CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority of EP 18154481.8 which was filed on January 31, 2018 and which is incorporated herein in its entirety by reference.

FIELD

[0002] The present invention relates to a method of determining an aberration map for a projection system and associated apparatus for carrying out the method.

BACKGROUND

[0003] A lithographic apparatus is a machine constructed to apply a desired pattern onto a substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). A lithographic apparatus may, for example, project a pattern at a patterning device (e.g., a mask) onto a layer of radiation-sensitive material (resist) provided on a substrate.

[0004] To project a pattern on a substrate a lithographic apparatus may use electromagnetic radiation. The wavelength of this radiation determines the minimum size of features which can be formed on the substrate. A lithographic apparatus, which uses extreme ultraviolet (EUV) radiation, having a wavelength within the range 4-20 nm, for example 6.7 nm or 13.5 nm, may be used to form smaller features on a substrate than a lithographic apparatus which uses, for example, radiation with a wavelength of 193 nm.

[0005] Radiation that has been patterned by the patterning device is focussed onto the substrate using a projection system. The projection system may introduce optical aberrations, which cause the image formed on the substrate to deviate from a desired image (for example a diffraction limited image of the patterning device).

[0006] It may be desirable to provide methods and apparatus for accurately determining such aberrations caused by a projection system such that these aberrations can be better controlled.

SUMMARY

[0007] According to a first aspect of the invention there is provided a method of determining an aberration map for a projection system, the method comprising: illuminating a patterning device with radiation, wherein the patterning device comprises a first patterned region arranged to receive at least a portion of the radiation and to form a plurality of first diffraction beams, the first diffraction beams being separated in a shearing direction; projecting, with the projection system, at least part of the plurality of first diffraction beams onto a sensor apparatus comprising: a second patterned region arranged to receive the first diffraction beams from the projection system and to form a plurality of second diffraction beams from each of the first diffraction beams; and a radiation detector arranged to receive at least a portion of the second diffraction beams, wherein the first and second patterned regions are matched such that at least some of the second diffraction beams formed from at least one of the first diffraction beams are spatially coherent with a second diffraction beam formed from at least one other first diffraction beam; moving at least one of the patterning device and the sensor apparatus in the shearing direction such that an intensity of radiation received by each part of the radiation detector varies as a function of the movement in the shearing direction so as to form an oscillating signal; determining from the radiation detector a phase of a harmonic of the oscillating signal at a plurality of positions on the radiation detector; and determining a set of coefficients that characterize the aberration map of the projection system from the phase of a harmonic of the oscillating signal at the plurality of positions on the radiation detector; wherein the set of coefficients that characterize the aberration map of the projection system are determined by equating the phase of a harmonic of the oscillating signal at each of the plurality of positions on the radiation detector to a combination of a plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system and solving to find the set of coefficients.

[0008] It will be appreciated that, since it involves moving at least one of the patterning device and the sensor apparatus in the shearing direction, the method according to the first aspect uses a phase stepping technique. The oscillating signal formed by the oscillation of the intensity of radiation received by each part of the radiation detector may be referred to as a phase stepping signal.

[0009] The method according to the first aspect is advantageous because it involves the combination of a plurality of differences in the aberration map between two positions in a pupil plane of the projection system and solving to find the set of coefficients. This allows more contributions to the harmonic of the oscillating signal (i.e. a greater number of interfering pairs of second diffraction beams) to be taken into account. In turn, this allows the method to be used for a greater range of grating geometries for the second patterned region. It will be appreciated that each such pair of two positions in a pupil plane of the projection system are separated in the shearing direction.

[00010] It will be appreciated that the matching of the first and second patterned regions (such that at least some of the second diffraction beams formed from at least one of the first diffraction beams are spatially coherent with a second diffraction beam formed from at least one other first diffraction beam) may be achieved by matching the pitches of the first and second patterned regions. It will be further appreciated that this matching of the pitches of the first and second patterned regions takes into account any reduction factor applied by the projection system. Taking this into account, in general, the pitch of the second patterned region may be an integer multiple of the pitch of the first patterned region or the pitch of the first patterned region may be an integer multiple of the pitch of the second patterned region.

[00011] The radiation detector may comprise a two dimensional array of sensing elements. Each sensing element may be referred to as a pixel of the radiation detector. It will be appreciated that the plurality of positions on the radiation detector at which the phase of a harmonic of the oscillating signal is determined may each correspond to different sensing element or pixel of the radiation detector. [00012] The harmonic of the oscillating signal which is equated to a combination of a plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system at each of the plurality of positions on the radiation detector may be any suitable harmonic.

[00013] The harmonic of the oscillating signal which is equated to a combination of a plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system at each of the plurality of positions on the radiation detector may be a first harmonic.

[00014] This may be beneficial since the first harmonic may generate the largest signal (or, alternatively, the largest signal to background). However, it will be appreciated that other harmonics may alternatively be used.

[00015] The set of coefficients that characterize the aberration map of the projection system may be determined by equating the phase of a harmonic of the oscillating signal at each of the plurality of positions on the radiation detector to a combination of a three or more differences in the aberration map between a pair of positions in a pupil plane of the projection system and solving to find the set of coefficients.

[00016] The pairs of positions in a pupil plane of the projection system may be separated in the shearing direction by a shearing distance which corresponds to the distance in the pupil plane between two adjacent first diffraction beams.

[00017] In contrast, prior art methods involve differences in the aberration map between pairs of positions in a pupil plane of the projection system are separated in the shearing direction by twice this shearing distance.

[00018] The radiation detector may comprise a plurality of distinct regions and, in the determination of the set of coefficients that characterize the aberration map of the projection system, a different set of a plurality of differences in the aberration map between two positions in the pupil plane of the projection system may be combined and equated to the phase of a harmonic of the oscillating signal for positions on the radiation detector in different distinct regions of the radiation detector.

[00019] It will be appreciated that the plurality of distinct regions of the radiation detector may correspond to different regions of the pupil plane of the projection system. In particular, the plurality of distinct regions of the radiation detector may correspond to regions of the pupil plane of the projection system that are within the numerical aperture of the projection system.

[00020] In general, the second diffraction beams can be considered to form a plurality of beams of radiation, each such beam of radiation being formed by a set of interfering second diffraction beams. Each such beam of radiation may be referred to herein as an interference beam. Each such interference beam formed by a plurality of interfering second interference beams may be considered to propagate in a different direction, such that the overlap of each interference beam at the radiation detector with a circle that represents the numerical aperture of the projection system is different. Although they may be considered to propagate in different directions and have a different overlap with a circle that represents the numerical aperture of the projection system, there is significant overlap between the different interference beams at the radiation detector. It will be appreciated that each of the plurality of distinct regions of the radiation detector may correspond to a region of the radiation detector which contains a different combination of these overlapping interference beams.

[00021] It will be appreciated that for a given position on the radiation detector the contribution to the harmonic of the oscillating signal from each interference beam corresponds to a difference in the aberration map between a different pair of positions in a pupil plane of the projection system.

[00022] For at least one of the plurality of positions on the radiation detector the phase of a harmonic of the oscillating signal may be equated to a combination of differences in the aberration map between pairs of positions in a pupil plane of the projection system which correspond to three or more interference beams.

[00023] For at least one of the plurality of positions on the radiation detector, at least two of the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system are combined with different interference strengths.

[00024] Such an arrangement is particularly advantageous, as now discussed. Allowing the at least two of the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system (each of which corresponds to a different interference beam) to be combined with different interference strengths represents a generalization of prior art techniques where only two interference beams are taken into account, with the same strength. This allows several effects to be taken into account, as now discussed.

[00025] First, it will be appreciated that the interference strength for an interference beam may be dependent on the geometries of the first and second patterned regions. Allowing the interference beams to be combined with different strengths may allow the method to be used for a greater range of grating geometries for the second patterned region (for example allowing a second patterned region comprising an array of circular apertures to be used without introducing significant errors).

[00026] Second, the inventors have realised that allowing the interference beams to be combined with different strengths may allow the method to take into account any non-uniform pupil fill (i.e. non uniformity of the pupil illumination of the first patterned region) and/or any non-uniformity of transmission (apodization) across the pupil plane of the projection system. It will be appreciated that taking into account non-uniform pupil fill and/or any non-uniformity of transmission across the pupil plane of the projection system may be in addition to taking into account differences in interference strengths which arise from complex geometries for the second patterned region (for example a second patterned region comprising an array of circular apertures). Alternatively, taking into account non- uniform pupil fill and/or any non-uniformity of transmission across the pupil plane of the projection system may be not in addition to taking into account differences in interference strengths which arise from complex geometries for the second patterned region (for this may be used to take into account these effects for arrangements that use a second patterned region comprising a checkerboard grating geometry). [00027] The method may further comprise, for each of the plurality of positions on the radiation detector: determining all pairs of second diffraction beams that interfere, which overlap with that position on the radiation detector, which contribute to the harmonic of the oscillating signal and which have an interference strength that is greater than a threshold value.

[00028] The method may further comprise, for each of the plurality of positions on the radiation detector: determining all interference beams, formed by a plurality of interfering second interference beams which propagate in the same direction and have the same overlap at the radiation detector with a circle that represents the numerical aperture of the projection system which overlap with that position on the radiation detector, which contribute to the harmonic of the oscillating signal and which have an interference strength that is greater than a threshold value.

[00029] It will be appreciated that this determination of either (a) all pairs of second diffraction beams or (b) all interference beams that which contribute to the harmonic of the oscillating signal at each point will be dependent on the geometries of the first and second patterned regions. It will be further appreciated that this determination of either (a) all pairs of second diffraction beams or (b) all interference beams that which contribute to the harmonic of the oscillating signal at each point will be dependent on which harmonic is used.

[00030] For example, the first harmonic of the oscillating signal (also referred to as the phase stepping signal) only depends on contributions that arise from the interference between spatially coherent diffraction beams (of the second patterned region) that originate from diffraction beams of the first patterned region that differ in order by ±1.

[00031] In some embodiments, the first patterned region may comprise a one-dimensional diffraction grating with a 50% duty cycle. With such a first patterned region, the efficiencies of the even diffraction orders (except the 0 ^th diffraction order) are zero. Therefore, the only two pairs of first diffraction beams that differ in order by ±1 (and therefore contribute to the first harmonic of such an oscillating phase- stepping signal) are the 0 ^th order beam with either the ± 1 ^sl order beams. Furthermore, with this geometry for the first patterned region, the scattering efficiencies are symmetric such that the efficiencies of the ±l ^st order diffraction beams are both the same. Therefore, the interference strengths for all pairs of second diffraction beams that contribute to the first harmonic of the oscillating phase stepping signal can be determined as follows. A second copy of the scattering efficiency plot for the second patterned region weighted by the scattering efficiency for the ±l ^st order diffraction beams of the first patterned region is overlaid with the scattering efficiency plot for the second patterned region but shifted in the shearing direction by 1 diffraction order (of the first diffraction grating). The product of the scattering efficiencies of these two overlaid scattering efficiencies plots is then determined. Each such interference strengths corresponds a different interference beam.

[00032] Each of the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system which are combined and equated to the phase of a harmonic of the oscillating signal may be a difference in the aberration map between a pair of positions in a pupil plane of the projection system from which a pair of second diffraction beams that interfere and which overlap with that position on the radiation detector originate.

[00033] The plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system may be combined weighted by an interference strength for that pair of second diffraction beams that interfere and which overlap with that position on the radiation detector.

[00034] In general, the tangent of the phase of a harmonic of the oscillating signal may be given by a ratio of (a) a weighted sum of the sines of the plurality of differences in the aberration map (weighted by the interference strength) to (b) a weighted sum of the cosines of the plurality of differences in the aberration map (weighted by the interference strength).

