is configured to computationally determine a wavefront of the test wave taking account of an n-dimensional aperiodic pattern which represents diffraction structures arranged on the diffractive optical element, where n has a value of 1 or 2, and in the process embed the n-dimensional aperiodic pattern in a higher-dimensional representation, in which the diffraction structures are represented by a periodic pattern. Furthermore, the measuring apparatus according to the invention comprises a detection device for capturing an interferogram generated by means of the test wave after interaction with the surface of the test object.
By evaluating the interferogram, it is possible to determine the shape of the surface of the test object when taking account of the wavefront of the test wave determined by computation; in particular, the measuring apparatus comprises an appropriately configured evaluation device.
The features indicated with regard to the abovementioned embodiments, exemplary embodiments or embodiment variants, etc., of the interferometric measurement method according to the invention can be correspondingly applied to the measuring apparatus according to the vention, and vice versa. These and other features of the embodiments according to the invention are explained in the description of the figures and in the claims. The individual features can be implemented, either separately or in combination, as embodiments of the invention. Furthermore, they can describe advantageous embodiments which are independently protectable and protection for which is claimed if appropriate only during or after pendency of the application.
The above and further advantageous features of the invention are illustrated in the following detailed description of exemplary embodiments according to the invention with reference to the accompanying schematic drawings. In the drawings:
Figure 1 shows an exemplary embodiment of a measuring apparatus for the interferometric determination of a shape of an optical surface, comprising a diffractive optical element for generating a test wave that is radiated onto the surface and an evaluation device which is configured to determine the surface shape on the basis of a wavefront of the test wave determined by computation by a wavefront determining device,
Figure 2 shows a schematic illustration elucidating the functionality of the wavefront determining device as per Figure 1,
Figure 3 shows a section of a plan view of an embodiment of diffraction structures of the diffractive optical element as per Figure 1, which represents a two-dimensional aperiodic pattern,
Figure 4 shows a pixelated version of the section as per Figure 3,
Figure 5 shows the profile of two one-dimensional aperiodic functions,
Figure 6 shows a unit cell of a two-dimensional periodic function generated by embedding the functions as per Figure 5,
Figure 7 shows the two-dimensional periodic function, the basis of which 1s the unit cell as per Figure 6,
Figure 8 shows the spectrum of the two-dimensional periodic function as per Figure 7 and
Figure 9 shows the spectra of the two one-dimensional aperiodic functions.
Detailed description of exemplary embodiments according to the invention
In the exemplary embodiments or embodiments or embodiment variants described below, elements which are functionally or structurally similar to one another are provided with the same or similar reference signs as far as possible. Therefore, for understanding the features of the individual elements of a specific exemplary embodiment, reference should be made to the description of other exemplary embodiments or the general description of the invention.
In order to facilitate the description, a Cartesian xyz-coordinate system is indicated in the drawing, from which system the respective positional relationship of the components illustrated in the figures is evident. In Figure 1, the y-direction extends perpendicularly to the plane of the drawing into said plane, the x-direction extends toward the right, and the z-direction extends upward.
Figure 1 elucidates an exemplary embodiment of a measuring apparatus 10 for determining the shape of an optical surface 12 of a test object 14 by interferometry. The measuring apparatus 10 can be used, in particular, to determine a deviation of the actual shape of the surface 12 from a target shape.
The test object 14 provided can be, for example, a mirror of a projection lens of a projection exposure apparatus for EUV microlithography having a non-spherical surface for reflecting EUV radiation at a wavelength of less than 100 nm, in particular a wavelength of approximately 13.5 nm or approximately 6.8 nm. The non-spherical surface of the mirror can have a free-form surface with a deviation from each rotation-symmetric asphere of more than 5 um and a deviation from each sphere of at least 1 mm.
The measuring apparatus 10 includes a light source 16 for providing a sufficiently coherent measurement radiation as an input wave 18. In this exemplary embodiment, the light source 16 comprises an optical waveguide having an exit surface 22. The optical waveguide 20 is connected to a radiation source (not illustrated in Figure 1), e.g., in the form of a laser. By 20 way of example, provision to this end can be made of a helium-neon laser with a wavelength of approximately 633 nm.
However, the illumination radiation may also have a different wavelength in the visible or non-visible wavelength range of electromagnetic radiation. The light source 16 with the optical waveguide 20 constitutes merely one example of a light source that may be used for the measuring apparatus 10. In alternative configurations, rather than the optical waveguide 20, an optical arrangement with lens elements, mirror elements or the like can be provided for providing a suitable input wave 18.
