Method of forming an illumination mode, method of exposing a substrate and illumination system

Abstract

A method of forming an illumination mode using an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane. The method comprises allocating different reflective elements of the array to direct radiation to different locations in the illumination mode by: determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency; applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; and using the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.

Description

METHOD OF FORMING AN ILLUMINATION MODECROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority of EP application 25152911.1 which was filed on 20 January 2025 and EP application 25184827.1 which was filed on 24 June 2025, and which are incorporated herein in its entirety by reference.FIELD

[0002] The present invention relates to a method forming a desired illumination mode. The method may be performed in a lithographic apparatus.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 (also often referred to as “design layout” or “design”) of a patterning device (e.g., a mask) onto a layer of radiation-sensitive material (resist) provided on a substrate (e.g., a wafer).

[0004] As semiconductor manufacturing processes continue to advance, the dimensions of circuit elements have continually been reduced while the amount of functional elements, such as transistors, per device has been steadily increasing over decades, following a trend commonly referred to as ‘Moore’s law’. To keep up with Moore’s law the semiconductor industry is chasing technologies that enable to create increasingly smaller features. 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 are patterned on the substrate. Typical wavelengths currently in use are 365 nm (i-line), 248 nm, 193 nm and 13.5 nm. A lithographic apparatus, which uses extreme ultraviolet (EUV) radiation, having a wavelength within a range of 4 nm to 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] An illumination system of a lithographic apparatus may be used to form a desired illumination mode. The term ‘illumination mode’ refers to a spatial distribution of exposure radiation in a pupil plane prior to the radiation being incident upon a patterning device. Properties of the illumination mode will affect a pattern that is exposed on a substrate by the lithographic apparatus.

[0006] It may be desirable to form an illumination mode using a method not taught or suggested by the prior art.SUMMARY

[0007] According to a first aspect of the invention there is provided a method of forming an illumination mode using an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane; wherein the method comprises allocating different reflective elements of the array to direct radiation to different locations in the illumination mode by: determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency; applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; and using the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.

[0008] Advantageously, the method allows for suppression of undesired frequencies. Suppression of low frequencies is particularly advantageous, for example, because this suppression reduces undesirable low frequency effects in images formed at the lithographic substrate.

[0009] The illumination mode may comprise a plurality of spatially separated illuminated areas, and wherein the different locations to which radiation is directed correspond with the spatially separated illuminated areas.

[0010] The spatially separated illuminated areas of the illumination mode may be poles.

[0011] The illumination mode may be a quadrupole mode.

[0012] The power spectral density function may include a low frequency cutoff.

[0013] A coherence size may be used to determine the low frequency cutoff of the power spectral density function.

[0014] The power spectral density function may include an additional and different low frequency cutoff used for reflective element allocation in an orthogonal direction on the array of reflective elements.

[0015] The low frequency cutoff may correspond with a spatial dimension of 4 mm or less.

[0016] The power spectral density function may include a high frequency cutoff.

[0017] Transforming the power spectral density function into the spatial domain signal may be performed using an inverse Fourier transform.

[0018] The different phases applied to the frequency components may be random phases.

[0019] According to a second aspect there is provided a method of forming an illumination mode using an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane; wherein the method comprises allocating different reflective elements of the array to direct radiation to different illuminated areas in the illumination mode which are separated by unilluminated regions, wherein the allocation of the reflective elements of the array is determined by:defining a cost function or merit function, which depends at least in part upon a distance between reflective elements of the array, which direct radiation to the same illumination region of the illumination mode;generating an initial matrix, which allocates reflective elements of the array to direct radiation to different illuminated areas of the illumination mode, and determining the cost function or merit function of the initial matrix; thena. selecting two locations in the matrix which represent reflective elements of the array, and swapping the illumination region allocations for those two locations in the matrix to generate a new matrix;b. determining the cost function or merit function of the new matrix; andc. if the new matrix provides a better cost function or merit function than the previous matrix then retaining the new matrix;wherein the method further comprises repeating a. to c. until a predetermined criterion is satisfied, and using the matrix to allocate reflective elements that direct radiation to different locations in the illumination mode.

[0020] Advantageously, the method allows a mirror allocation to be obtained, which is configured to satisfy any desired combination of properties.

[0021] Optionally, the method further comprises, if the new matrix provides a worse cost function or merit function than the previous matrix, then retaining the new matrix if a number generated using a random number generator satisfies a predetermined test.

[0022] The merit function or cost function may take into account one or more of the following: a size of a radiation coherence cell at the array of reflective elements, a shape of a radiation coherence cell at the array of reflective elements, distance between locations in the pupil plane, which receive radiation from neighbouring mirrors or mirrors within the same coherence cell, and a comparison of the resulting illumination mode with a desired illumination mode.

[0023] The predetermined criterion may be one or more of: the cost function falling below a threshold, the cost function falling below a threshold normalized with respect to the cost function of the initial matrix, the merit function moving above a threshold, the merit function moving above a threshold normalized with respect to the merit function of the initial matrix.

[0024] The predetermined criterion may be one or more of: a predetermined number of iterations of a. to c. have taken place, the illumination mode corresponds with a desired illumination mode to within a desired accuracy, a predetermined processing time has elapsed, a new matrix has been rejected less than a threshold number of times.

[0025] According to a third aspect of the invention there is provided a method of exposing a substrate using a lithographic apparatus, the method comprising providing a beam of radiation using an illumination system; using the method of the first aspect to form an illumination mode using the beam of radiation; directing the beam of radiation with the illumination mode onto a patterning device toimpart the radiation beam with a pattern in its cross-section; and projecting the patterned radiation beam onto a target portion of a substrate.

[0026] According to a fourth aspect of the invention there is provided an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane, and further comprising a processor, wherein the processor is configured to allocate different reflective elements of the array to direct radiation to different locations in an illumination mode by determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency; applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; and using the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.

[0027] Advantageously, the illumination system is able to suppress undesired frequencies. Suppression of low frequencies is particularly advantageous, for example, because this suppression reduces undesirable low frequency effects in images formed at the lithographic substrate.

[0028] The illumination mode may comprise a plurality of spatially separated illuminated areas, and wherein the different locations to which radiation is directed correspond with the spatially separated illuminated areas.

[0029] The spatially separated illuminated areas of the illumination mode may be poles.

[0030] The illumination mode may be a quadrupole mode.

[0031] The power spectral density function may include a low frequency cutoff.