[00035] The plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system may be combined as a weighted sum, weighted by the interference strengths for the corresponding pairs of second diffraction beams that interfere and which overlap with that position on the radiation detector.

[00036] For example, the phase of a harmonic of the oscillating signal may be approximated by a ratio of (a) a weighted sum of the plurality of differences in the aberration map (weighted by the interference strength) to (b) a sum of the interference strengths. This may be considered to be a linear approximation, which may be accurate for sufficiently small shearing angles or distances.

[00037] According to a second aspect of the invention there is provided a computer readable medium carrying a computer program comprising computer readable instructions configured to cause a computer to carry out a method according to the first aspect of the invention.

[00038] According to a third aspect of the invention there is provided a computer apparatus comprising: a memory storing processor readable instructions, and a processor arranged to read and execute instructions stored in said memory, wherein said processor readable instructions comprise instructions arranged to control the computer to carry out the method according to the first aspect of the invention.

[00039] According to a fourth aspect of the invention there is provided a measurement system for determining an aberration map for a projection system, the measurement system comprising: a patterning device; an illumination system arranged to illuminate the patterning device with radiation, the patterning device comprising a first patterned region arranged to receive a radiation beam and to form a plurality of first diffraction beams, the first diffraction beams being separated in a shearing direction; a sensor apparatus comprising a second patterned region and a radiation detector; the projection system being configured to project the first diffraction beams onto the sensor apparatus, the second patterned region being arranged to receive the first diffraction beams from the projection system and to form a plurality of second diffraction beams from each of the first diffraction beams; a positioning apparatus configured to move at least one of the patterning device and the sensor apparatus in the shearing direction; and a controller configured to: control the positioning apparatus so as to move at least one of the first patterning device and the sensor apparatus in the shearing direction such that an intensity of radiation received by each part of the radiation detector varies as a function of the movement in the shearing direction so as to form an oscillating signal; determine from the radiation detector a phase of a harmonic of the oscillating signal at a plurality of positions on the radiation detector; and determine a set of coefficients that characterize the aberration map of the projection system from the phase of a harmonic of the oscillating signal at the plurality of positions on the radiation detector; wherein the set of coefficients that characterize the aberration map of the projection system are determined by equating the phase of a harmonic of the oscillating signal at each of the plurality of positions on the radiation detector to a combination of a plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system and solving to find the set of coefficients.

[00040] The measurement system according to the fourth aspect of the invention may be operable to implement any of the steps of the method according to the first aspect of the invention.

[00041] The controller may be configured such that the harmonic of the oscillating signal which is equated to a combination of a plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system at each of the plurality of positions on the radiation detector is a first harmonic.

[00042] The controller may be configured to determine the set of coefficients that characterize the aberration map of the projection system by equating the phase of a harmonic of the oscillating signal at each of the plurality of positions on the radiation detector to a combination of a three or more differences in the aberration map between a pair of positions in a pupil plane of the projection system and solving to find the set of coefficients.

[00043] The pairs of positions in a pupil plane of the projection system may be separated in the shearing direction by a shearing distance which corresponds to the distance in the pupil plane between two adjacent first diffraction beams.

[00044] The radiation detector may comprise a plurality of distinct regions and the controller may be operable such that in the determination of the set of coefficients that characterize the aberration map of the projection system, a different set of a plurality of differences in the aberration map between two positions in the pupil plane of the projection system are combined and equated to the phase of a harmonic of the oscillating signal for positions on the radiation detector in different distinct regions of the radiation detector.

[00045] The controller may be configured such that for at least one of the plurality of positions on the radiation detector the phase of a harmonic of the oscillating signal is equated to a combination of differences in the aberration map between pairs of positions in a pupil plane of the projection system which correspond to three or more interference beams.

[00046] For at least one of the plurality of positions on the radiation detector, the controller may be operable to combine at least two of the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system with different interference strengths. [00047] The controller may be further operable, for each of the plurality of positions on the radiation detector, to: determine all pairs of second diffraction beams that interfere, which overlap with that position on the radiation detector, which contribute to the harmonic of the oscillating signal and which have an interference strength that is greater than a threshold value.

[00048] The controller may be further operable, for each of the plurality of positions on the radiation detector, to: determine all interference beams, formed by a plurality of interfering second interference beams which propagate in the same direction and have the same overlap at the radiation detector with a circle that represents the numerical aperture of the projection system which overlap with that position on the radiation detector, which contribute to the harmonic of the oscillating signal and which have an interference strength that is greater than a threshold value.

[00049] Each of the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system which are combined and equated to the phase of a harmonic of the oscillating signal may be a difference in the aberration map between a pair of positions in a pupil plane of the projection system from which a pair of second diffraction beams that interfere and which overlap with that position on the radiation detector originate.

[00050] The controller may be configured such that the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system are combined weighted by an interference strength for that pair of second diffraction beams that interfere and which overlap with that position on the radiation detector.

[00051] The controller may be configured such that the plurality of differences in the aberration map between a pair of positions in a pupil plane of the projection system may be combined as a weighted sum, weighted by the interference strengths for the corresponding pairs of second diffraction beams that interfere and which overlap with that position on the radiation detector.

[00052] According to a fifth aspect of the invention there is provided a lithographic apparatus comprising the measurement system of the fourth aspect of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[00053] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which:

Figure 1 depicts a lithographic system comprising a lithographic apparatus and a radiation source;

Figure 2 is a schematic illustration of a measurement system according to an embodiment of the invention;

Figures 3A and 3B are schematic illustrations of a patterning device and a sensor apparatus which may form part of the measurement system of Figure 2; Figure 4 is a schematic illustration of a measurement system according to an embodiment of the invention, the measurement system comprising a first patterned region and a second patterned region, the first patterned region arranged to receive

radiation and to form a plurality of first diffraction beams;

Figures 5 A to 5C each shows a different set of second diffraction beams formed by the second patterned region of the measurement system shown in Figure 4, that set of second diffraction beams having been produced by a different first diffraction beam formed by the first patterned region;

Figure 6 A shows the scattering efficiency for one dimensional diffraction grating with a 50% duty cycle and which may represent the first patterned region of the measurement system shown in Figure 4;

Figure 6B shows the scattering efficiency for two dimensional diffraction grating of the form of a checkerboard with a 50% duty cycle and which may represent the second patterned region of the measurement system shown in Figure 4;

Figure 6C shows an interference strength map for the measurement system shown in Figure 4 when employing the first patterned region shown in Figure 6A and the second patterned region shown in Figure 6B, each of the interference strengths shown representing the second interference beams which contribute to the first harmonic of the oscillating phase-stepping signal and which have a different overlap, at the radiation detector, with a circle that represents the numerical aperture of the projection system;

Figures 7A, 7B and 7C show the portion of the numerical aperture of the projection system of the measurement system shown in Figure 4 that is filled by the three different first diffraction beams shown in Figure 4;

Figures 8A-8C show a portion of the radiation detector of the measurement system shown in Figure 4 which corresponds to the numerical aperture of the projection system of the measurement system and which is filled by three second diffraction beams which originate from the first diffraction beam represented by Figure 7B;

Figures 9A-9C show a portion of the radiation detector of the measurement system shown in Figure 4 which corresponds to the numerical aperture of the projection system of the measurement system and which is filled by three second diffraction beams which originate from the first diffraction beam represented by Figure 7A;

Figures 10 A- 10C show a portion of the radiation detector of the measurement system shown in Figure 4 which corresponds to the numerical aperture of the projection system of the measurement system and which is filled by three second diffraction beams which originate from the first diffraction beam represented by Figure 7C;

Figure 11 A show a portion of the radiation detector of the measurement system shown in Figure 4 which corresponds to the numerical aperture of the projection system of the measurement system and which represents the overlap between the second diffraction beams shown in Figures 8B and 9A and the overlap between the second diffraction beams shown in Figures 8A and 10B;

Figure 1 IB show a portion of the radiation detector of the measurement system shown in Figure 4 which corresponds to the numerical aperture of the projection system of the measurement system and which represents the overlap between the second diffraction beams shown in Figures 8B and 10C and the overlap between the second diffraction beams shown in Figures 8C and 9B;

Figure 12 shows a unit cell of a grating comprising an array of circular pinholes and having a 50% (by area) duty cycle;

Figure 13A shows the scattering efficiency for one dimensional diffraction grating with a 50% duty cycle and which may represent the first patterned region of the measurement system shown in Figure 4;

Figure 13B shows the scattering efficiency for two dimensional diffraction comprising the unit cell of Figure 12 and which may represent the second patterned region of the measurement system shown in Figure 4;

Figure 13C shows an interference strength map for the measurement system shown in Figure 4 when employing the first patterned region shown in Figure 13 A and the second patterned region shown in Figure 13B, each of the interference strengths shown representing the second interference beams which contribute to the first harmonic of the oscillating phase- stepping signal and which have a different overlap, at the radiation detector, with a circle that represents the numerical aperture of the projection system;

Figure 14A is a representation of 21 second diffraction beams generated by a second patterned region having a unit cell as shown in Figure 12, the second diffraction beams corresponding to diffraction efficiencies contained within the white dotted line in Figure 13B;

Figure 14B is a representation of 16 interference beams, each generated by the interference of pairs of the second diffraction beams shown in Figure 14 A, each of the interference beam corresponding to an interference strength contained within the white dotted line in Figure 13C;

Figure 15 shows a map of a radiation detector within a white dashed circular line, which corresponds to the numerical aperture of the projection system, the map indicating how many of the 20 interference beams (from within the white dotted line in Figure 13C) overlap for each position on the radiation detector;

Figure 16 shows a single discrete region of the radiation detector from the map of Figure 15 within which all 20 of the interference beams overlap;

Figure 17 A shows 4 discrete regions of the radiation detector from the map of Figure 15 within which 19 of the 20 interference beams overlap;

Figure 17B is a representation of the permutations of the 20 interference beams which correspond to the 4 regions shown in Figure 17 A; Figure 18 A shows 18 discrete regions of the radiation detector from the map of Figure 15 within which 18 of the 20 interference beams overlap;

Figure 18B is a representation of the permutations of the 20 interference beams which correspond to the 18 regions shown in Figure 18 A;

Figure 19 A shows all of the regions of the radiation detector within which one or more of the 20 interference beams overlap; and

Figure 19B is a representation of the permutations of the 20 interference beams which correspond to the all of the regions shown in Figure 19A.

DETAILED DESCRIPTION

[00054] Figure 1 shows a lithographic system comprising a radiation source SO and a lithographic apparatus LA. The radiation source SO is configured to generate an EUV radiation beam B and to supply the EUV radiation beam B to the lithographic apparatus LA. The lithographic apparatus LA comprises an illumination system IL, a support structure MT configured to support a patterning device MA (e.g., a mask), a projection system PS and a substrate table WT configured to support a substrate W.

[00055] The illumination system IL is configured to condition the EUV radiation beam B before the EUV radiation beam B is incident upon the patterning device MA. Thereto, the illumination system IL may include a facetted field mirror device 10 and a facetted pupil mirror device 11. The faceted field mirror device 10 and faceted pupil mirror device 11 together provide the EUV radiation beam B with a desired cross-sectional shape and a desired intensity distribution. The illumination system IL may include other mirrors or devices in addition to, or instead of, the faceted field mirror device 10 and faceted pupil mirror device 11.

[00056] After being thus conditioned, the EUV radiation beam B interacts with the patterning device MA. As a result of this interaction a patterned EUV radiation beam B’ is generated. The projection system PS is configured to project the patterned EUV radiation beam B’ onto the substrate W. For that purpose, the projection system PS may comprise a plurality of mirrors 13, 14 which are configured to project the patterned EUV radiation beam B’ onto the substrate W held by the substrate table WT. The projection system PS may apply a reduction factor to the patterned EUV radiation beam B’ thus forming an image with features that are smaller than corresponding features on the patterning device MA. For example, a reduction factor of 4 or 8 may be applied. Although the projection system PS is illustrated as having only two mirrors 13, 14 in Figure 1, the projection system PS may include a different number of mirrors (e.g. six or eight mirrors).