The illumination radiation provided by the light source 16 exits the exit surface 22 of the optical waveguide 20 in the form of an input wave 18 having a spherical wavefront and propagates divergently along a propagation axis 42 that is directed at a diffractive optical element 24. In the process, the input wave 18 first passes through a beam splitter and then through the diffractive optical element 24. In alternative configurations, a collimator for producing an input wave 18 having a plane wavefront can be provided between the beam splitter 38 and the diffractive optical element 24.
The diffractive optical element 24 serves to generate a test wave 26, a reference wave 28 and three calibration waves 50, 52 and 53 from the input wave 18. To this end, the diffractive optical element 24 is configured in the form of a complex encoded CGH and includes diffraction structures 34, which are generally formed by at least two diffractive individual patterns that are arranged superposed on one another in a plane. In the case of two superposed diffractive individual patterns, these can be formed, e.g., by a first diffractive individual pattern in the form of a basic grating and a second diffractive individual pattern in the form of a super grating.
Depending on the number k of diffractive individual patterns arranged in superposed fashion, the resultant diffraction pattern is referred to as a k-times encoded diffraction pattern. In the exemplary embodiment illustrated in Figure 1, the diffraction structures 34 are formed by a 5-times encoded diffraction pattern, i.e., the diffraction pattern is composed of five diffractive individual patterns superposed on one another.
A first of the diffractive individual patterns generates the test wave 26 with a wavefront 54 at least partly adjusted to a target shape of the optical surface 12 in transmission from the input wave 18. Expressed differently, the first diffractive individual pattern is configured in such a way that a target wavefront of the test wave 26 generated by the diffractive optical element 24 corresponds or is at least approximated to the target shape of the optical surface 12. The wavefront of the test wave 26 is therefore adjusted in such a way that the test wave 26 is incident in perpendicular or approximately perpendicular fashion on the surface 12 with a target shape at each location thereof and reflected back on itself. As will be explained in detail below, the wavefront of the test wave 26 incident on the surface 12 1s determined by computation, wherein the computationally determined wavefront is denoted by the reference sign 54a.
The test wave 26 propagates in the direction of the test object 14 and is incident on the optical surface 12 of the test object 14. The test wave 26 is reflected by the surface 12 back to the diffractive optical element 24 and is diffracted again upon passage through the diffraction structures 34. In the process, the reflected test wave 26 is transformed back into an approximately spherical wave, wherein the wavefront thereof has, due to deviations of the surface 12 of the test object 14 from the target shape, corresponding deviations from a spherical wavefront.
A second diffractive individual pattern of the diffraction structures 34 generates the reference wave 28 with a plane wavefront in transmission from the input wave. Furthermore, the measuring apparatus 10 comprises a reflective optical element 30, which is embodied as a plane mirror and which serves to reflect back the reference wave 28 with a plane wavefront. In another configuration, the reference wave 28 can have a spherical wavefront and the reflective optical element can be configured as a spherical mirror.
The reflected reference wave 28 again passes through the diffractive optical element 24 and is diffracted again. In the process, the reflected reference wave 28 1s transformed back into a spherical wave.
The measuring apparatus 10 furthermore includes a capture device 36 having a beam splitter 38 for guiding the combination of the reflected test wave 26 and the reflected reference wave 28 out of the beam path of the input wave 18, and an interferometer camera 40 for capturing an interferogram produced by superposing the reference wave 28 on the test wave 26. In the process, the measuring apparatus 10 is in a so-called test mode, in which the three calibration waves 50, 52 and 53 are blocked from impinging on respective calibration mirrors 70, 71 and 72 by means of respective shutters 74, 75 and 76. Alternatively, the calibration mirrors 70, 71 and 72 can also be removed from the beam paths of the calibration waves 50, 52 and 53 in the test mode so that the latter are not reflected back to the diffractive optical element 24.
The diffractive optical element 24 therefore also serves for superposing the reflected reference wave 28 on the reflected test wave 26.
Both waves are incident on the beam splitter 38 as convergent beams and are reflected thereby in the direction of the interferometer camera 40. Both convergent beams 44 travel through an eyepiece 46 and are ultimately incident on a capture area 48 of the interferometer camera 40. The interferometer camera 40 can be configured for example in the form of a
CCD sensor and captures an interferogram 41 produced by the interfering waves. Arranged in the focus of the convergent beams 44 can be a stop (not illustrated in Figure 1) as a spatial filter for reducing scattered radiation.