[0032] A coherence size may be used to determine the low frequency cutoff of the power spectral density function.

[0033] The power spectral density function may include an additional and different low frequency cutoff used for reflective element allocation in an orthogonal direction on the array of reflective elements.

[0034] The low frequency cutoff may correspond with a spatial dimension of 4 mm or less.

[0035] According to a fifth aspect of the invention there is provided an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane, and further comprising a processor, wherein the processor is configured to allocate different reflective elements of the array to direct radiation to different locations in a pupil plane; by allocating different reflective elements of the array to direct radiation to different illuminated areas in the illumination mode which are separated by unilluminated regions, wherein the allocation of the reflective elements of the array is determined by:defining a cost function or merit function, which depends at least in part upon a distance between reflective elements of the array, which direct radiation to the same illumination region of the illumination mode;generating an initial matrix, which allocates reflective elements of the array to direct radiation to different illuminated areas of the illumination mode, and determining the cost function or merit function of the initial matrix; thena. selecting two locations in the matrix which represent reflective elements of the array, and swapping the illumination region allocations for those two locations in the matrix to generate a new matrix;b. determining the cost function or merit function of the new matrix; andc. if the new matrix provides a better cost function or merit function than the previous matrix then retaining the new matrix;wherein the method further comprises repeating a. to c. until a predetermined criterion is satisfied, and using the matrix to allocate reflective elements that direct radiation to different locations in the illumination mode.

[0036] Advantageously, the illumination system allows a mirror allocation to be obtained, which is configured to satisfy any desired combination of properties.

[0037] The illumination system of the fifth aspect of the invention, wherein if the new matrix provides a worse cost function or merit function than the previous matrix, then the processor is configured to retain the new matrix if a number generated using a random number generator satisfies a predetermined test.

[0038] The merit function or cost function may take into account one or more of the following: a size of a radiation coherence cell at the array of reflective elements, a shape of a radiation coherence cell at the array of reflective elements, distance between locations in the pupil plane which receive radiation from neighbouring mirrors or mirrors within the same coherence cell, and a comparison of the resulting illumination mode with a desired illumination mode.

[0039] The predetermined criterion may be one or more of: the cost function falling below a threshold, the cost function falling below a threshold normalized with respect to the cost function of the initial matrix, the merit function moving above a threshold, the merit function moving above a threshold normalized with respect to the merit function of the initial matrix.

[0040] The predetermined criterion may be one or more of: a predetermined number of iterations of a. to c. have taken place, the illumination mode corresponds with a desired illumination mode to within a desired accuracy, a predetermined processing time has elapsed, a new matrix has been rejected less than a threshold number of times.

[0041] Features of different aspects of the invention may be combined together.BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which:Figure 1 schematically depicts a lithographic apparatus;Figure 2 schematically depicts operation of an array of individually controllable reflective elements, which forms part of an illumination system of the lithographic apparatus;Figure 3 schematically depicts a quadrupole illumination mode formed using the array of individually controllable reflective elements;Figure 4 is a chart depicting a power spectral density function used by an embodiment of the invention;Figure 5 schematically depicts spatial frequency components for an array of mirrors, which may form part of an embodiment of the invention;Figure 6 depicts a mirror allocation scheme obtained using a method according to an embodiment of the invention for the power spectral density function depicted in Figure 4;Figure 7 depicts an alternative mirror allocation scheme obtained using the method according to an embodiment of the invention for the power spectral density function depicted in Figure 4;Figure 8 is a chart depicting an alternative power spectral density function used by an embodiment of the invention;Figure 9 depicts a mirror allocation scheme obtained using a method according to an embodiment of the invention for the power spectral density function depicted in Figure 8;Figure 10 depicts a numerical matrix representing an initial mirror allocation scheme for a method of mirror allocation according to an alternative embodiment of the invention;Figure 11 depicts a mirror allocation after 20 iterations of the method;Figure 12 depicts a mirror allocation after 10,000 iterations of the method;Figure 13 depicts progression of a normalized cost function during the 10,000 iterations of the method; andFigure 14 depicts progression of the normalized cost function during a final 1,200 iterations of the method.DETAIEED DESCRIPTION

[0043] In the present document, the terms “radiation” and “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g., with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g., having a wavelength in the range of about 5-100 nm). The term “reticle”, “mask” or “patterning device” as employed in this text may be broadly interpreted as referring to a generic patterning device that can be used to endow an incoming radiation beam with a patterned cross-section, corresponding to a pattern that is to be created in a target portion of the substrate. The term “light valve” can also be used in this context. Besides the classic mask(transmissive or reflective, binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include a programmable mirror array and a programmable LCD array.

[0044] Figure 1 schematically depicts a lithographic apparatus LA. The lithographic apparatus LA includes an illumination system (also referred to as illuminator) IL configured to condition a radiation beam B (e.g., UV radiation, DUV radiation or EUV radiation), a mask support (e.g., a mask table) MT constructed to support a patterning device (e.g., a mask) MA and connected to a first positioner PM configured to accurately position the patterning device MA in accordance with certain parameters, a substrate support (e.g., a wafer table) WT constructed to hold a substrate (e.g., a resist coated wafer) W and connected to a second positioner PW configured to accurately position the substrate support in accordance with certain parameters, and a projection system (e.g., a refractive projection lens system) PS configured to project a pattern imparted to the radiation beam B by patterning device MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.

[0045] In operation, the illumination system IL receives a radiation beam from a radiation source SO, e.g., via a beam delivery system BD. The illumination system IL may include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic, and / or other types of optical components, or any combination thereof, for directing, shaping, and / or controlling radiation. The illumination system IL may be used to condition the radiation beam B to have a desired spatial and angular intensity distribution in its cross section at a plane of the patterning device MA. The illumination system IL may be used to form a desired illumination mode in a pupil plane of the lithographic apparatus.

[0046] The term “projection system” PS used herein should be broadly interpreted as encompassing various types of projection system, including refractive, reflective, catadioptric, anamorphic, magnetic, electromagnetic and / or electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, and / or for other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term “projection lens” herein may be considered as synonymous with the more general term “projection system” PS.

[0047] The lithographic apparatus LA may be of a type, wherein at least a portion of the substrate may be covered by a liquid having a relatively high refractive index, e.g., water, so as to fill a space between the projection system PS and the substrate W - which is also referred to as immersion lithography. More information on immersion techniques is given in US6952253, which is incorporated herein by reference.