[00057] The substrate W may include previously formed patterns. Where this is the case, the lithographic apparatus LA aligns the image, formed by the patterned EUV radiation beam B’, with a pattern previously formed on the substrate W. [00058] A relative vacuum, i.e. a small amount of gas (e.g. hydrogen) at a pressure well below atmospheric pressure, may be provided in the radiation source SO, in the illumination system IL, and/or in the projection system PS.

[00059] The radiation source SO may be a laser produced plasma (LPP) source, a discharge produced plasma (DPP) source, a free electron laser (FEL) or any other radiation source that is capable of generating EUV radiation.

[00060] In general, the projection system PS has an optical transfer function which may be non- uniform, which can affect the pattern which is imaged on the substrate W. For unpolarized radiation such effects can be fairly well described by two scalar maps, which describe the transmission (apodization) and relative phase (aberration) of radiation exiting the projection system PS as a function of position in a pupil plane thereof. These scalar maps, which may be referred to as the transmission map and the relative phase map, may be expressed as a linear combination of a complete set of basis functions. A particularly convenient set is the Zernike polynomials, which form a set of orthogonal polynomials defined on a unit circle. A determination of each scalar map may involve determining the coefficients in such an expansion. Since the Zernike polynomials are orthogonal on the unit circle, the Zernike coefficients may be obtained from a measured scalar map by calculating the inner product of the measured scalar map with each Zernike polynomial in turn and dividing this by the square of the norm of that Zernike polynomial. In the following, unless stated otherwise, any reference to Zernike coefficients will be understood to mean the Zernike coefficients of a relative phase map (also referred to herein as an aberration map). It will be appreciated that in alternative embodiments other sets of basis functions may be used. For example some embodiments may use Tatian Zernike polynomials, for example for obscured aperture systems.

[00061] The wavefront aberration map represents the distortions of the wavefront of light approaching a point in an image plane of the projection system PS from a spherical wavefront (as a function of position in the pupil plane or, alternatively, the angle at which radiation approaches the image plane of the projection system PS). As discussed, this wavefront aberration map W(x, y ) may be expressed as a linear combination of Zernike polynomials:

where x and y are coordinates in the pupil plane, Z _n (x, y) is the nth Zernike polynomial and c _n is a coefficient. It will be appreciated that in the following, Zernike polynomials and coefficients are labelled with an index which is commonly referred to as a Noll index. Therefore, Z _n ( x , y) is the Zernike polynomial having a Noll index of n and c _n is a coefficient having a Noll index of n. The wavefront aberration map may then be characterized by the set of coefficients c _n in such an expansion, which may be referred to as Zernike coefficients.

[00062] It will be appreciated that only a finite number of Zernike orders are taken into account. Different Zernike coefficients of the phase map may provide information about different forms of aberration which are caused by the projection system PS. The Zernike coefficient having a Noll index of 1 may be referred to as the first Zernike coefficient, the Zernike coefficient having a Noll index of 2 may be referred to as the second Zernike coefficient and so on.

[00063] The first Zernike coefficient relates to a mean value (which may be referred to as a piston) of a measured wavefront. The first Zernike coefficient may be irrelevant to the performance of the projection system PS and as such may not be determined using the methods described herein. The second Zernike coefficient relates to the tilt of a measured wavefront in the x-direction. The tilt of a wavefront in the x-direction is equivalent to a placement in the x-direction. The third Zernike coefficient relates to the tilt of a measured wavefront in the y-direction. The tilt of a wavefront in the y-direction is equivalent to a placement in the y-direction. The fourth Zernike coefficient relates to a defocus of a measured wavefront. The fourth Zernike coefficient is equivalent to a placement in the z- direction. Higher order Zernike coefficients relate to other forms of aberration which are caused by the projection system (e.g. astigmatism, coma, spherical aberrations and other effects).

[00064] Throughout this description the term“aberrations” should be intended to include all forms of deviation of a wavefront from a perfect spherical wavefront. That is, the term“aberrations” may relate to the placement of an image (e.g. the second, third and fourth Zernike coefficients) and/or to higher order aberrations such as those which relate to Zernike coefficients having a Noll index of 5 or more. Furthermore, any reference to an aberration map for a projection system may include all forms of deviation of a wavefront from a perfect spherical wavefront, including those due to image placement.

[00065] The transmission map and the relative phase map are field and system dependent. That is, in general, each projection system PS will have a different Zernike expansion for each field point (i.e. for each spatial location in its image plane).

[00066] As will be described in further detail below, the relative phase of the projection system PS in its pupil plane may be determined by projecting radiation from an object plane of the projection system PS (i.e. the plane of the patterning device MA), through the projection system PS and using a shearing interferometer to measure a wavefront (i.e. a locus of points with the same phase). The shearing interferometer may comprise a diffraction grating, for example a two dimensional diffraction grating, in an image plane of the projection system (i.e. the substrate table WT) and a detector arranged to detect an interference pattern in a plane that is conjugate to a pupil plane of the projection system PS.

[00067] The projection system PS comprises a plurality of optical elements (including mirrors 13, 14). As already explained, although the projection system PS is illustrated as having only two mirrors 13, 14 in Figure 1, the projection system PS may include a different number of mirrors (e.g. six or eight mirrors). The lithographic apparatus LA further comprises adjusting means PA for adjusting these optical elements so as to correct for aberrations (any type of phase variation across the pupil plane throughout the field). To achieve this, the adjusting means PA may be operable to manipulate optical elements within the projection system PS in one or more different ways. The projection system may have a co-ordinate system wherein its optical axis extends in the z direction (it will be appreciated that the direction of this z axis changes along the optical path through the projection system, for example at each mirror or optical element). The adjusting means PA may be operable to do any combination of the following: displace one or more optical elements; tilt one or more optical elements; and/or deform one or more optical elements. Displacement of optical elements may be in any direction (x, y, z or a combination thereof). Tilting of optical elements is typically out of a plane perpendicular to the optical axis, by rotating about axes in the x or y directions although a rotation about the z axis may be used for non-rotationally symmetric optical elements. Deformation of an optical element may be performed for example by using actuators to exert force on sides of the optical element and/or by using heating elements to heat selected regions of the optical element. In general, it may not be possible to adjust the projection system PS to correct for apodizations (transmission variation across the pupil plane). The transmission map of a projection system PS may be used when designing masks MAs for the lithographic apparatus LA.

[00068] In some embodiments, the adjusting means PA may be operable to move the support structure MT and/or the substrate table WT. The adjusting means PA may be operable to displace (in any of the x, y, z directions or a combination thereof) and/or tilt (by rotating about axes in the x or y directions) the support structure MT and/or the substrate table WT.

[00069] A projection system PS which forms part of a lithographic apparatus may periodically undergo a calibration process. For example, when a lithographic apparatus is manufactured in a factory the optical elements (e.g. mirrors) which form the projection system PS may be set up by performing an initial calibration process. After installation of a lithographic apparatus at a site at which the lithographic apparatus is to be used, the projection system PS may once again be calibrated. Further calibrations of the projection system PS may be performed at regular intervals. For example, under normal use the projections system PS may be calibrated every few months (e.g. every three months).

[00070] Calibrating a projection system PS may comprise passing radiation through the projection system PS and measuring the resultant projected radiation. Measurements of the projected radiation may be used to determine aberrations in the projected radiation which are caused by the projection system PS. Aberrations which are caused by the projection system PS may be determined using a measurement system. In response to the determined aberrations, the optical elements which form the projection system PS may be adjusted so as to correct for the aberrations which are caused by the projection system PS.

[00071] Figure 2 is a schematic illustration of a measurement system 10 which may be used to determine aberrations which are caused by a projection system PS. The measurement system 10 comprises an illumination system IL, a measurement patterning device MA’, a sensor apparatus 21 and a controller CN. The measurement system 10 may form part of a lithographic apparatus. For example, the illumination system IL and the projection system PS which are shown in Figure 2 may be the illumination system IL and projection system PS of the lithographic apparatus which is shown in Figure 1. For ease of illustration additional components of a lithographic apparatus are not shown in Figure 2. [00072] The measurement patterning device MA’ is arranged to receive radiation from the illumination system IL. The sensor apparatus 21 is arranged to receive radiation from the projection system PS. During normal use of a lithographic apparatus, the measurement patterning device MA’ and the sensor apparatus 21 which are shown in Figure 2 may be located in positions that are different to the positions in which they are shown in Figure 2. For example, during normal use of a lithographic apparatus a patterning device MA which is configured to form a pattern to be transferred to a substrate W may be positioned to receive radiation from the illumination system IL and a substrate W may be positioned to receive radiation from the projection system PS (as is shown, for example, in Figure 1). The measurement patterning device MA’ and the sensor apparatus 21 may be moved into the positions in which they are shown in Figure 2 in order to determine aberrations which are caused by the projection system PS. The measurement patterning device MA’ may be supported by a support structure MT, such as the support structure which is shown in Figure 1. The sensor apparatus 21 may be supported by a substrate table, such as the substrate table WT which is shown in Figure 1. Alternatively the sensor apparatus 21 may be supported by a measurement table (not shown) which may be separate to the substrate table WT.

[00073] The measurement patterning device MA’ and the sensor apparatus 21 are shown in more detail in Figures 3 A and 3B. Cartesian co-ordinates are used consistently in Figures 2, 3 A and 3B. Figure 3A is a schematic illustration of the measurement patterning device MA’ in an x-y plane and Figure 3B is a schematic illustration of the sensor apparatus 21 in an x-y plane.

[00074] The measurement patterning device MA’ comprises a plurality of patterned regions 15a- 15c. In the embodiment which is shown in Figures 2 and 3A the measurement patterning device MA’ is a reflective patterning device MA’. The patterned regions 15a- 15c each comprises a reflective diffraction grating. Radiation which is incident on the patterned regions 15a-15c of the measurement patterning device MA’ is at least partially scattered by thereby and received by the projection system PS. In contrast, radiation which is incident on the remainder of the measurement patterning device MA’ is not reflected or scattered towards the projection system PS (for example, it may be absorbed by the measurement patterning device MA’).

[00075] The illumination system IL illuminates the measurement patterning device MA’ with radiation. Whilst not shown in Figure 2, the illumination system IL may receive radiation from a radiation source SO and condition the radiation so as to illuminate the measurement patterning device MA’ . For example, the illumination system IL may condition the radiation so as to provide radiation having a desired spatial and angular distribution. In the embodiment which is shown in Figure 2, the illumination system IL is configured to form separate measurement beams 17a-17c. Each measurement beam 17a-17c illuminates a respective patterned region 15a-15c of the measurement patterning device MA’.

[00076] In order to perform a determination of aberrations which are caused by the projection system PL, a mode of the illumination system IL may be changed in order to illuminate the measurement patterning device MA’ with separate measurement beams 17a-17c. For example, during normal operation of a lithographic apparatus, the illumination system IL may be configured to illuminate a patterning device MA with a slit of radiation. However the mode of the illumination system IL may be changed such that the illumination system IL is configured to form separate measurement beams 17a- 17c in order to perform a determination of aberrations caused by the projection system PL. In some embodiments different patterned regions 15a-15c may be illuminated at different times. For example, a first subset of the patterned regions 15a-15c may be illuminated at a first time so as to form a first subset of measurement beams 17a-17c and a second subset of patterned regions 15a-15c may be illuminated at a second time so as to form a second subset of measurement beams 17a-17c.