An evaluation device 78 of the measuring apparatus 10 determines the actual shape of the optical surface 12 of the test object 14 from the captured interferogram 41 or from a plurality of such interferograms 41. In the process, the evaluation device 78 additionally takes account of the wavefront 54a of the test wave 26 determined by computation, which is made available to the evaluation device 78 by a wavefront determining device 56. This relates to the wavefront of the test wave 26 after the generation thereof at the diffraction structures 34 of the diffractive optical element 24, 1.e., the wavefront of the test wave 26 before the latter is incident on the optical surface 12.
In the determination of the surface shape, the evaluation device 78 in particular takes into account a result of a calibration measurement of the diffractive element 24. As a result of the calibration measurement described in more detail below, it is possible to take account of changes in the optical properties of the diffractive optical element 24, which changes are caused by changes in temperature, for example. Such changes influence the wavefront of the test wave 26 generated by the diffractive optical element 24.
According to one embodiment, the evaluation device 78 uses the result of the calibration measurement for correcting the wavefront 54a provided by the wavefront determining device 56. According to an alternative embodiment, the result of the calibration measurement is already taken into account by the wavefront determining device 56 when calculating the wavefront 54.
As an alternative or in addition to the evaluation device 78, the measuring apparatus 10 can have a data memory or an interface with a network to make possible a determination of the surface shape using the interferogram 41 that is stored or transmitted via the network by way of an external evaluation unit.
To carry out the calibration measurement, the measuring apparatus 10 is operated in a calibration mode in which the test object 14 is not arranged in the beam path of the test wave 26 or the beam path of the test wave 26 is blocked by means of a shutter. The shutters 74, 75 and 76 in the beam paths of the calibration waves 50, 52 and 53 are successively opened and closed again, one at a time.
When the shutter 74 is opened, the calibration wave 50, which has a plane wavefront, is reflected back on itself by the plane calibration mirror 70. The superposition of the reflected calibration wave 50 with the reflected reference wave 28 generates a first calibration interferogram on the capture area 48 of the interferometer camera 40. When the shutter 75 is opened, the converging calibration wave 52, which has a spherical wavefront, is reflected back on itself by the convex spherical calibration mirror 71. The superposition of the reflected calibration wave 52 with the reflected reference wave 28 generates a second calibration interferogram on the capture area 48 of the interferometer camera 40. When the shutter 76 is opened, the expanding calibration wave 53, which has a spherical wavefront, is reflected back on itself by the concave spherical calibration mirror 72. The superposition of the reflected calibration wave 53 with the reflected reference wave 28 generates a third calibration interferogram on the capture area 48 of the interferometer camera 40. By evaluating the three calibration interferograms, the aforementioned changes in the optical properties of the diffractive optical element 24 are ascertained as a result of the calibration measurement.
The wavefront determining device 56 determines the wavefront 54 from a one-dimensional or two-dimensional aperiodic pattern 58 and a topography 60 of the diffraction structures 34. The pattern 58 corresponds to the contours visible in a plan view of the diffraction structures 34 or the aforementioned diffraction pattern resulting from the diffractive individual patterns arranged in superposed fashion. The two-dimensional pattern 58 is aperiodic in both dimensions on account of the superposition of five diffractive individual patterns. The topography 60 of the diffraction structures 34 should be understood to mean, in particular, a height distribution of the diffraction structures 34 as a function of the two- dimensional area in which the pattern 58 extends.
The wavefront determining device 56 for determining the wavefront 54 by computation can be part of the measuring apparatus 10.
Alternatively, the latter can also be provided as external computing unit. In this case, the measuring apparatus 10 has an interface for transmitting the wavefront to the evaluation device 78.
As elucidated in Figure 2 and described in detail below, the wavefront determining device 56 is configured to initially embed the aperiodic two- dimensional pattern 58 in a higher-dimensional representation (step S1), in which the diffraction structures 34 are represented by a periodic pattern 64.
Expressed differently, the periodic pattern is available in a three- dimensional or higher-dimensional representation of the two-dimensional pattern 58. In particular, the pattern 58 is embedded in the higher- dimensional representation by means of a dimension converter module 62.
Furthermore, the wavefront determining device 56 is configured to represent the periodic pattern 64 by means of a Fourier series 66 (step S2).
In particular, this process is implemented by means of an FFT module 65.
Furthermore, the wavefront determining device 56 1s configured to convert the aperiodic pattern 58 into a Fourier representation 68 of the aperiodic pattern by means of expansion coefficients contained in the Fourier series 66 (step S3). In particular, this process is implemented by means of a conversion module 67.