[0048] The lithographic apparatus LA may also be of a type having two or more substrate supports WT (also named “dual stage”). In such “multiple stage” machine, the substrate supports WT may be used in parallel, and / or steps in preparation of a subsequent exposure of the substrate W may be carried out on the substrate W located on one of the substrate support WT while another substrate W on the other substrate support WT is being used for exposing a pattern on the other substrate W.

[0049] In addition to the substrate support WT, the lithographic apparatus LA may comprise a measurement stage. The measurement stage is arranged to hold a sensor and / or a cleaning device. Thesensor may be arranged to measure a property of the projection system PS or a property of the radiation beam B. The measurement stage may hold multiple sensors. The cleaning device may be arranged to clean part of the lithographic apparatus, for example a part of the projection system PS or a part of a system that provides the immersion liquid. The measurement stage may move beneath the projection system PS when the substrate support WT is away from the projection system PS.

[0050] In operation, the radiation beam B is incident on the patterning device, e.g., mask, MA which is held on the mask support MT, and is patterned by the pattern (design layout) present on patterning device MA. Having traversed the mask MA, the radiation beam B passes through the projection system PS, which focuses the beam onto a target portion C of the substrate W. With the aid of the second positioner PW and a position measurement system IF, the substrate support WT can be moved accurately, e.g., so as to position different target portions C in the path of the radiation beam B at a focused and aligned position. Similarly, the first positioner PM and possibly another position sensor (which is not explicitly depicted in Figure 1) may be used to accurately position the patterning device MA with respect to the path of the radiation beam B. Patterning device MA and substrate W may be aligned using mask alignment marks Ml, M2 and substrate alignment marks Pl, P2. Although the substrate alignment marks Pl, P2 as illustrated occupy dedicated target portions, they may be located in spaces between target portions. Substrate alignment marks Pl, P2 are known as scribe-lane alignment marks when these are located between the target portions C.

[0051] To clarify the invention, a Cartesian coordinate system is used. The Cartesian coordinate system has three axis, i.e., an x-axis, a y-axis and a z-axis. Each of the three axis is orthogonal to the other two axis. A rotation around the x-axis is referred to as an Rx-rotation. A rotation around the y-axis is referred to as an Ry-rotation. A rotation around about the z-axis is referred to as an Rz -rotation. The x-axis and the y-axis define a horizontal plane, whereas the z-axis is in a vertical direction. The Cartesian coordinate system is not limiting the invention and is used for clarification only. Instead, another coordinate system, such as a cylindrical coordinate system, may be used to clarify the invention. The orientation of the Cartesian coordinate system may be different, for example, such that the z-axis has a component along the horizontal plane.

[0052] The illumination system IL may be used to form a desired illumination mode in a pupil plane PP of the lithographic apparatus, as schematically depicted in Figure 2. A source 31 (equivalent to SO in FIG. 1) outputs a relatively narrow, collimated radiation beam which passes through a shutter 11. The radiation beam then passes through beam divergence optics 32, which expand the beam to a size that corresponds to the size of an array 33 of mirrors 33a-e. Ideally, the radiation beam divergence optics 32 should output a collimated beam. Preferably, the size of the expanded radiation beam generally corresponds with the size of the array 33 of mirrors. For example, the size of the radiation beam may be sufficient that the radiation beam is incident at substantially all mirrors 33a-e. The array 33 of mirrors may be a two-dimensional array. The arrangement depicted in Figure 2 is merely an example. In other examples, the source 31 may provide a beam, which is not narrow and / or collimated, and the shutter 11may be absent. In general, optics may be provided which generally collimate the radiation beam before it is incident upon the array 33 of mirrors.

[0053] In FIG. 2, by way of example, three sub-beams of the radiation beam are shown. A first subbeam is incident at mirror 33b. Like the other mirrors 33a-e of the array 33, the mirror 33b can be controlled to adjust its orientation so that the sub-beam is reflected in a desired predetermined direction. The sub-beam is incident at a mirror 35, which is located in a pupil plane PP of the lithographic apparatus. The other sub-beams shown in FIG. 2 are reflected by the mirrors 33c, 33d and are incident at other points in the pupil plane PP. By controlling the orientations of the mirrors 33a to 33e, almost any spatial intensity distribution in the pupil plane PP can be produced. In other words, almost any desired illumination mode may be formed. In some embodiments there is no mirror at the pupil plane PP. A mirror may instead be located further downstream from the pupil plane PP.

[0054] Although FIG. 2 shows only five mirrors 33a-e, the array 33 may include many more mirrors, for example arranged in a two-dimensional grid. For example, the array 33 may include 1024 (e.g., 32x32) mirrors, or 4096 (e.g., 64x64) mirrors, or any other suitable number of mirrors. An array of mxn mirrors, where m is an integer and n is an integer, may be used (e.g., with m=n or m^n). The array may for example comprise m mirrors along one direction, where m = 2k(k being an integer), and may for example comprise n mirrors along an orthogonal direction, where n = 2J(j being an integer). More than one array of mirrors may be used. For example a group of four mirror arrays having 32x32 mirrors each may be used. In the following text, the term ‘array’ may mean a single array or a group of mirror arrays.

[0055] The orientation of each mirror of the array 33 may be adjusted separately. The orientations of the mirrors may be controlled by a controller CT (see FIG. 1).

[0056] Figure 3 schematically depicts a quadrupole illumination mode (referred to hereafter as a quadrupole mode). The quadrupole mode comprises four areas of radiation, which are separated from one another and are located adjacent a numerical aperture perimeter 38 of the illumination system (the areas of radiation are within the numerical aperture). Each area of radiation is referred to as a pole (poles 1-4). Each pole of radiation is formed by orienting mirrors of the array 33 depicted in Figure 2 to direct radiation to that pole. Pole 1 is schematically depicted as being formed by areas of radiation 40. Although only a small number of areas of radiation 40 are depicted, in practice many more areas of radiation may be used to form the pole (e.g., 50 areas of radiation or more). Poles 2-4 are formed in the same way.

[0057] An embodiment of the invention may determine which mirrors of the array 33 are used to direct radiation to each of Poles 1-4 of a quadrupole mode. An embodiment of the invention allows a spatial frequency of mirrors that direct radiation to each of Poles 1-4 to be selected or controlled. Other embodiments may determine, which mirrors 33a-e of the array 33 are used to direct radiation to parts of other illumination modes (and may allow selection or control of spatial frequencies of those mirrors).