[00077] In other embodiments the mode of the illumination system IL may be unchanged in order to perform a determination of aberrations caused by the projection system PL. For example, the illumination system IL may be configured to illuminate the measurement patterning device MA’ with a slit of radiation (e.g. which substantially corresponds with an illumination area used during exposure of substrates). Separate measurement beams 17a-17c may then be formed by the measurement patterning device MA’ since only the patterned regions 15a-15c reflect or scatter radiation towards the projection system PS.

[00078] In the Figures the Cartesian co-ordinate system is shown as being conserved through the projection system PS. However, in some embodiments the properties of the projection system PS may lead to a transformation of the co-ordinate system. For example, the projection system PS may form an image of the measurement patterning device MA’ which is magnified, rotated and/or mirrored relative to the measurement patterning device MA’. In some embodiments the projection system PS may rotate an image of the measurement patterning device MA’ by approximately 180° around the z- axis. In such an embodiment the relative positions of a first measurement beam 17a and a third measurement beam 17c which are shown in Figure 2, may be swapped. In other embodiments the image may be mirrored about an axis which may lie in an x-y plane. For example, the image may be mirrored about the x-axis or about the y-axis.

[00079] In embodiments in which the projection system PS rotates an image of the measurement patterning device MA’ and/or the image is mirrored by the projection system PS, the projection system is considered to transform the co-ordinate system. That is, the co-ordinate system which is referred to herein is defined relative to an image which is projected by the projection system PS and any rotation and/or mirroring of the image causes a corresponding rotation and/or mirroring of the co-ordinate system. For ease of illustration, the co-ordinate system is shown in the Figures as being conserved by the projection system PS. However, in some embodiments the co-ordinate system may be transformed by the projection system PS.

[00080] The patterned regions 15a-15c modify the measurement beams 17a-17c. In particular, the patterned regions 15a-15c cause a spatial modulation of the measurement beams 17a-17c and cause diffraction in the measurement beams 17a-17c. In the embodiment which is shown in Figure 3B the patterned regions 15a-15c each comprise two distinct portions. For example, a first patterned region 15a comprises a first portion 15a’ and a second portion 15a”. The first portion 15a’ comprises a diffraction grating which is aligned parallel to a u-direction and the second portion 15a” comprises a diffraction grating which is aligned parallel to a v-direction. The u and v-directions are depicted in Figure 3A. The u and v-directions are both aligned at approximately 45° relative to both the x and y- directions and are aligned perpendicular to each other. Second 15b and third 15c patterned regions which are shown in Figure 3A are identical to the first patterned region 15a and each comprise first and second portions whose diffraction gratings are aligned perpendicular to each other.

[00081] The first and second portions of the patterned regions 15a-15c may be illuminated with the measurement beams 17a- 17c at different times. For example, the first portions of each of the patterned regions 15a-15c may be illuminated by the measurement beams 17a-17c at a first time. At a second time the second portions of each of the patterned regions 15a-15c may be illuminated by the measurement beams 17a- 17c. As was mentioned above in some embodiments different patterned regions 15a-15c may be illuminated at different times. For example, the first portions of a first subset of patterned regions 15a- 15c may be illuminated at a first time and the first portions of a second subset of patterned regions 15a-15c may be illuminated at a second time. Second portions of the first and second subsets of patterned regions may be illuminated at the same or different times. In general any schedule of illuminating different portions of patterned regions 15a-15c may be used.

[00082] The modified measurement beams 17a-17c are received by the projection system PS. The projection system PS forms an image of the patterned regions 15a-15c on the sensor apparatus 21. The sensor apparatus 21 comprises a plurality of diffraction gratings 19a- 19c and a radiation detector 23. The diffraction gratings 19a- 19c are arranged such that each diffraction grating 19a- 19c receives a respective modified measurement beam 17a- 17c which is output from the projection system PL. The modified measurement beams 17a-17c which are incident on the diffraction gratings 19a-19c are further modified by the diffraction gratings 19a-19c. The modified measurement beams which are transmitted at the diffraction gratings 19a- 19c are incident on the radiation detector 23.

[00083] The radiation detector 23 is configured to detect the spatial intensity profile of radiation which is incident on the radiation detector 23. The radiation detector 23 may, for example, comprise an array of individual detector elements or sensing elements. For example, the radiation detector 23 may comprise an active pixel sensor such as, for example, a CMOS (complementary metal-oxide- semiconductor) sensor array. Alternatively, the radiation detector 23 may comprise a CCD (charge- coupled device) sensor array. The diffraction gratings 19a- 19c and portions of the radiation sensor 23 at which the modified measurement beams 17a- 17c are received form detector regions 25a-25c. For example, a first diffraction grating 19a and a first portion of the radiation sensor 23 at which a first measurement beam 17a is received together form a first detector region 25a. A measurement of a given measurement beam 17a- 17c may be made at a respective detector region 25a-25c (as depicted). As was described above, in some embodiments the relative positioning of the modified measurement beams 17a-17c and the co-ordinate system may be transformed by the projection system PS.

[00084] The modification of the measurement beams 17a-17c which occurs at the patterned regions 15a-15c and the diffraction gratings 19a-19c of the detector regions 25a-25c results in interference patterns being formed on the radiation detector 23. The interference patterns are related to the derivative of the phase of the measurement beams and depend on aberrations caused by the projection system PS. The interference patterns may therefore be used to determine aberrations which are caused by the projection system PS.

[00085] In general, the diffraction gratings 19a-19c of each of the detector regions 25a-25c comprises a two-dimensional transmissive diffraction grating. In the embodiment which is shown in Figure 3B the detector regions 25a-25c each comprise a diffraction grating 19a- 19c which is configured in the form of a checkerboard. As described further below, embodiments of the present invention have particular application to arrangements where the detector regions 25a-25c each comprises a two- dimensional transmissive diffraction grating 19a- 19c that is not configured in the form of a checkerboard.

[00086] Illumination of the first portions of the patterned regions 15a- 15c may provide information related to aberrations in a first direction and illumination of the second portions of the patterned regions 15a- 15c may provide information related to aberrations in a second direction.

[00087] In some embodiments, the measurement patterning device MA’ and/or the sensor apparatus 21 is sequentially scanned and/or stepped in two perpendicular directions. For example, the measurement patterning device MA’ and/or the sensor apparatus 21 may be stepped relative to each other in the u and v-directions. The measurement patterning device MA’ and/or the sensor apparatus 21 may be stepped in the u-direction whilst the second portions 15a”-15c” of the patterned regions 15a- 15c are illuminated and the measurement patterning device MA’ and/or the sensor apparatus 21 may be stepped in the v-direction whilst the first portions 15a’-15c’ of the patterned regions 15a-15c are illuminated. That is, the measurement patterning device MA’ and/or the sensor apparatus 21 may be stepped in a direction which is perpendicular to the alignment of a diffraction grating which is being illuminated.

[00088] The measurement patterning device MA’ and/or the sensor apparatus 21 may be stepped by distances which correspond with a fraction of the grating period of the diffraction gratings. Measurements which are made at different stepping positions may be analysed in order to derive information about a wavefront in the stepping direction. For example, the phase of the first harmonic of the measured signal (which may be referred to as a phase stepping signal) may contain information about the derivative of a wavefront in the stepping direction. Stepping the measurement patterning device MA’ and/or the sensor apparatus 21 in both the u and v-directions (which are perpendicular to each other) therefore allows information about a wavefront to be derived in two perpendicular directions (in particular, it provides information about a derivative of the wavefront in each of the two perpendicular directions), thereby allowing the full wavefront to be reconstructed.

[00089] In addition to stepping of the measurement patterning device MA’ and/or the sensor apparatus 21 in a direction which is perpendicular to the alignment of a diffraction grating which is being illuminated (as was described above), the measurement patterning device MA’ and/or the sensor apparatus 21 may also be scanned relative to each other. Scanning of the measurement patterning device MA’ and/or the sensor apparatus 21 may be performed in a direction which is parallel to the alignment of a diffraction grating which is being illuminated. For example, the measurement patterning device MA’ and/or the sensor apparatus 21 may be scanned in the u-direction whilst the first portions 15a’-15c’ of the patterned regions 15a-15c are illuminated and the measurement patterning device MA’ and/or the sensor apparatus 21 may be scanned in the v-direction whilst the second portions 15a”- 15c” of the patterned regions 15a- 15c are illuminated. Scanning of the measurement patterning device MA’ and/or the sensor apparatus 21 in a direction which is parallel to the alignment of a diffraction grating which is being illuminated allows measurements to be averaged out across the diffraction grating, thereby accounting for any variations in the diffraction grating in the scanning direction. Scanning of the measurement patterning device MA’ and/or the sensor apparatus 21 may be performed at a different time to the stepping of the measurement patterning device MA’ and/or the sensor apparatus 21 which was described above.

[00090] It will be appreciated that a variety of different arrangements of the patterned regions 15a- 15c and the detector regions 25a-25c may be used in order to determine aberrations caused by the projection system PS. The patterned regions 15a-15c and/or the detector regions 25a-25c may comprise diffraction gratings. In some embodiments the patterned regions 15a-15c and/or the detector regions 25a-25c may comprise components other than a diffraction grating. For example, in some embodiments the patterned regions 15a-15c and/or the detector regions may comprise a single slit or a pin-hole opening through which at least a portion of a measurement beam 17a-17c may propagate. In general the patterned regions and/or the detector regions may comprise any arrangement which serves to modify the measurement beams.

[00091] The controller CN receives measurements made at the sensor apparatus 21 and determines, from the measurements, aberrations which are caused by the projection system PS. The controller may be configured to control one or more components of the measurement system 10. For example, the controller CN may control a positioning apparatus PW which is operable to move the sensor apparatus 21 and/or the measurement patterning device MA’ relative to each other. The controller may control an adjusting means PA for adjusting components of the projection system PS. For example, the adjusting means PA may adjust optical elements of the projection system PS so as to correct for aberrations which are caused by the projection system PS and which are determined by the controller CN. [00092] In some embodiments, the controller CN may be operable to control the adjusting means PA for adjusting the support structure MT and/or the substrate table WT. For example, the adjusting means PA may adjust support structure MT and/or substrate table WT so as to correct for aberrations which are caused by placement errors of patterning device MA and/or substrate W (and which are determined by the controller CN).

[00093] Determining aberrations (which may be caused by the projection system PS or by placement errors of the patterning device MA or the substrate W) may comprise fitting the measurements which are made by the sensor apparatus 21 to Zernike polynomials in order to obtain Zernike coefficients. Different Zernike coefficients may provide information about different forms of aberration which are caused by the projection system PS. Zernike coefficients may be determined independently at different positions in the x and/or the y-directions. For example, in the embodiment which is shown in Figure 2, 3A and 3B, Zernike coefficients may be determined for each measurement beam 17a-17c.

[00094] In some embodiments the measurement patterning device MA’ may comprise more than three patterned regions, the sensor apparatus 21 may comprise more than three detector regions and more than three measurement beams may be formed. This may allow the Zernike coefficients to be determined at more positions. In some embodiments the patterned regions and the detector regions may be distributed at different positions in both the x and y-directions. This may allow the Zernike coefficients to be determined at positions which are separated in both the x and the y-directions.

[00095] Whilst, in the embodiment which is shown in Figures 2, 3A and 3B the measurement patterning device MA’ comprises three patterned regions 15a-15c and the sensor apparatus 21 comprises three detector regions 25a-25c, in other embodiments the measurement patterning device MA’ may comprise more or less than three patterned regions 15a-15c and/or the sensor apparatus 21 may comprise more or less than three detector regions 25a-25c.

[00096] Methods for determining aberrations caused by a projection system PS are now described with reference to Figure 4.

[00097] In general, measurement patterning device MA’ comprises at least one first patterned region 15a-15c and the sensor apparatus 21 comprises at least one second patterned region 19a-19c.