Furthermore, using an algorithm 69 on the basis of the Fourier representation 68 of the aperiodic pattern 58 and the topography 60 of the diffraction structures 34, the wavefront determining device 56 is configured to determine the wavefront 54 generated by means of the diffractive optical element 24. In particular, this process is implemented by means of a module provided to this end. By way of example, the algorithm 69 can be based on the RCWA (rigorous coupled wave analysis) method known to a person skilled in the art.
In summary, the wavefront determining device 56 is configured to carry out the following steps:
Step S1 — embedding the aperiodic pattern in a higher-dimensional representation, in which the former is represented by a periodic pattern;
Step S2 — representing the periodic pattern by means of a Fourier series;
Step S3 — converting the aperiodic pattern into a Fourier representation by means of expansion coefficients contained in the Fourier series; and
Step S4 — determining the wavefront generated by means of the diffractive optical element on the basis of the Fourier representation of the aperiodic pattern and the topography of the diffraction structures.
The determination carried out by means of the wavefront determining device 56 of the wavefront 54 generated by the diffractive optical element 24 taking account of the topography 60 of the diffraction structures 34 can be used as follows for the purposes of producing a diffractive optical element 24: Initially, a structure pattern 58 is determined for the diffraction structures 34 of the diffractive optical element 24 to be produced, which should be configured to generate a specified target wavefront, on the basis of a simulation calculation that only takes the layout of the diffraction structures 34 into account. Thereupon, the wavefront generated by means of the diffractive optical element 24 is determined by computation on the basis of the sequence of steps S1 to S4 described above, in which the topography 60 of the diffraction structures 34 arranged on the diffractive optical element 24 is taken into account. Furthermore, a correction for the structure pattern is determined on the basis of a wavefront difference between the actual wavefront and the target wavefront for the purposes of minimizing the wavefront difference. Finally, the diffractive optical element 24 is produced on the basis of the corrected structure pattern.
To carry out the individual steps, the wavefront determining device 56 may comprise the aforementioned modules; alternatively, a plurality of steps or all steps could be carried out in one module.
As explained above, the diffraction structures 34 as per Figure 1 are a 5-times encoded diffraction pattern, which is composed of five diffractive individual patterns superposed on one another. The aperiodic pattern 58 provided to the wavefront determining device 56 corresponds to a section from the 5-times encoded diffraction pattern or the n-times encoded diffraction pattern as a whole.
Figure 3 shows an example of the aperiodic pattern 58 in the form of a 10 um x 10 um section of the 5-times encoded diffraction pattern. In
Figure 3, white regions represent air-filled regions of the diffractive optical element 24 embodied as a phase CGH, while the black regions correspond to areas with glass. As emerges from the representation, the aperiodic pattern 581s a two-dimensional aperiodic pattern in this embodiment. Figure 4 elucidates a pixelated version of the aperiodic pattern 58 as per Figure 3 with pixel dimensions of 250 nm. This version is similar to the real structure of an exemplary embodiment of a diffraction pattern generated by means of a mask writer with a beam cross section of 250 nm.
In general form, the two-dimensional aperiodic pattern 58 formed by a 5-times encoded diffraction grating can be defined by up to 5 grating vectors, and for example arise from the following binarization rule: fG)= i Zn=19n cos(2r 7, ¥) > 0 where ¥ = (x,y) and ¥ = (yy, vy) QO 0 otherwise
To make things clearer, the method for ascertaining the Fourier representation 68 of the aperiodic pattern 58, which is initially carried out by the wavefront determining device 56 during the computational determination of the wavefront 54 of the test wave 26, is initially explained below on the basis of a one-dimensional aperiodic pattern 58e and subsequently extended to the two-dimensional aperiodic pattern 58 as per
Figure 3 or Figure 4. By way of example, a one-dimensional aperiodic pattern 58e can be formed by a 2-times encoded diffraction pattern, in which the diffraction only occurs in a plane.