[0058] An embodiment of the invention provides a method of determining allocation of mirrors of an array of individually controllable mirrors in a manner, which provides control of a distribution of the mirrors that contribute to each pole of the quadrupole mode. In other words, the method provides control of spatial frequencies present in a distribution of mirrors of the reflective array, which contributes to the quadrupole illumination mode.

[0059] A first part of the method comprises establishing a desired power spectral density (PSD) for the spatial frequencies of the mirrors, which direct radiation to each pole of the quadrupole mode. An example of a desired power spectral density is depicted in Figure 4. In this example, which is provided for illustrative purposes, the array is a 1 x 64 array of mirrors. A minimum frequency corresponds with a maximum spatial wavelength of the radiation at the mirror array. That is, a frequency of 1 / 64 (which may be referred to as a relative frequency of 1 / 64). A maximum frequency corresponds with a minimum spatial wavelength of the radiation at the mirror array. That is, a frequency of 64 / 64, i.e., a frequency of 1 (which may be referred to as a relative frequency of 1).

[0060] Examples of spatial wavelengths and associated spatial frequencies are schematically depicted in Figure 5. A 1 x 64 micro mirror array 50 is depicted at the bottom of Figure 5, as an example. A single wavelength, which has a length corresponding with a length of the mirror array 50, is depicted as chart A. In this example the frequency is f = 1 / 64.

[0061] In chart B of Figure 5, the spatial frequency has been doubled to equal 2 / 64. The length of the micro mirror array 50 corresponds with two wavelengths. In chart C the frequency is 3 / 64, meaning that three wavelengths fit on the array 50. In chart D the frequency is 8 / 64 , meaning that eight wavelengths fit on the array 50. In chart E the frequency is 16 / 64, meaning that sixteen wavelengths fit on the array 50. Although a selection of frequencies is shown in Figure 5, the micro mirror array 50 is capable of providing every spatial frequency from f=l / 64 to f=64 / 64. That is, for a given pole the mirrors of the micro mirror array 50, which point towards that pole, may have any spatial frequency from f=l / 64 to f=64 / 64. The mirrors of the micro mirror array 50, which point towards the pole, may have a combination of spatial frequencies from f= 1 / 64 to f=64 / 64. In general, the frequency ranges from 1 / N to N / N, with N being an integer. Preferably N=2k, with k being an integer.

[0062] Referring again to Figure 4, the power spectral density indicates the relative amplitude of each spatial frequency component of a distribution of mirrors pointing at a given quadrupole mode (i.e., the relative amplitudes of the frequency components schematically depicted in part in Figure 5). The amplitude is a relative amplitude expressed in arbitrary units. The frequency is expressed as a relative frequency, the right hand end of the chart being a frequency of 1. The scales of the chart are logarithmic.

[0063] The power spectral density may be defined by a function, which provides a bandpass characteristic:PSDBP(n = PSD(f; blt C1) ■ PSD(f; b2,C2)where bi is the cutoff frequency of the bandpass at low frequencies and a determines how quickly frequencies below the cutoff frequency are dampened (the slope of roll-off at low frequencies), and where b is the cutoff frequency of the bandpass at high frequencies and -C2 determines how quickly frequencies above the cutoff frequency are dampened (the slope of roll-off at high frequencies). Although not in the above power spectral density, the power spectral density may include an amplitude term.

[0064] The power spectral density depicted in Figure 4 was generated using the above function, with a cutoff frequency at low frequencies of 4 / 64 (bi = 4 / 64) and a cutoff frequency at high frequencies of 50 / 64. The cutoff frequency of 4 / 64 is labelled as a high pass frequency, indicated by a dashed line, because the filter passes frequencies above this cutoff frequency. The cutoff frequency of 50 / 64 (62 = 50 / 64) is labelled as a low pass frequency, indicated by a dotted line, because the filter passes frequencies below this cutoff frequency. In the power spectral density depicted in Figure 4, the roll-off parameter is 10 (i.e., ci = 10 and C2 = 10).

[0065] As depicted in Figure 4, in the power spectral density the lowest frequencies are suppressed. In addition, the highest frequencies are suppressed.

[0066] Suppression of the lowest frequencies is particularly advantageous because this suppression reduces undesirable low frequency effects in images formed at the lithographic substrate. In other words, low frequency wavy lines which may be formed in an image on a substrate may be suppressed (i.e., the wavy lines may be not formed or may be formed with a reduced amplitude). These low frequency wavy lines may be due to spatial coherence of the radiation used to form images. The radiation, when incident at the mirror array, may include areas of spatial coherence which are referred to herein as coherence cells. The coherence cells may for example extend over several mirrors of the mirror array (e.g., around 4 mirrors). If the mirrors of a given coherence cell all direct radiation to a particular pole of the quadrupole mode, then the spatial coherence of the coherence cell is retained and this may generate undesirable wavy lines in an image. This is because the illumination mode for that specific coherence cell is very limited in size, and as a result the coherence cell fully consists of low frequency content. This low frequency content generates the undesirable low frequency wavy lines in the image.

[0067] If the mirrors of the coherence cell direct radiation to different poles of the quadrupole mode, then a substantial part of the low-frequency content is sent to the high-frequency domain. The high frequency content does not generate low frequency wavy lines in the image. High frequency wavy linesmay be generated, but high frequency wavy lines are removed or suppressed during processing of the substrate (unlike low frequency wavy lines).

[0068] Although the effect of the invention is described above in connection with a quadrupole mode, the same effect applies for other illumination modes. In general, an embodiment of the invention may have an illumination mode with a plurality of spatially separated illuminated areas (e.g., separated by dark unilluminated areas). Radiation from a coherence cell may be directed to locations, which lie in different spatially separated illuminated areas. The spatially separated illuminated areas of the illumination mode may be poles.

[0069] The power spectral density function of Figure 4 suppresses low frequencies of the spatial frequency component of a distribution of mirrors pointing at a given quadrupole mode. In other words, the power spectral density function of Figure 4 reduces the probability of mirrors with a separation of 4 mirrors or less all pointing at the same pole of the quadrupole mode.