[00098] Figure 4 is a schematic illustration of a measurement system 30 which may be used to determine aberrations which are caused by a projection system PS. Measurement system 30 may be the same as the measurement system 10 shown in Figure 2, however, it may have a different number of first patterned regions (on measurement patterning device MA’) and second patterned regions (in the sensor apparatus 21). Therefore, the measurement system 30 shown in Figure 4 may include any features of the measurement system 10 shown in Figure 2 described above and these features will not be further described below.

[00099] In Figure 4, only a single first patterned region 31 is provided on the measurement patterning device MA’ and a single second patterned region 32 is provided in the sensor apparatus 21. [000100] The measurement patterning device MA’ is irradiated with radiation 33 from the illumination system IL. For ease of understanding only a single line (which may, for example, represent a single ray, for example the chief ray, of an incident radiation beam) is shown in Figure 4. However, it will be appreciated that the radiation 33 will comprises a range of angles incident on the first patterned region 31 of the measurement patterning device MA’ . That is, each point on the first patterned region 31 of the measurement patterning device MA’ may be illuminated by a cone of light. In general, each point is illuminated by substantially the same range of angles, this being characterized by the intensity of radiation in a pupil plane of the illumination system IL (not shown).

[000101] The first patterned region 31 is arranged to receive the radiation 33 and to form a plurality of first diffraction beams 34, 35, 36. A central first diffraction beam 35 corresponds to a 0 ^th order diffraction beam of first patterned region 31 and the other two first diffraction beams 34, 36 correspond to the ±l ^st order diffraction beams of first patterned region 31. It will be appreciated that more, higher order diffraction beams will, in general, also be present. Again for ease of understanding, only three first diffraction beams 34, 35, 36 are shown in Figure 4.

[000102] It will also be appreciated that, as the incoming radiation 33 comprises a cone of radiation converging on a point on the first patterned region 31, each of the first diffraction beams 34, 35, 36 also comprises a cone of radiation diverging from that point on the first patterned region 31.

[000103] To achieve the generation of the first diffraction beams 34, 35, 36, the first patterned region 31 may be of the form of a diffraction grating. For example, the first patterned region 31 may be generally of the form of the patterned region 15a shown in Figure 3A. In particular, at least a portion of the first patterned region 31 may be of the form of the first portion 15a’ of the patterned region 15a shown in Figure 3A, i.e. a diffraction grating which is aligned parallel to a u-direction (note that Figure 4 is shown in the z-v plane). Therefore, the first diffraction beams 34-36 are separated in a shearing direction, which is the v-direction.

[000104] The first diffraction beams 34-36 are at least partially captured by the projection system PS, as now described. How much of the first diffraction beams 34-36 is captured by the projection system PS will be dependent on: the pupil fill of the incident radiation 33 from the illumination system IL; the angular separation of the first diffraction beams 34-36 (which in turn is dependent on the pitch of the first patterned region 31 and the wavelength of the radiation 33); and the numerical aperture of the projection system PS.

[000105] The measurement system 30 may be arranged such that first diffraction beam 35 that corresponds to the 0 ^th order diffraction beam substantially fills the numerical aperture of the projection system PS, which may be represented by a circular region of a pupil plane 37 of the projection system PS, and the first diffraction beams 34, 36 that correspond to the ±l ^st order diffraction beams overlap significantly with the first diffraction beam 35 that corresponds to the 0 ^th order diffraction beam. With such an arrangement, substantially all of the first diffraction beam 35 that corresponds to the 0 ^th order diffraction beam and most of the first diffraction beams 34, 36 that correspond to the ± 1 ^sl order diffraction beams is captured by the projection system PS and projected onto the sensor apparatus 21. (Furthermore, with such an arrangement a large number of diffraction beams generated by the first patterned region 31 are at least partially projected onto the sensor apparatus 21).

[000106] The role of the first patterned region 31 is to introduce spatial coherence, as now discussed.

[000107] In general, two rays of radiation 33 from the illumination system IL that are incident on the same point of the measurement patterning device MA’ at different angles of incidence are not coherent. By receiving the radiation 33 and forming a plurality of first diffraction beams 34, 35, 36, the first patterned region 31 may be considered to form a plurality of copies of the incident radiation cone 33 (the copies having, in general different phases and intensities). Within any one of these copies, or first diffraction beams 34, 35, 36, two rays of radiation which originate from the same point on the measurement patterning device MA’ but at different scattering angles, are not coherent (due to the properties of the illumination system IL). However, for a given ray of radiation within any one of the first diffraction beams 34, 35, 36 there is a corresponding ray of radiation in each of the other first diffraction beams 34, 35, 36 that is spatially coherent with that given ray. For example, the chief rays of each of the first diffraction beams 34, 35, 36 (which correspond to the chief ray of the incident radiation 33) are coherent and could, if combined, interfere at the amplitude level.

[000108] This coherence is exploited by the measurement system 30 to determine an aberration map of the projection system PS.

[000109] The projection system PS projects part of the first diffraction beams 34, 35, 36 (which is captured by the numerical aperture of the projection system) onto the sensor apparatus 21.

[000110] In Figure 4, the sensor apparatus 21 comprises the single second patterning region 32. As described further below (with reference to Figures 5A-5C), second patterned region 32 is arranged to receive these first diffraction beams 34-36 from the projection system PS and to form a plurality of second diffraction beams from each of the first diffraction beams. In order to achieve this, the second patterning region 32 comprises a two-dimensional transmissive diffraction grating. In Figure 4, all radiation that is transmitted by the second patterning region 32 is represented as a single arrow 38. This radiation 38 is received by a detector region 39 of the radiation detector 23 and is used to determine the aberration map.

[000111] Each of the first diffraction beams 34-36 that is incident on the patterning region 32 will diffract to from a plurality of second diffraction beams. Since the second patterning region 32 comprises a two-dimensional diffraction grating, from each incident first diffraction beam, a two dimensional array of secondary diffraction beams is produced (the chief rays of these secondary diffraction beams being separated in both the shearing direction (v-direction) and the direction perpendicular thereto (the u-direction). In the following, a diffraction order that is n ^lh order in the shearing direction (the v- direction) and m ^th order in the non-shearing direction (the u-direction) will be referred to as the (n, m) ^lh diffraction order of the second patterned region 32. In the following, where it is not important what order a second diffraction beam is in the non-shearing direction (the u-direction), the (n, m) ^lh diffraction order of the second patterned region 32 may be referred to simply as the n ^lh order second diffraction beam.

[000112] Figures 5A to 5C show a set of second diffraction beams produced by each of the first diffraction beams 34-36. Figure 5A shows a set of second diffraction beams 35a-35e produced by the first diffraction beam 35 that corresponds to the 0 ^th order diffraction beam of first patterned region 31. Figure 5B shows a set of second diffraction beams 36a-36e produced by the first diffraction beam 36 that corresponds to the -1 ^st order diffraction beam of first patterned region 31. Figure 5C shows a set of second diffraction beams 34a-34e produced by the first diffraction beam 34 that corresponds to the +l ^st order diffraction beam of first patterned region 31.

[000113] In Figure 5A, second diffraction beam 35a corresponds to the 0 ^th order diffraction beam (of second patterned region 32, and in the shearing direction), whereas second diffraction beams 35b, 35c correspond to the ±l ^st order diffraction beams and second diffraction beams 35d, 35e correspond to the ±2 ^nd order diffraction beams. It will be appreciated that Figures 5A-5C are shown in the v-z plane and the shown second diffraction beams may, for example, correspond to 0 ^th order diffraction beam of second patterned region 32 in the non-shearing direction (i.e. the u-direction). It will be further appreciated that there will be a plurality of copies of these second diffraction beams, representing higher order diffraction beams in the non-shearing direction that are into or out of the page of Figures 5A-5C.

[000114] In Figure 5B, second diffraction beam 36a corresponds to the 0 ^th order diffraction beam (of second patterned region 32, and in the shearing direction), whereas second diffraction beams 36b, 36c correspond to the ±l ^st order diffraction beams and second diffraction beams 36d, 36e correspond to the ±2 ^nd order diffraction beams.

[000115] In Figure 5C, second diffraction beam 34a corresponds to the 0 ^th order diffraction beam (of second patterned region 32, and in the shearing direction), whereas second diffraction beams 34b, 34c correspond to the ±l ^st order diffraction beams and second diffraction beams 34d, 34e correspond to the ±2 ^nd order diffraction beams.

[000116] It can be seen from Figures 5A-5C that several of the second diffraction beams spatially overlap with each other. For example, the second diffraction beam 35b that corresponds to the -1 ^st order diffraction beam of second patterned region 32, which originates from the 0 ^th order diffraction beam 35 of first patterned region 31 overlaps with the second diffraction beam 36a that corresponds to the 0 ^th order diffraction beam of second patterned region 32, which originates from the -1 ^st order diffraction beam 36 of first patterned region 31. All of the lines in Figures 4 and 5A-5C may be considered to represent a single ray of radiation that originates from a single input ray 33 from the illumination system IL. Therefore, as explained above, these lines represent spatially coherent rays that, if spatially overlapping at radiation detector 23 will produce an interference pattern. Furthermore, the interference is between rays which have passed though different parts of the pupil plane 37 of the projection system PS (which are separated in the shearing direction). Therefore, the interference of radiation that originates from a single input ray 33 is dependent on phase differences between two different parts of the pupil plane.

[000117] This spatial overlapping and spatial coherence of the second diffraction beams at radiation detector 23 is achieved by matching the pitches of the first and second patterned regions 31, 32. It will be appreciated that this matching of the pitches of the first and second patterned regions 31, 32 takes into account any reduction factor applied by the projection system PS. Taking this into account, either the pitch of the second patterned region 32 should be an integer multiple of the pitch of the first patterned region 31 or the pitch of the first patterned region 31 should be an integer multiple of the pitch of the second patterned region 32. In the example shown in Figures 5A-5C, the pitches of the first and second patterned regions 31, 32 are substantially equal (taking into account any reduction factor).

[000118] As can be seen from Figures 5A-5C, each point on the detector region 39 of the radiation detector 23 will, in general, receive several contributions that are summed coherently. For example, the point on the detector region 39 which receives the second diffraction beam 35b that corresponds to the -1 ^st order diffraction beam of second patterned region 32, which originates from the 0 ^th order diffraction beam 35 of first patterned region 31 overlaps with both: (a) the second diffraction beam 36a that corresponds to the 0 ^th order diffraction beam of second patterned region 32, which originates from the -1 ^st order diffraction beam 36 of first patterned region 31; and (b) the second diffraction beam 34d that corresponds to the -2 ^nd order diffraction beam of second patterned region 32, which originates from the +l ^st order diffraction beam 34 of first patterned region 31. It will be appreciated that when higher order diffraction beams of the first patterned region 31 are taken into account there will be more beams that should be summed coherently at each point on the detector region 39 in order to determine the intensity of radiation as measured by that part of the detector region 39 (for example a corresponding pixel in a two dimensional array of sensing elements).

[000119] In general, a plurality of different second diffraction beams contributes to the radiation received by each part of the detector region 39. The intensity of radiation from such a coherent sum is given by:

I = DC + å _{pairs { i]} Yi cos(Ac h), (2) where DC is a constant term (which is equivalent to the incoherent sum of the different diffraction beams), the sum is over all pairs of different second diffraction beams, is an interference strength for that pair of second diffraction beams and Df _; is a phase difference between that pair of second diffraction beams.

[000120] The phase difference Df _; between a pair of second diffraction beams is dependent on two contributions: (a) a first contribution relates to the different part of the pupil plane 37 of the from which they originate; and (b) a second contribution relates to the position within the unit cells of each of the first and second patterned regions 31, 32 from which they originate. [000121] The first of these contributions can be understood to arise from the fact that the different coherent radiation beams have passed through different parts of the projection system PS and are therefore related to the aberrations that it is desired to determine (in fact they are related to a difference between two points in the aberration map that are separated in the shearing direction).