For the purposes of explaining the ascertainment of the Fourier representation 68 of the aperiodic pattern 58, the one-dimensional aperiodic pattern 58e is defined by the following binarization rule: f(x) = @(cos(27x) + acos(27Bx)) (2) 1 x>0 with the step function (x) = i x=10 (3) 0 otherwise
It is obvious that f(x) is aperiodic for irrational fB, for example 68 = v3. However, it is possible to determine the Fourier spectrum of this function by embedding the latter into a 2-D function (step S1) q(x, y) = 8(cos(2mx) + acos(2my)) (4)
The function f(x) is obtained from g{x,y) by a projection on a line f(x) = q(x, Bx) (5)
The function g(x,y) = q(x + 1,y) = q(x, y + 1) is periodic both along the x- and the y-coordinate and therefore represents the periodic pattern 64 (cf. Figure 7). Therefore, the function g(x,y) can be represented as a Fourier series 66 (step S2) q(x, Y) = Time—os Cam e Tm) (6) with the expansion coefficients
Cam = IN IN q (x,y)e Frm) dxdy (7)
What follows from the identity of Equation (5) is that f(x) also has a
Fourier representation, with f(x) = Znm=-0 Cam e~iznntmp)x (8)
Expression (8) corresponds to the aforementioned Fourier representation 68 of the aperiodic pattern 58. Expression (8) is ascertained by means of the expansion coefficients Cnm of Expression (7) contained in the Fourier series 66 —cf. step S3.
In comparison with a periodic function, the spectrum of f(x) is defined on a non-equidistant grid. This becomes clearer if the following
Fourier representation is used: f(x) = No Ep e 27 where k = n+ Bm (9) where k corresponds to an irrational index.
The trick above shows how it is possible to calculate the Fourier representation of a binarized function without the introduction of a length factor.
The method described above for calculating the Fourier representation 68 of a one-dimensional aperiodic pattern 58e in the form of the general representation specified under (2) and (3) is explained below on the basis of a specific example. Two functions which each represent a one- dimensional aperiodic pattern 58e fix) =0 (cos(27x) + 0.8cos(VZ27x)) (10) fox) = @(cos(27x) + 0.8cos(2x)) (11) are considered in this example. These follow from Equation (2) by the choice ofa=08andf =V2orp = x . Figure 5 shows the course of the two functions. The spectrum of the two functions can be determined by the latter being embedded in a two-dimensional periodic function qx, y) = 0(cos(2mx) + 0.8cos(2my)) (12)
Figure 6 shows the unit cell of g(x,y). The two functions f; (x) and f(x) are obtained by projecting g(x,y) on in each case a line with fix) =q(xv2x) ‚and (13) x _ x 14
LG) =q(x-) (14)
The lines plotted in Figure 7 elucidate these circumstances. The spectrum of g(x, vy)
Cam = ff; q (x,y) Zr dxdy (15) can be efficiently determined by a Fast Fourier Transform (FFT) and is illustrated in Figure 8. The Fourier representations 68 in the form of the spectra of f(x) and f,(x) emerge from the Fourier coefficients Cam as described in Equation (9).
Table 1 below lists the 13 strongest Fourier coefficients and their respective index k for f; (x).
ooo es ae [owe ee [a a [os [| a [aa 3
Table 1
The spectra of the two functions f(x) and f;(x) are shown in Figure 9. Although the spectra look very different, both can be constructed by simple sorting from the spectrum of q(x, y), as illustrated above.
The present example exhibits the properties described below. Firstly, the Fourier coefficients do not depend on the choice of the grating vectors £, but only on other parameters of the binarization rule. Typically, a diffractive optical element in the form of a CGH is defined with a fixed binarization rule, with only the grating vectors changing over the surface. Therefore, the calculation of the spectrum is reduced to the one-time determination of the spectrum of the embedded function q(x, y) and a fast assignment.
The spectrum is dense. That is to say, any direction k can be approximated as closely as desired by the sum k = n + fm (see Equation 5 (9). By way of example, for B = v2, k = 7 can be approximated by m =
1004248 and n = —1420218 to five-digit accuracy with |-1420218 + 1004248v2 — | < 1075.
To extend the method for ascertaining the Fourier representation 68 of the aperiodic pattern, described above on the basis of a one-dimensional aperiodic pattern 58e, to the two-dimensional aperiodic pattern 58 as per
Figure 3 in the form of diffraction structures 34 of a five-times encoded diffractive optical element 24, the spectrum of the following function is calculated according to one embodiment: f@) = 6 (Ther gn cos(2n7, 3) (16) where gn > 0, net gn = 1, X= (x, y) and y = (Vo Yy) (17)
To this end, a periodic function is a) = (EE 88) = 0 (Eni gu cos (28) (18) with the expansion coefficients
Comus = J. q (E)e rem dg (19) is defined according to one embodiment, the volume integral being calculated over a unit cube.