[0070] In an embodiment, mirror orientation allocations are determined using a power spectral density with the following method:• Each of the spatial frequencies of mirror orientation, which may be present at the 1 x 64 mirror array, is provided with an amplitude according to a desired power spectral density. Referring to Figure 5, this means that the amplitude of each depicted frequency (and other frequencies not depicted) is applied. In the case of the power spectral density of Figure 4, the amplitude of f=l / 64 is strongly suppressed, the amplitude of f=2 / 64 also suppressed, etc.• The square root of each amplitude for each frequency is obtained to obtain a set of frequency components.• Different phases (e.g., between -n and n) are applied to different frequency components. For example, random phases (including pseudo-random phases) may be applied. Referring to Figure 5, this is schematically indicated by the double-headed arrow 52. Applying a random phase to each frequency component corresponds with translating each sinewave to the left or right by a random amount. Each of the frequency components along with their associated phases form a frequency domain signal.• An inverse Fourier transform is used to convert the frequency domain signal into a spatial domain signal. That is, a signal that varies as a function of position on the 1 x 64 micro mirror array. An example of the spatial domain signal is depicted in Figure 6. Although the spatial domain signal may appear to be random with no particular frequency distribution, the spatial domain signal includes a reduced amount of low frequency components..• The spatial domain signal is used to allocate mirrors to each pole of the quadrupole, as indicated.In this example, data points of the spatial domain signal with the lowest vertical axis value are allocated to the first pole. Since there are 64 mirrors in this example, the 16 mirrors associated with the lowest values are allocated to the first pole. The 16 mirrors associated with the lowest remainingvalues are allocated to the second pole. The 16 mirrors associated with the lowest remaining values are allocated to the third pole. There are then 16 remaining mirrors (they are the mirrors with the highest values). These are allocated to the fourth pole. Because the power spectral density depicted in Figure 4 was used, allocation of mirrors to each of the poles of the quadrupole mode is provided using a spatial domain signal in which low spatial frequencies are suppressed. From looking at Figure 6 it can be seen that there are only two instances in which four mirrors all point to the same pole (mirrors 18-21 point to pole 1, and mirrors 25-29 point to pole 4), and there are no instances in which five or more mirrors point to the same pole.

[0071] The random allocation of phases to each frequency component means that running the same method again will provide a different mirror orientation allocation. To illustrate this, the method was run again using the same power spectral density parameters, i.e., / ? / = 4 / 64, b = 50 / 64, ci = 10, C2 = 10. The spatial domain signal obtained by running the method again is depicted in Figure 7. Again, low frequency components of the mirror orientation can be seen to be suppressed. There is only one instance in which four mirrors all point to the same pole (mirrors 25-28 point to pole 2), and there are no instances in which five or more mirrors point to the same pole.

[0072] Running the same method again will generate a different mirror orientation allocation if different phases (e.g., random phases) are applied to the frequencies. Low spatial frequencies will again be suppressed in the mirror allocation. If the phases used for the frequencies are not changed then the same mirror allocation will be generated by the method.

[0073] A mirror allocation generated for one lithographic apparatus may be used for another lithographic apparatus. However, there may be differences between the lithographic apparatuses, such as variations of mirror reflectivity or variations of incident light intensity. If there are differences between lithographic apparatuses then it may be preferable to generate a new mirror allocation.

[0074] A method according to an embodiment of the invention may be used to apply any desired power spectral density to the allocation of mirror orientations. The desired power spectral density may be selected based upon desired properties of images formed at a substrate by the lithographic apparatus. For example, it may be desirable to supress low frequency modulations in images (low frequency wavy lines) and this may be achieved by suppressing low frequencies in the mirror allocations as explained above. Higher frequency modulations in an image may be acceptable in some cases because the higher frequency modulations may be removed during processing of the substrate after it has been exposed by the radiation.

[0075] In another example it may be desirable to suppress high frequency modulations. A power spectral density, which suppresses high frequency mirror allocations, is depicted in Figure 8. In this case there is a high frequency cutoff at bi = 4 / 64 with the roll-off being ci = 10. There is no low frequency cutoff. With this power spectral density the higher frequencies are suppressed. This may be useful if there is a particularly undesirable high frequency intensity variation of radiation. In one example, a high frequency intensity variation may be caused by variation of beam pointing from theradiation source to the illumination system. This undesired high frequency intensity variation may be suppressed via selection of a power spectral density which suppresses high frequencies. The intensity variation may be suppressed via selection of a power spectral density, which suppresses the frequency or frequencies of the intensity variation.

[0076] Figure 9 depicts a spatial domain signal obtained using the power spectral density of Figure 8. As can be seen, the spatial domain signal does not contain high frequency components, but instead contains only low frequencies (the spatial domain signal varies slowly as a function of spatial position). Allocation of poles using the spatial domain signal is shown. In this case, the lowermost 16 values of the spatial domain signal are clustered in two groups. Thus, two groups of mirrors direct radiation to the first pole. There are four groups of mirrors directing radiation to the second pole. Two of the groups of mirrors are quite large (6 mirrors and 7 mirrors), but two groups are small (2 mirrors and 1 mirror). The third and fourth poles of the mode are also formed predominantly from mirrors in large groups.

[0077] As noted above, the power spectral density function can be selected as desired to suppress unwanted frequency components in a mirror allocation for an illumination mode.

[0078] For simplicity the above examples are explained in relation to a 1 x 64 array of mirrors. In practice the array of mirrors may be a 2-dimensional array (e.g., 32 x 32 mirrors, 64 x 64 mirrors, or some other size). The method can be extended to 2-dimensions for a 2-dimensional array. In 2D, the power spectral density can accept multiple frequencies that account for the different spatial directions of the array, for example the horizontal and the vertical directions. The power spectral density can be selected to suppress horizontal and vertical frequencies differently. This allows for more detailed tuning of the frequencies present in patterns exposed on a substrate. Random phases may be applied for each frequency combination. Transformation from the frequency to the spatial domain may be performed using a 2D inverse Fourier transform.

[0079] In the above embodiments the frequency domain signal is transformed to the spatial domain using an inverse Fourier transform. However, any suitable transform from the frequency domain to the spatial domain may be used.

[0080] In the above embodiments the illumination mode is a quadrupole mode. However, embodiments of the invention may be used in connection with any illumination mode.

[0081] The application of different (e.g., random) phases to the frequency components before transforming to the spatial domain advantageously avoids introducing undesirable frequency components, which may otherwise be present. Thus, embodiments of the invention allow suppression of undesired frequency components, whilst avoiding accidental introduction of frequency components that could arise for example if a repeating pattern of mirror allocations were to be used.