[000122] The second of these contributions can be understood to arise from the fact that the relative phases of multiple rays of radiation that arise from a single ray incident on a diffraction grating will depend on which part of the unit cell of that grating the ray was incident. This therefore does not contain information relating to the aberrations. As explained above, in some embodiments, the measurement patterning device MA’ and/or the sensor apparatus 21 are sequentially scanned and/or stepped in the shearing direction. This causes the phase differences between all of pairs of interfering radiation beams received by the radiation detector 23 to change. As the measurement patterning device MA’ and/or the sensor apparatus 21 are sequentially stepped in the shearing direction by an amount that is equivalent to a fraction of the pitches of the first and second patterned regions 31, 32, in general, the phase differences between pairs of second diffraction beams will all change. If the measurement patterning device MA’ and/or the sensor apparatus 21 are stepped in the shearing direction by an amount that is equivalent to an integer multiple of the pitches of the first and second patterned regions 31, 32 the phase differences between pairs of second diffraction beams will remain the same. Therefore, as the measurement patterning device MA’ and/or the sensor apparatus 21 are by sequentially scanned and/or stepped in the shearing direction, the intensity received by each part of the radiation detector 23 will oscillate. The first harmonic of this oscillating signal (which may be referred to as a phase-stepping signal), as measured by the radiation detector 23, is dependent on the contributions to equation (1) that arise from adjacent first diffraction beams 34-36, i.e. first diffraction beams that differ in order by ±1. Contributions that arise from first diffraction beams that differ in order by a different amount will contribute to higher order harmonics of the signal determined by the radiation detector 23 due to such phase stepping techniques.

[000123] For example, of the three overlapping second diffraction beams discussed above (35b, 36a and 34d) only two of the three possible pairs of these diffraction beams will contribute to the first harmonic of the phase stepping signal: (a) second diffraction beams 35b and 36a (which originate from the 0 ^th order diffraction beam 35 and the -1 ^st order diffraction beam 36 of first patterned region 31 respectively); and (b) second diffraction beams 35a and 34d (which originate from the 0 ^th order diffraction beam 35 and the +l ^st order diffraction beam 34 of first patterned region 31 respectively).

[000124] Each pair of second diffraction beams will result in an interference term of the form shown in equation (2), which contributes to the first harmonic of the phase stepping signal, i.e. an interference term of the form:

2p (3) g cos( - v + AW) where g is an amplitude of the interference term, p is the pitch of the first and second patterned regions 31, 32, v parameterizes the relative positions of the first and second patterned regions 31, 32 in the shearing direction and AW is a difference between the value of the aberration map at two positions in the pupil plane of the projection system PS, the two positions corresponding to the positions from which the two second diffraction beams originate. The amplitude g of the interference term is proportional to the product of the compound scattering efficiencies of the two second diffraction beams, as discussed further below. The frequency of the first harmonic of the phase stepping signal is given by the inverse of the pitch p of the first and second patterned regions 31, 32 in the shearing direction. The phase of the phase stepping signal is given by AW (the difference between the values of the aberration map at two positions in the pupil plane of the projection system PS, the two positions corresponding to the positions from which the two second diffraction beams originate).

[000125] The interference strength for a pair of second diffraction beams is proportional to the product of the compound scattering efficiencies of the two second diffraction beams, as now discussed.

[000126] In general, the scattering efficiency of the diffraction beams produced by a diffraction grating will depend on the geometry of the grating. These diffraction efficiencies, which may be normalised to the efficiency of a 0 ^th order diffraction beam, describe the relative intensities of the diffraction beams. As used herein, the compound scattering efficiency of a second diffraction beam is given by the product of the scattering efficiency of the first diffraction beam from which it originates and the scattering efficiency for the diffraction order of the second patterned region 32 to which it corresponds.

[000127] In the above description of the embodiments shown in Figures 3A to 5C, where the first portion 15a’ of the patterned region 15a shown in Figure 3A is illuminated, the shearing direction corresponds to the v-direction and the non-shearing direction corresponds to the u-direction. It will be appreciated that when the second portion 15a” of the patterned region 15a shown in Figure 3A is illuminated, the shearing direction corresponds to the u-direction and the non-shearing direction corresponds to the v-direction. Although in these above-described embodiments, the u and v-directions (which define the two shearing directions) are both aligned at approximately 45° relative to both the x and y-directions of the lithographic apparatus LA, it will be appreciated that in alternative embodiments the two shearing directions may be arranged at any angle to the x and y-directions of the lithographic apparatus LA (which may correspond to non-scanning and scanning directions of the lithographic apparatus LA). In general, the two shearing directions will be perpendicular to each other. In the following, the two shearing directions will be referred to as the x-direction and the y-direction. However, it will be appreciated that these shearing directions may be arranged at any angle relative to both the x and y-directions of the lithographic apparatus LA.

[000128] Figure 6 A shows the scattering efficiency for a first patterned region 31 that is of the form of the first portion 15a’ of the patterned region 15a shown in Figure 3A, having a 50% duty cycle. The horizontal axis represents the diffraction order in the shearing direction. The diffraction efficiencies shown in Figure 6A are normalised to the efficiency of a 0 ^th order diffraction beam, such that the efficiency of the 0 ^th order diffraction beam is 100%. With this geometry (a 50% duty cycle), the efficiencies of the even diffraction orders (except the 0 ^th diffraction order) are zero. The efficiencies of the ±l ^st order diffraction beams are 63.7%.

[000129] Figure 6B shows the scattering efficiency for a second patterned region 32 that is of the form of the diffraction grating 19a shown in Figure 3B, i.e. in the form of a checkerboard with a 50% duty cycle. The horizontal axis represents the diffraction order in the shearing direction. The vertical axis represents the diffraction order in the non-shearing direction. The diffraction efficiencies shown in Figure 6B are normalised to the efficiency of the (0, 0) ^th order diffraction beam, such that the efficiency of the (0, 0) ^th order diffraction beam is 100%.

[000130] As explained above, the first harmonic of the oscillating phase- stepping signal only depends on the contributions to equation (1) from first diffraction beams that differ in order by ±1. As can be seen from Figure 6A, with a 50% duty cycle grating on the measurement patterning device MA’, the only two pairs of first diffraction beams that differ in order by ±1 are the 0 ^th order beam with either the ±l ^st order beams. Furthermore, with this geometry for the first patterned region 31, the scattering efficiencies are symmetric such that the efficiencies of the ±l ^st order diffraction beams are both the same (63.7%). Therefore, the interference strengths for all pairs of second diffraction beams that contribute to the first harmonic of the oscillating phase-stepping signal can be determined as follows. A second copy of the scattering efficiency plot for the second patterned region 32 shown in Figure 6B is weighted by the scattering efficiency for the ±l ^st order diffraction beams of the first patterned region 31 and then overlaid with the scattering efficiency plot for the second patterned region 32 shown in Figure 6B but shifted in the shearing direction by the separation of 1 pair of diffraction orders (of the first patterned region 31). Here, the pitches of the first and second patterned regions 31, 32 are equal (taking into account any reduction factor applied by the projection system PS) and therefore, in this example, the second copy of the scattering efficiency plot for the second patterned region 32 is a shifted in the shearing direction by 1 diffraction order of the second patterned region 31. The product of the scattering efficiencies of these two overlaid scattering efficiencies plots is then determined. Such a plot of the interference strengths for all pairs of second diffraction beams that contribute to the first harmonic of the oscillating phase- stepping signal is shown in Figure 6C.

[000131] Note that each of the interference strengths shown in Figure 6C actually represents two different pairs of second diffraction beams. For example, the left hand pixel shown in Figure 6C represents both: (a) interference between second diffraction beams 35a and 34b and (b) second diffraction beam 35b and 36a. Similarly, the right hand pixel shown in Figure 6C represents both: (a) interference between second diffraction beam 35a and 36c and (b) second diffraction beam 35c and 34a. In general, each pixel of such a map represents two pairs of second diffraction beams: (a) a first pair of second diffraction beams that include one second diffraction beam that originated from the first diffraction beam 35 corresponding to the 0 ^th diffraction order of first patterning device 31 and another second diffraction beam that originated from the first diffraction beam 34 corresponding to the +1^ order diffraction order of first patterned region 31 ; and (b) a second pair of second diffraction beams that include one second diffraction beam that originated from the first diffraction beam 35 corresponding to the 0 ^th diffraction order of first patterning device 31 and another second diffraction beam that originated from the first diffraction beam 36 corresponding to the -1 ^st order diffraction order of first patterned region 31.

[000132] In general, each of the interference strengths y _; shown in Figure 6C represents two different pairs of second diffraction beams: (a) one pair comprising an n ^th order second diffraction beam produced by the first diffraction beam 35 (that corresponds to the 0 ^th order diffraction beam of first patterned region 31); and (b) another pair comprising an (n+l) ⁴ order second diffraction beam produced by the first diffraction beam 35. Therefore, each interference strength y _; may be characterized by the two diffraction orders of the first diffraction beam 35 ((n, m) ^th and (n+1, m) ^lh ) that contribute, and may be denoted as Ύ _h,h+ i _, ·p _i · In the following, where it is clear that m=0, or the value of m is unimportant, this interference strength may be denoted as y _{n n+1} .

[000133] Although each of the interference strengths y _; (or y _{n,n+1 .m )} shown in Figure 6C represents two different pairs of second diffraction beams, each of the interference strengths y _; shown in Figure 6C represents the second diffraction beams which contribute to the first harmonic of the oscillating phase-stepping signal and which have a different overlap, at the radiation detector 23, with a circle that represents the numerical aperture of the projection system PS, as now described.

[000134] Figures 7A, 7B and 7C show the portion of the pupil plane 37 of the projection system PS which corresponds to the numerical aperture of the projection system PS that is filled by first diffraction beams 34, 35, 36 respectively. In each of Figures 7A, 7B and 7C the numerical aperture of the projection system PS is represented by a circle 40 and the portion of the pupil plane 37 of the projection system PS that is filled by first diffraction beams 34, 35, 36 is shown by a shaded region of this circle 40 in Figures 7A, 7B and 7C respectively. As can be seen from Figure 7B, in the example shown, the central first diffraction beam 35 which corresponds to a 0 ^th order diffraction beam substantially fills the numerical aperture of the projection system PS. As can be seen from Figure 7A and 7C, each of the two first diffraction beams 34, 36 which correspond to the ±l ^st order diffraction beams of first patterned region 31 have been shifted such that they only partially fill the numerical aperture. It will be appreciated that this shift of the first order first diffraction beams 34, 36 relative to the numerical aperture is in practice very small and has been exaggerated here for ease of understanding.

[000135] Figures 8 A- 10C show the portion of the radiation detector 23 that is filled by various second diffraction beams. In each of Figures 8 A- 10C the numerical aperture of the projection system PS is represented by a circle 40 and the portion of this circle that is filled by the second diffraction beams is shown by a shaded region of this circle 40. Figures 8A-8C show the portion of the circle 40 that is filled by (-1, 0) ^lh , (0, 0) ^lh and (1, ()) ^lh order diffraction beams 35b, 35a, 35c which originate from the first diffraction beam 35 which corresponds to a 0 ^th order diffraction beam of the first patterned region 31. Figures 9A-9C show the portion of the circle 40 that is filled by (-1, ())"', (0, 0) ^th and (1, ())"' order diffraction beams 34b, 34a, 34c which originate from the first diffraction beam 34 which corresponds to the 1 ^st order diffraction beam of the first patterned region 31. Figures 10A-10C show the portion of the circle 40 that is filled by (-1, ())"', (0, ())"' and (1, 0) ^th order diffraction beams 36b, 36a, 36c which originate from the first diffraction beam 36 which corresponds to the -1 ^st order diffraction beam of the first patterned region 31.