From the identity [GE =q(¥, 2.7, %.¥, 2.7, XV, X75 %), (20) which corresponds to a projection on a two-dimensional plane, the Fourier representation 68 in the form of the sought-after spectrum emerges immediately with f@) = > Commu € BA EF EF BF BF, FX) (ump) nm, Lu = > cz e-iènk where k = 7, Vy Vai ¥yi¥s) nml, uv) k (21)
As already described above in relation to the one-dimensional aperiodic pattern, the Fourier coefficients of q(Ê) do not depend on the choice of the grating vectors Y, but are only the property of further parameters of the binarization rule. Hence, the Fourier integrals in
Equation (19) need only be calculated once per diffractive optical element 24 if the binarization rule does not change over the diffractive surface.
The pixelation of the aperiodic pattern 58 as per Figure 3 illustrated in Figure 4 can be taken into account in the higher-dimensional space by means of two additional dimensions.
A person skilled in the art considers the equations above easily extendible to two-, three-, four-, six- or more-times encoded diffractive optical elements CGH.
The algorithm 69 for determining the wavefront 54 generated by means of the diffractive optical element 24 on the basis of the Fourier representation 68 and the topography 60 of the diffraction structures 34, implemented in step S4, is based on Fourier modal methods in one embodiment, for example on the RCWA (rigorous coupled wave analysis) method known to a person skilled in the art.
Fourier modal methods for solving Maxwell's equations are formulated for periodic structures using reciprocal grating vectors y = (var Yy). For this reason, the electromagnetic field coefficients E (x,y,z) and
H (x,y,z) and the permittivity £(x,y) are approximated by the finite Fourier series thereof with
NM ey) =D OD Came Ennn) (22) n=-Nmz=-M
NM
E92) zei» > Uk, (ennai) (23) n=-Nm=-M
NM
H‚(y,2) zei > Sne ers) (24) n=-Nm=-M in all of these methods, where
EES
Com = f= fr € (x, y)ezn(nvxtmyyy) gy (25)
By using these approximations, Maxwell's equations become algebraic equations or normal differential equations in the z-coordinate, which can be solved by known methods.
Fourier modal methods fail in the case of aperiodic diffraction structures. According to one embodiment according to the invention, a change in the formulation which allows Maxwell's equations to be calculated is specified. To this end, the integrals for calculating the Fourier coefficients from Equation 25 are replaced by the following integrals: zE Im Loft ft i2nk-® ¢ (k) - Jim 412 ff eye (26)
For periodic structures, the two integrals supply identical results with t(n,m) = Cam (27)
and for aperiodic CGH structures, a new aperiodic approximation of the fields and the permittivity is obtained: (x,y) = Egt (K)e k= ’ (28) where the vectors k are no longer defined on an equidistant grid. The preceding paragraphs showed how it is possible to efficiently calculate the
Fourier coefficients. Using these coefficients and the reformulated RCWA equations, it is possible to approximate and solve Maxwell's equations as conventional differential equations.
The above description of exemplary embodiments, embodiments or embodiment variants is to be understood to be by way of example. The disclosure effected thereby firstly enables the person skilled in the art to understand the present invention and the advantages associated therewith, and secondly encompasses alterations and modifications of the described structures and methods that are also obvious in the understanding of the person skilled in the art. Therefore, all such alterations and modifications, insofar as they fall within the scope of the invention in accordance with the definition in the accompanying claims, and equivalents are intended to be covered by the protection of the claims.
List of reference signs 10 Measuring apparatus 12 Optical surface 14 Test object 16 Light source 18 Input wave 20 Optical waveguide 22 Exit surface 24 Diffractive optical element 26 Test wave 28 Reference wave 30 Reflective optical element 34 Diffraction structures 36 Capture device 88 Beam splitter 40 Interferometer camera 41 Interferogram 42 Propagation axis 44 Convergent beam 46 Eyepiece 48 Capture area 50 Plane calibration wave 52 Spherical calibration wave 53 Spherical calibration wave 54 Wavefront of the test wave 54a Wavefront of the test wave determined computationally 56 Wavefront determining device 58 Aperiodic pattern of the diffraction structures 58e One-dimensional aperiodic pattern 60 Topography of the diffraction structures
62 Dimension converter module 64 Periodic pattern in a higher-dimensional representation 65 FFT module 66 Fourier series of the periodic pattern 67 Conversion module 68 Fourier representation of the aperiodic pattern 69 Algorithm 70 Plane calibration mirror 71 Spherical calibration mirror 72 Spherical calibration mirror 74 Shutter 75 Shutter 76 Shutter 78 Evaluation device
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