[0082] In an embodiment, the spatial coherence of radiation may be defined in terms of a coherence size pcoh, being the maximum distance over which coherence exists in the radiation. The coherence cell may be a disk with a diameter corresponding with the coherence size pcoh. In a lithographic apparatus, the coherence size pcoh, is determined at least in part by properties of the source SO, 31 , which may bea laser. The coherence size pcoh of the radiation beam at the micro mirror array may be used to determine the power spectral density function that is used to determine the mirror allocation. For example, the cutoff frequency below which low frequencies are suppressed may be selected with reference to the coherence size pcoh. In one example, the micro mirror array comprises 64 x 64 mirrors and measures 5 x 5 cm2. In this example, each mirror may be square with a length of 0.78 mm. The coherence size pcoh of the radiation beam at the mirror array may be around 3 mm, which corresponds with slightly less than 4 mirrors. The low frequency cutoff for the power spectral density function may be set at 4 / 64. In general, the low frequency cutoff of the power spectral density function may be set at a value, which corresponds to a distance greater than the coherence size pcoh of the radiation at the micro mirror array.

[0083] It may be possible to measure the coherence size pcoh of the radiation beam at the micro mirror array by removing the micro mirror array and replacing it with an image sensor. The image sensor may be used to obtain an image for a single pulse of incident radiation. This image will include intensity fluctuations due to coherence effects. The spatial size of the intensity fluctuations may correspond with the coherence size pcoh. Multiple images may be used when measuring the coherence size pcoh.

[0084] Alternatively, a series of substrate exposures may be performed with mirrors allocated using different power spectral density functions, and modulations present in the exposed substrates may be used to determine the coherence size pcoh of the radiation beam.

[0085] In an alternative approach, a series of substrate exposures may be performed with the low frequency cutoff being increased gradually until undesired low frequency modulations are no longer present in exposed images or have a negligible contribution.

[0086] In an embodiment, the coherence cell may be non-circular. For example, the coherence cell may be elliptical. An elliptical coherence cell may be caused for example due to properties of a laser SO, 31 used to generate the radiation beam. Where this is the case, a two dimensional power density function used to determine mirror allocation may include different cutoff frequencies in the first and second directions. The cutoff frequencies may be selected for example through knowledge of the shape and size of coherence cells of the radiation. The cutoff frequencies may be selected for example through generating a series of exposures with mirrors allocated using different power spectral density functions in each dimension, and selecting power spectral density functions which provide desired image properties (e.g., no low frequency modulations or low frequency modulations below a desired threshold).

[0087] An alternative embodiment uses a Monte Carlo approach with a cost function to determine a mirror allocation, instead of using a power density function. The cost function may relate to the size of coherence cells of the radiation, and may take into account coherence cell shape. The cost function may relate to any parameter relevant to performance of the lithographic apparatus. The cost function may include multiple parameters.

[0088] The embodiment may be used for example in a lithographic apparatus as described further above. The embodiment may be used in a lithographic apparatus in which an array of individuallycontrollable reflective elements (referred to hereafter as mirrors of an array) are arranged to direct portions of a radiation beam to desired locations in a pupil plane. The embodiment may be useful when a coherence cell of radiation extends over multiple mirrors of the array. The embodiment includes an assumption that the illumination mode being formed using the mirror array includes at least two poles, which are separated by dark (unilluminated) regions. The illumination mode may comprise more than two poles.