[000136] It can be seen from Figures 8B, 9A, 8A and 10B that the region of the radiation detector which receives a contribution from both: (a) interference between second diffraction beams 35a and 34b and (b) second diffraction beam 35b and 36a is the region 41 shown in Figure 11 A. Similarly, It can be seen from Figures 8B, 10C, 8C and 9B that the region of the radiation detector which receives a contribution from both: (a) interference between second diffraction beam 35a and 36c and (b) second diffraction beam 35c and 34a is the region 42 shown in Figure 1 IB.

[000137] In general, each of the interference strengths y _; shown in Figure 6C may be considered to represent a beam of radiation formed by a plurality of interfering second interference beams, each such beam of radiation formed by a plurality of interfering second interference beams propagating in a different direction, such that the overlap of each such beam of radiation at the radiation detector 23, with a circle that represents the numerical aperture of the projection system PS is different.

[000138] In general, the second diffraction beams can be considered to form a plurality of beams of radiation, each such beam of radiation being formed by a set of interfering second diffraction beams. Each such beam of radiation may be referred to herein as an interference beam. Each such interference beam formed by a plurality of interfering second interference beams may be considered to propagate in a different direction, such that the overlap of each interference beam at the radiation detector 23 with a circle that represents the numerical aperture of the projection system PS is different. Although they may be considered to propagate in different directions and have a different overlap with a circle that represents the numerical aperture of the projection system PS, there is significant overlap between the different interference beams at the radiation detector 23. Each of the interference strengths y _; shown in Figure 6C may be considered to represent a different interference beam (formed by a plurality of interfering second interference beams).

[000139] As previously described, each of the interference strengths y _; (or y _n,n+i;m ) shown in Figure 6C represents two different pairs of second diffraction beams. However, for a given position on the radiation detector, both of these pairs of contributing second diffraction beams comprise two interfering rays that originate from the same two points in the pupil plane 37 of the projection system PS. In particular, for a position (x, y) on the radiation detector (these co-ordinates corresponding to co ordinates of the pupil plane 37 of the projection system PS and the x direction corresponding to the shearing direction), the two pairs of interfering second diffraction beams that contribute have an interference strengths y _{n n+ 1;m} each comprise a ray of a second diffraction beam that originated from a position (x-ns, y-ms) in the pupil plane 37 and a ray of a second diffraction beam that originated from a position (x-(n+l)s, y-ms) in the pupil plane 37, where s is a shearing distance. The shearing distance s corresponds to the distance in the pupil plane 37 between two coherent rays of adjacent first diffraction beams 34-36. Therefore, both pairs of contributing second diffraction beams give rise to an interference term of the form of expression (3), where AW is a difference between the value of the aberration map at these two positions in the pupil plane 37.

[000140] It can be seen from Figure 6C that with a second patterned region 32 which is of the form of a 50% duty cycle checkerboard there are only two sets of second diffraction beams that contribute to the first harmonic of the phase stepping signal, both with an interference strength (y_ _{l 0} , 7o _,+i ) of 25.8%. This is due to the checkerboard geometry, which, as can be seen in from Figure 6 A, results in a diffraction efficiency plot where, with the exception of the (-1, 0) ^th , (0, 0) ^lh and (1, 0) ^lh order diffraction beams, moving in the shearing direction, every other diffraction beam has a diffraction efficiency of 0%. That is, the grating efficiencies of the (n, m) ^lh diffraction orders wherein n+m is an even number are all zero, except the (0, 0) ^lh diffraction order. As a result of these grating efficiencies being zero, all of the interference strengths which contribute to the first harmonic of the phase stepping signal are zero except for interference strengths y_ _{l 0} and y _0,+1 .

[000141] For the overlap between the two regions 41, 42 shown in Figures 11A and 11B (that this overlap region will form the majority of circle 40 for small shearing angles) the first harmonic of the oscillating phase-stepping signal will be proportional to the sum of two cosines (cf. equation (2) and expression (3)):

I = DC + y_ _{l 0} cosCWli - W ₀ ) + y _{0 +1} cos(W ₀ - W ₊₁ ) (4) where the first cosine is of a difference in the aberration map between a first two points in the pupil plane and the second cosine is of a difference in the aberration map between a second two points in the pupil plane (here the phase stepping terms have been omitted for clarity of understanding). In particular, for a given position (x, y) on the radiation detector (x referring to the shearing direction), the first two points include a corresponding point in the pupil plane (x, y) (represented as W ₀ in equation (4)) and another point which is shifted in a first direction along the shearing direction by the shearing distance (x-s, y) (represented as W_ ₁ in equation (4)). Similarly, the second two points include a corresponding point in the pupil plane (x, y) (represented as W ₀ in equation (4)) and another point which is shifted in a second direction along the shearing direction by the shearing distance (x+s, y) (represented as W ₊₁ in equation (4)).

[000142] Existing wavefront reconstruction techniques exploit the fact that the two interference strengths in equation (4) are equal such that this sum of two cosines can be re-written using trigonometric identities as a cosine of the difference in the aberration map between two positions that are separated in the shearing direction by twice the shearing distance, i.e. cos(VF_ ₁ — W ₊₁ ), multiplied by a factor that is approximately 1 for small shearing distances. Therefore, such known techniques involve the determination of a set of Zernike coefficients by equating the phase of the first harmonic of a phase stepping signal to a difference in the aberration map between positions in the pupil plane that are separated in the shearing direction by twice the shearing distance. Recall that the aberration map depends on the Zernike coefficients (see equation (1)). This is done for a plurality of positions on the radiation sensor (for example at a plurality of pixels or individual sensing elements in an array) first for a first shearing direction and then subsequently for a second, orthogonal direction. These constraints for the two shearing orthogonal directions are simultaneously solved to find the set of Zernike coefficients.

[000143] As discussed above, the combination of a first patterned region 31 comprising a linear grating and a second patterned region 32 comprising a two-dimensional checkerboard is advantageous (since only two interference beams contribute to the first harmonic of the phase stepping signal). Due to the geometry of the checkerboard, checkerboard gratings typically comprise an optical transmissive carrier or support layer. However, EUV radiation is strongly absorbed by most materials and therefore no good transmissive materials exist for EUV radiation. Furthermore, such a transmissive carrier is not favourable in a wafer production environment of an EUV lithographic system, since the transmissive carrier will rapidly become contaminated in such an environment. This would render the transmissive carrier untransmissive for EUV. Such contamination problems which could only be addressed by regular cleaning actions that would impact system availability and therefore the throughput of the lithographic system. For the above-mentioned reasons a checkerboard grating arrangement is difficult to implement for lithographic systems that use EUV radiation.

[000144] For this reason, existing aberration measurement systems for EUV radiation use, as a second patterned device 32, a geometry which uses an array of circular pinholes. Figure 12 shows the unit cell 50 of such a grating, having a 50% (by area) duty cycle. The unit cell 50 comprises a circular aperture 51 provided in an EUV absorbing membrane 52. The circular aperture 51 is a through aperture which represents a void in the EUV absorbing membrane 52 through which EUV radiation is transmitted. However, such a pinhole array geometry (as shown in Figure 12) generates unwanted interference beams that contribute to the first harmonic of the phase stepping signal, as now discussed with reference to Figures 13A-13B.

[000145] Figure 13 A shows the scattering efficiency for a first patterned region 31 that is of the form of the first portion 15a’ of the patterned region 15a shown in Figure 3A, having a 50% duty cycle (of the same geometry as that shown in Figure 6A). Again, the diffraction efficiencies are normalised to the efficiency of a 0 ^th order diffraction beam, such that the efficiency of the 0 ^th order diffraction beam is 100%. Figure 13B shows the scattering efficiency for a second patterned region 32 that is of the form of a pinhole array having the unit cell 50 shown in Figure 12. The diffraction efficiencies shown in Figure 13B are normalised to the efficiency of the (0, 0) ^lh order diffraction beam, such that the efficiency of the (0, 0) ^th order diffraction beam is 100%. [000146] Figure 13C is a plot of the interference strengths y _{n n+1} for interference beams that contribute to the first harmonic of the oscillating phase- stepping signal (this is constructed from the scattering efficiencies of Figures 13A and 13B in an analogous manner to the construction of Figure 6C from the scattering efficiencies of Figures 6A and 6B).

[000147] It can be seen from Figure 13C that with a second patterned region 32 which has the unit cell 50 shown in Figure 12, in addition to the two main interference beams (with interference strengths 7-I, _Q , 7o _,+i °f 25.2%), there are a number of additional interference beams with small, but non-zero, interference strengths y _{n n+1} . Since the interference strengths for these additional interference beams are not the same, for regions of the radiation detector 23 where multiple interference beams overlap, the first harmonic of the oscillating phase-stepping signal will be proportional to a weighted sum of a plurality of cosines (cf. equation (4)), the cosines having different weights. As a result, they cannot be combined easily using trigonometric identities. However, since the interference strengths y _{n n+1} for the additional interference beams are small (in comparison to the interference strengths y_ _{1 0} , 7o _,+i ), such known existing aberration measurement systems for EUV radiation neglect these terms (i.e. assume they are zero) in the reconstruction of the wavefront to find the set of Zernike coefficients.

[000148] This assumption impacts the accuracy of the wavefront measurement. In turn, this has a negative impact on system imaging, overlay and focus performance. Embodiments of the present invention have been devised to at least partially address the above-described problems for aberration measurement systems for EUV radiation.

[000149] Embodiments of the invention relate to new methods for reconstructing a wavefront or aberration map, as now described.

[000150] According to some embodiments of the invention, a new method for reconstructing a wavefront or aberration map involves equating the phase of a first harmonic of the oscillating signal at each of the plurality of positions on the radiation detector to a combination of a plurality of differences in the aberration map between pairs of positions (separated in the shearing direction) in a pupil plane of the projection system. These constraints are simultaneously solved to find the set of Zernike coefficients. This new general technique will now be described with reference to Figures 14A-19B.

[000151] In general, for a given region of the radiation detector 23 multiple interference beams overlap and therefore the first harmonic of the oscillating phase- stepping signal will be proportional to a weighted sum of a plurality of cosines (cf. equation (4)), the cosines having different weights. The inventors of the present invention have realised that each of these cosines has the same dependence on the relative positions of the first and second patterned regions 31, 32 in the shearing direction (cf. equation (3), where v parameterizes the relative positions of the first and second patterned regions 31, 32 in the shearing direction) and can therefore be combined as the addition of a plurality of phasors, as now explained. [000152] In particular, the first harmonic of the phase stepping signal for a given position on the radiation detector 23 (for example a given sensing element or pixel) can be written as:

where M and AW are, respectively, an amplitude and a phase of first harmonic of the phase stepping signal, the sum is over all interference beams which overlap with this position on the radiation detector 23 and which contribute to the first harmonic of the oscillating phase- stepping signal, y _; is the interference strength of the ith interference beam and A W _; is a difference between the value of the aberration map at two positions in the pupil plane of the projection system PS, the two positions corresponding to the positions from which the two second diffraction beams that contribute to the ith interference beam originate. Although M is the amplitude of the first harmonic of the phase stepping signal, it may alternatively be referred to as the modulation.