[0089] An example of the embodiment is set out below. The example relates to an array of 4096 mirrors, which are used to form a quadrupole illumination mode. In other examples, an array of mirrors may have a different number of mirrors, and the illumination mode may have a different number of poles.1. The method starts with a numerical matrix M with a size which corresponds with the mirror array (e.g., 64x64 = 4096 mirrors). The values in the array indicate which pole the mirror is assigned to, e.g., index 1 assigned to pole 1, index 2 to pole 4, index 3 to pole 2, and index 4 to pole 3. As an initial condition for the method in the following steps, the array may be filled in a simple manner, so the first 1024 mirrors point to pole 1, the following 1024 to pole 2, and so on. This is schematically depicted in Figure 10, which shows the mirror allocation as four stipes, one stripe for each pole of the illumination mode. In the depicted example, the number of mirrors pointing to each pole is equal. However, the number of mirrors pointing to each pole may vary (for example if a pole or poles is desired to have a different intensity than another pole). In an alternative starting numerical matrix, the poles may be scrambled or distributed across the numerical matrix M.2. A cost function W (M, ... ), is defined, i.e., a function that takes at least the matrix M from the previous step as input and returns a high value in the case of unwanted behavior, or a low value for preferred behavior. For example:a. High frequency variation in the mirror allocation may be preferred for the reasons set out further above, so it may be preferred to have neighboring mirrors pointing to different poles. Therefore: the cost function may increase when neighboring mirrors point towards the same pole.b. The cost function may increase when neighboring mirrors, or mirrors within a region of interest, are pointing to pupil locations very close to each other (e.g., within a predetermined distance of each other).c. The cost function may include a weighting factor related to the coherence cell size. This may be expressed in multiples of the mirror size. The shape of the coherence cell may be taken into account. That is, the weighting factor may have a first value in a first direction, and may have a different value in an orthogonal direction.d. The cost function may include a factor related to the total intensity that each pole in the pupil receives from the mirror array. This can be used, e.g., in case not all mirrors of thearray receive an equal amount of light from the source, or in the case of differences in reflectivity between the mirrors. For example, a large intensity difference between the poles may increase the cost function, meaning that a small difference is preferred.e. An example of the way that the cost function may be applied is: each mirror is considered, and the method looks in the neighborhood of that mirror. Consider mirror i. If a mirror j, which is close to mirror i, (e.g., is a neighboring mirror) points to the same pole, increase the value of the cost function with some value, e.g., A = 1. Repeat this for every mirror in the MMA.i. Additionally, the cost function can be increased with a value which decreases as a function of the distance between mirrors i and j, e.g., A oc — - — j. The function may (rj-rt)be an inverse of the square of the distance, or may have a different form.ii. A region of interest around each mirror may have the dimensions of a coherence cell.For example, this can be a square region of a size of 5x5 mirrors, centered around mirror i. In another example, the region of interest may be elongate (e.g., if the coherence cell is elongate). The value of the cost function may be increased if mirror j is within the region of interest.iii. This cost function example can be implemented by zero-padding the array M on all sides, and using the circshift operation. That is, zero-padding is used, and allows the contents of the array M to be shifted by one mirror each time. This allows the comparison of the same indices using vector operations. This way, the cost function consists of elementary array operations that work on the zero-padded array, which can be done very efficiently.f. The cost function applied to a mirror allocation may involve the distance between locations in the pupil plane which receive radiation from two array mirrors in close proximity (e.g., neighboring mirrors or mirrors within the same coherence cell). For example, array mirrors i and j point toward (Jx i, oy iand axj,ayj, respectively in the pupil plane. The distance between them is calculated asHere, to promote high-frequency speckle behavior (which is preferred over low-frequency speckle behavior for the reasons explained further above), a term may be added to the cost function that includes the reciprocal of this distance, i.e., l / r^. Thus, when the distancein the pupil plane increases the cost function reduces. The distance referred to here is different to the distance referred to further above (see 2.e.i). Here, the distance is in thepupil plane, in a units (i.e., location relative to the numerical aperture), whereas the previous distance was the physical distance on the mirror array (which may for example be expressed in multiples of the mirror size).i. Alternatively, only the distance between the mirrors in the pupil plane in a particular direction may be taken into account. A term may be included in the cost function to account for that, e.g.,g. A cost function may be defined to include a rendered version of the illumination pupil itself.This allows for the inclusion of pupil performance parameters such as pole imbalance, telecentricity, etc. in the cost function.An advantage of the embodiment is that it provides freedom in defining the cost function. That is, the cost function may be a combination of some or all of the above. The cost function may comprise other parameters specific to the system type, pupil performance, or imaging layer, in order to promote desired behaviour.3. An initial value Woof the cost function is defined for an initial mirror allocation matrix M.When the initial mirror allocation is as depicted in Figure 10, the initial value of the cost function is VF0« 51863. Note that the numerical values of different cost functions are arbitrary, and therefore they are not comparable. In this example, the cost function is a simple one; the cost function increases if neighbouring mirrors point to the same pole.4. The following loop is then used:a. Using a random number generator, generate two (different) random integers that represent two distinct locations in the mirror array.b. Swap the values (i.e., the assigned pole in the illumination plane) at these locations of the mirror allocation matrix, and calculate the value of the cost function of the candidate matrix Mc.c. Additionally, using a random generator, generate a random number 0 < r < 1.d. Accept the candidate matrix Mcif:i. Cost function W has decreased (this indicates preferred behavior); or optionally, ii. The random number is smaller than some pre-defined threshold r < rthresfl0id. This step is optional and is to prevent the method getting accidentally stuck in a local minimum of the cost function. The method may work without this optional step, but may provide an output, which is a local minimum of the cost function. In general, the new matrix is used if a number generated using a random number generator satisfies a predetermined test.iii. Otherwise, reject the candidate matrix Mcand revert back to the matrix M that was used at the start of this iteration.e. Repeat steps a-c until a stopping criterion has been reached. Such criterion can be, for example:i. The cost function W has converged below a certain threshold.ii. The cost function W, normalized by the value of the cost function of the initial condition Wo, converged below a certain threshold.iii. A certain number of iterations have been performed.iv. Calculated pupil performance is good enough.v. Calculation time has expired.vi. The number of rejected candidate matrices in the last N iterations exceeds a threshold value. When a candidate matrix is rejected, this means that the current matrix has a lower cost function W than the candidate matrix. Rejection of for example 90 out of 100 candidate matrices may indicate that a sufficiently good matrix M has been achieved. The threshold value and the number of iterations may be selected as desired. Rejected candidate matrices are depicted as dots in Figure 14 (as explained further below).vii. The number of accepted candidate matrices in the last N iterations is below a threshold value. As with vi above, the threshold value and the number of iterations may be selected as desired.

[0090] An intermediate result of the approach, after 20 iterations, is shown in Figure 11. It can be seen that a number of mirror swaps have been accepted, because they lowered the value of the cost function, compared to the initial cost function. The value of the cost function W is 50,866, which is lower than the value of the initial cost function Wo = 51,863.

[0091] Another result of the approach, after 10,000 iterations, is shown in Figure 12. In this example, this is the end result because the stopping criterion was 10,000 iterations. The cost function W has been reduced to 10,507.

[0092] The convergence behaviour of the cost function is shown in Figure 13. The value of the cost function in Figure 13 in this case has been normalized with the value of the initial cost function Wo- Although the method was stopped after 10,000 iterations, the method could instead have been stopped based on behaviour / con vergence of the cost function. Figure 14 shows a final portion of the cost function of Figure 13. The dots are values of the cost function of candidate arrays, and the dark line indicates the value of the cost function of accepted arrays. It can be seen that the cost function increased at around 9,600 steps. This was due to the effect of the random number threshold condition explained above at step 4.d.i

[0093] The depicted example relates to an illumination mode with four poles. However, in other embodiments the illumination mode may have a different number of poles. In general, embodiments ofthe invention may be used in connection with illuminated areas which are separated by dark (unilluminated) regions. These may be referred to as isolated features. For example, the illumination mode may consist of two isolated features, three isolated features, five isolated features, or more (e.g., 10 or more isolated features).

[0094] The calculation may be accelerated by applying checks, which are used to discard some potential mirror swaps without calculating the effect of the swap. For example, if two mirrors to be swapped are within a predetermined distance of each other then that swap may be discarded without calculating the effect of the swap.

[0095] In a modified embodiment, instead of minimising a cost function, a merit function may be maximised. The merit function M may be the inverse of the cost function, i.e., M = 1 / W.

[0096] In some cases, it is possible that certain mirror allocations have a negative impact on pupil performance, e.g., intensity variations may lead to pole imbalance, or energetic telecentricity. To reduce a potential negative effect on pupil performance, some mirrors may be unconstrained (i.e., not included in the cost function determination of the mirror allocation). These unconstrained mirrors may be used to reduce pupil performance issues. For example, the mirrors may be used to reduce a pole imbalance by directing more radiation to a pole which has less intensity than other poles. Alternatively, pupil performance metrics may be included as part of the cost function (e.g., the cost function may be increased if there is a difference between pole intensities).

[0097] The above embodiment is described in terms of illumination poles. However, the embodiment may be used in connection with any illumination mode which comprises illuminated areas, which are separated by unilluminated regions.

[0098] A mirror allocation for an illumination mode may be determined according to an embodiment of the invention and then used during exposure of substrates by a lithographic apparatus. The illumination mode may be modified, for example to improve imaging performance. When the illumination mode is modified, a new mirror allocation may be determined according to an embodiment of the invention, and then used during exposure of subsequent substrates by the lithographic apparatus.