[000153] The inventors of the present invention have realised that since each of the cosines in the sum in equation (5) has the same dependence on the relative positions of the first and second patterned regions 31, 32 in the shearing direction they can be combined as the addition of a plurality of phasors, such that:

[000154] Furthermore, the A W _; is a difference between the value of the aberration map at two positions in the pupil plane of the projection system PS, the two positions corresponding to the positions from which the two second diffraction beams that contribute to the ith interference beam originate. Recall that, where x denotes the shearing direction, for a position (x, y) on the radiation detector, the two pairs of interfering second diffraction beams that contribute have an interference strengths y _{n n+ 1;m} each comprise a ray of a second diffraction beam that originated from a position (x-ns, y-ms) in the pupil plane 37 and a ray of a second diffraction beam that originated from a position (x-(n+l)s, y-ms) in the pupil plane 37, where s is a shearing distance. Therefore, for x as the shearing direction, differences in the aberration map AM/ in equation (6) can be expressed in terms of the Zernike polynomials (see equation (1)) as:

ms)] (7)

where x and y are coordinates of the radiation detector (which correspond to coordinates of the pupil plane 37 of the projection system PS). Similarly, for y as the shearing direction, differences in the aberration map AWJ in equation (6) can be expressed in terms of the Zernike polynomials (see equation (1)) as:

ns)] (8) where again x and y are coordinates of the radiation detector (which correspond to coordinates of the pupil plane 37 of the projection system PS).

[000155] For small shearing angles (or, equivalently, small shearing distances), equation (6) can be re-expressed (using a Taylor or small angle expansion for the trigonometric functions) as:

[000156] Embodiments of the present invention involve equating the phase (AW) of the first harmonic of the oscillating phase- stepping signal as measured by the radiation detector 23 to a combination of a plurality of differences (either AW* or AWJ depending which shearing direction is currently being used) in the aberration map between a pair of positions in the pupil plane 37 of the projection system PS. This combination may either use the exact relationship of equation (6) (by taking the inverse tangent of both sides) or may use the linear approximation of equation (9). Each of the differences (either AW* or AWJ) in the aberration map between a pair of positions in the pupil plane 37 of the projection system PS can be represented as a linear combination of Zernike coefficients c _n using equation (7) or equation (8). Therefore, for a single measured phase AW of the first harmonic of the oscillating phase-stepping signal can be considered to represent a single equation with N unknowns, N being the number of Zernike coefficients c _n .

[000157] This procedure (equating the measured phase AW to a combination of a plurality of differences AW* or AWJ) is performed for a plurality of positions on the radiation detector 23, for example at each sensing element or pixel of the radiation detector 23, in each of the two orthogonal shearing directions. This results in 2M equations with N unknowns, M being the number of positions on the radiation detector 23 and N being the number of Zernike coefficients c _n . These equations are then solved to find the set of Zernike coefficients c _n . Note that the equations from the two orthogonal shearing directions are solved simultaneously to find the set of Zernike coefficients c _n . It will be appreciated that these equations may be solved numerically, using one or more techniques that are known to the skilled person.

[000158] This new method is advantageous because it involves the combination of a plurality of differences in the aberration map between two positions in a pupil plane of the projection system PS and solving to find the set of coefficients c _n . This allows more contributions to the first harmonic of the oscillating signal (i.e. a greater number of interfering pairs of second diffraction beams) to be taken into account. In turn, this allows the method to be used for a greater range of grating geometries for the second patterned region 32. It will be appreciated that each such pair of two positions in a pupil plane of the projection system are separated in the shearing direction.

[000159] Furthermore, this new method allows contributions from a plurality of interference beams to be combined with different interference strengths. This is advantageous because it allows such different interference strengths that may arise from the grating geometry for the second patterned region 32 to be taken into account. However, advantageously, in addition this may also allow any non-uniform pupil fill (i.e. non-uniformity of the pupil illumination of the first patterned region 31) and/or any non uniformity of transmission (apodization) across the pupil plane 37 of the projection system PS to be taken into account. This may be achieved by modifying the interference strengths in dependence to the pupil fill and/or apodization, either of which may, for example, be measured within the lithographic apparatus LA.

[000160] Note that using this new method the pairs of positions in a pupil plane 37 of the projection system PS are separated in the shearing direction by a shearing distance (which corresponds to the distance in the pupil plane 37 between two adjacent first diffraction beams 34-36). This is in contrast to known reconstruction methods, which involve differences in the aberration map between pairs of positions in a pupil plane of the projection system which are separated in the shearing direction by twice this shearing distance.

[000161] It will be appreciated from the above discussion that each difference (AW _; ) in the aberration map is between two positions in the pupil plane 37, these two positions corresponding to the positions in the pupil plane 37 from which the pairs of interfering second diffraction beams that contribute to the ith interference beam originate. It will be further appreciated that the plurality of differences (AW _; ) in the aberration map that are combined at any given position on the radiation detector 23 correspond to the interference beams which overlap with this position on the radiation detector 23 and which contribute to the first harmonic of the oscillating phase-stepping signal. Furthermore, in general at different positions on the radiation detector 23, different combinations of interference beams contribute, as now discussed with reference to Figures 14A and 14B.

[000162] Figure 14A is a representation of 21 second diffraction beams generated by a second patterned region 32 having a unit cell 50 as shown in Figure 12. In particular, these may be considered to be the second diffraction beams generated by such a second patterned region 32 and originating from the 0 ^th order first diffraction beam 35. Figure 14 A shows the diffraction efficiencies of these second diffraction beams and these correspond to the diffraction efficiencies contained within the white dotted line in Figure 13B. Actually the white dotted line in Figure 13B corresponds to 25 second diffraction beams, however, the diffraction efficiencies for 4 of these are negligible and therefore they have not been included on Figure 14A. In addition, in Figure 14A a white dashed circular line indicates a region on the radiation detector 23 which corresponds to the numerical aperture of the projection system PS.

[000163] Figure 14B is a representation of 16 interference beams, each generated by the interference of pairs of the second diffraction beams shown in Figure 14A with second diffraction beams generated by this second patterned region 32 and originating from the ± 1 ^sl order first diffraction beams 34, 36. Figure 14B shows the interference strengths of these interference beams and these correspond to the interference strengths contained within the white dotted line in Figure 13C. Again, the white dotted line in Figure 13C actually corresponds to 20 different second diffraction beams, however, the interference strengths for 4 of these are negligible and therefore they have not been included on Figure 14B. In addition, in Figure 14B a white dashed circular line indicates a region on the radiation detector 23 which corresponds to the numerical aperture of the projection system PS.

[000164] It is apparent from Figure 14B that there is significant overlap between each of these interference beams at the radiation sensor 23. Figure 15 shows a map of the radiation detector 23 within a white dashed circular line, which corresponds to the numerical aperture of the projection system PS. The map of Figure 15 shows how many of the 20 interference beams (from within the white dotted line in Figure 13C) overlap for each position on the radiation detector 23.

[000165] In some embodiments of the method of the present invention, the radiation detector 23 is considered to comprise a plurality of distinct regions. In the determination of the set of Zernike coefficients c _n that characterize the aberration map of the projection system PS, a different set of a plurality of differences AW _; in the aberration map are combined for positions on the radiation detector in different distinct regions of the radiation detector. It will be appreciated that each of the plurality of distinct regions of the radiation detector may correspond to a region of the radiation detector 23 which contains a different combination of these overlapping interference beams.

[000166] There is a single discrete region 55 of the radiation detector within which all 20 of the interference beams overlap. This is a central portion of the radiation detector 23, which is reproduced in Figure 16.

[000167] There are 4 discrete regions 56 of the radiation detector 23 within which 19 of the 20 interference beams overlap. These 4 regions 56 are reproduced in Figure 17A. The permutations of the 20 interference beams for each of these regions 56 are represented in Figure 17B. In Figure 17B, each column represents a different one of the interference beams and each row represents a different permutation of the interference beams. In any given row, the pixels which are black represent the absence of the interference beam corresponding to that column from the permutation. Similarly, there are 18 discrete regions 57 of the radiation detector 23 within which 18 of the 20 interference beams overlap. These 18 regions 57 are reproduced in Figure 18A. The permutations of the 20 interference beams for each of these regions 56 are represented in Figure 18B. Figure 19A shows all of the discrete regions of the radiation detector within which one or more of the 20 interference beams overlap. The permutations of the 20 interference beams for all these discrete regions of the radiation detector are shown in Figure 19B.

[000168] It will be appreciated that in some embodiments the method for determining the aberration map of a projection system PS may comprise, for each of the plurality of positions on the radiation detector 23 (for example for each sensing element or pixel), determining all interference beams which overlap with that position on the radiation detector 23 and which contribute to the first harmonic of the oscillating phase stepping signal. Alternatively (and equivalently), all pairs of second diffraction beams that interfere and which overlap with that position on the radiation detector 23 and contribute to the first harmonic of the oscillating signal may be determined. In the above-described example, wherein the second patterned region 32 that is of the form of a pinhole array having the unit cell 50 shown in Figure 12, it will be appreciated that there are an infinite number of interference beams. However, most of these have negligible interference strengths (the interference strengths decreasing as the diffraction orders of the second diffraction beams that contribute to the interference beams increases). Therefore, for practical implementations only those interference beams which have an interference strength that is greater than a threshold value may be taken into account. Alternatively, a threshold may be imposed on the diffraction orders of the second interference beams which can contribute (as was imposed in the above-described example where only 20 interference beams were considered). Although in the example described above with reference to Figures 14A to 19B only 25 diffraction orders were taken into account (in particular the 25 second diffraction beams that correspond to the diffraction efficiencies contained within the white dotted line in Figure 13B) it will be appreciate that this was merely by way of an example to explain how the method works. It will be further appreciated that, in alternative embodiments, more, or fewer, diffraction orders may be taken into account. It will be further appreciated that the number of diffraction orders taken into account may be chosen in dependence on the geometries of the first and second patterned regions 31, 32.

[000169] It will be appreciated that this determination of either (a) all pairs of second diffraction beams or (b) all interference beams that which contribute to the first harmonic of the oscillating signal at each position on the radiation detector 23 will be dependent on the geometries of the first and second patterned regions 31, 32.

[000170] It will be appreciated that some or all of the steps of the above mentioned method may be performed by the controller CN (see Figure 2). The method may, for example, be stored on a computer readable medium carrying a computer program comprising computer readable instructions configured to cause a computer (for example controller CN) to carry out the above-described method. A computer apparatus for implementing the above-described method may comprise a memory storing processor readable instructions, and a processor (for example the controller CN) arranged to read and execute instructions stored in said memory. Said processor readable instructions may comprise instructions arranged to control the computer to carry out the above-described method.

[000171] Although the above described embodiments use the first harmonic of the a phase stepping signal it will be appreciated that in alternative embodiments higher harmonics of the phase stepping signal may alternatively be used.

[000172] Although the above described embodiments use a first patterned region 31 comprising a one-dimensional diffraction grating 31 with a 50% duty cycle it will be appreciated that in alternative embodiments other the first patterned region 31 may use different geometries. For example, in some embodiments, the first patterned region 31 may comprise a two-dimensional checkerboard diffraction grating with a 50% duty cycle.

[000173] Although specific reference may be made in this text to the use of lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications. Possible other applications include the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat-panel displays, liquid- crystal displays (LCDs), thin-film magnetic heads, etc.

[000174] Although specific reference may be made in this text to embodiments of the invention in the context of a lithographic apparatus, embodiments of the invention may be used in other apparatus. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus that measures or processes an object such as a wafer (or other substrate) or mask (or other patterning device). These apparatus may be generally referred to as lithographic tools. Such a lithographic tool may use vacuum conditions or ambient (non- vacuum) conditions.

[000175] Where the context allows, embodiments of the invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the invention may also be implemented as instructions stored on a machine -readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine-readable medium may include read only memory (ROM); random access memory (RAM); magnetic storage media; optical storage media; flash memory devices; electrical, optical, acoustical or other forms of propagated signals (e.g. carrier waves, infrared signals, digital signals, etc.), and others. Further, firmware, software, routines, instructions may be described herein as performing certain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc. and in doing that may cause actuators or other devices to interact with the physical world.

[000176] While specific embodiments of the invention have been described above, it will be appreciated that the invention may be practiced otherwise than as described. The descriptions above are intended to be illustrative, not limiting. Thus it will be apparent to one skilled in the art that modifications may be made to the invention as described without departing from the scope of the claims set out below.