[0099] As explained further above, the orientations of the mirrors may be controlled by a controller CT (see FIG. 1). The controller, which may form part of the illumination system, may include a processor configured to perform a method according to an embodiment of the invention.

[0100] Embodiments of the invention are described in terms of mirrors and a mirror array. However, mirrors are merely examples of reflective elements. Other reflective elements may be used.

[0101] Although specific reference may be made in this text to the use of a 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, liquidcrystal displays (LCDs), thin-film magnetic heads, etc.

[0102] 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.

[0103] 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.

[0104] Embodiments include the following numbered clauses:1. A method of forming an illumination mode using an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane; wherein the method comprises allocating different reflective elements of the array to direct radiation to different locations in the illumination mode by:determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency;applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; andusing the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.2. The method of clause 1 , wherein the illumination mode comprises a plurality of spatially separated illuminated areas, and wherein the different locations to which radiation is directed correspond with the spatially separated illuminated areas.3. The method of clause 2, wherein the spatially separated illuminated areas of the illumination mode are poles.4. The method of clause 3, wherein the illumination mode is a quadrupole mode.5. The method of any preceding clause, wherein the power spectral density function includes a low frequency cutoff.6. The method of clause 5, wherein a coherence size is used to determine the low frequency cutoff of the power spectral density function.7. The method of clause 5 or clause 6, wherein the power spectral density function includes an additional and different low frequency cutoff used for reflective element allocation in an orthogonal direction on the array of reflective elements.8. The method of any of clauses 5 to 7, wherein the low frequency cutoff corresponds with a spatial dimension of 4 mm or less.9. The method of any preceding clause, wherein the power spectral density function includes a high frequency cutoff.10. The method of any preceding clause, wherein transforming the power spectral density function into the spatial domain signal is performed using an inverse Fourier transform.11. The method of any preceding clause, wherein the different phases applied to the frequency components are random phases.12. A method of forming an illumination mode using an illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane; wherein the method comprises allocating different reflective elements of the array to direct radiation to different illuminated areas in the illumination mode which are separated by unilluminated regions, wherein the allocation of the reflective elements of the array is determined by: defining a cost function or merit function which depends at least in part upon a distance between reflective elements of the array which direct radiation to the same illumination region of the illumination mode;generating an initial matrix which allocates reflective elements of the array to direct radiation to different illuminated areas of the illumination mode, and determining the cost function or merit function of the initial matrix; thena. selecting two locations in the matrix which represent reflective elements of the array, and swapping the illumination region allocations for those two locations in the matrix to generate a new matrix; b. determining the cost function or merit function of the new matrix; andc. if the new matrix provides a better cost function or merit function than the previous matrix then retaining the new matrix;wherein the method further comprises repeating a. to c. until a predetermined criterion is satisfied, and using the matrix to allocate reflective elements that direct radiation to different locations in the illumination mode.13. The method of clause 12, wherein if the new matrix provides a worse cost function or merit function than the previous matrix, then retaining the new matrix if a number generated using a random number generator satisfies a predetermined test.14. The method of clause 12 or clause 13, wherein the merit function or cost function takes into account one or more of the following: a size of a radiation coherence cell at the array of reflective elements, a shape of a radiation coherence cell at the array of reflective elements, distance between locations in the pupil plane which receive radiation from neighbouring mirrors or mirrors within the same coherence cell, and a comparison of the resulting illumination mode with a desired illumination mode.15. The method of any of clauses 12 to 14, wherein the predetermined criterion is one or more of: the cost function falling below a threshold, the cost function falling below a threshold normalized with respect to the cost function of the initial matrix, the merit function moving above a threshold, the merit function moving above a threshold normalized with respect to the merit function of the initial matrix.16. The method of any of clauses 12 to 15, wherein the predetermined criterion is one or more of: a predetermined number of iterations of a. to c. have taken place, the illumination mode corresponds with a desired illumination mode to within a desired accuracy, a predetermined processing time has elapsed, a new matrix has been rejected less than a threshold number of times.17. A method of exposing a substrate using a lithographic apparatus, the method comprising: providing a beam of radiation using an illumination system;using the method of clause 1 or clause 12 to form an illumination mode using the beam of radiation; directing the beam of radiation with the illumination mode onto a patterning device to impart the radiation beam with a pattern in its cross-section; andprojecting the patterned radiation beam onto a target portion of a substrate.18. An illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane, and further comprising a processor, wherein the processor is configured to allocate different reflective elements of the array to direct radiation to different locations in an illumination mode by:determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency;applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; andusing the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.19. The illumination system of clause 18, wherein the illumination mode comprises a plurality of spatially separated illuminated areas, and wherein the different locations to which radiation is directed correspond with the spatially separated illuminated areas.20. The illumination system of clause 19, wherein the spatially separated illuminated areas of the illumination mode are poles.21. The illumination system of clause 20, wherein the illumination mode is a quadrupole mode 22. The illumination system of any of clauses 18 to 21, wherein the power spectral density function includes a low frequency cutoff.23. The illumination system of clause 22, wherein a coherence size is used to determine the low frequency cutoff of the power spectral density function.24. The illumination system of clause 22 or clause 23, wherein the power spectral density function includes an additional and different low frequency cutoff used for reflective element allocation in an orthogonal direction on the array of reflective elements.25. The illumination system of any of clauses 22 to 24, wherein the low frequency cutoff corresponds with a spatial dimension of 4 mm or less.26. An illumination system comprising an array of individually controllable reflective elements arranged to direct portions of a radiation beam to desired locations in a pupil plane, and further comprising a processor, wherein the processor is configured to allocate different reflective elements of the array to direct radiation to different locations in a pupil plane; by allocating different reflective elements of the array to direct radiation to different illuminated areas in the illumination mode which are separated by unilluminated regions, wherein the allocation of the reflective elements of the array is determined by:defining a cost function or merit function which depends at least in part upon a distance between reflective elements of the array which direct radiation to the same illumination region of the illumination mode;generating an initial matrix which allocates reflective elements of the array to direct radiation to different illuminated areas of the illumination mode, and determining the cost function or merit function of the initial matrix; thena. selecting two locations in the matrix which represent reflective elements of the array, and swapping the illumination region allocations for those two locations in the matrix to generate a new matrix; b. determining the cost function or merit function of the new matrix; andc. if the new matrix provides a better cost function or merit function than the previous matrix then retaining the new matrix;wherein the method further comprises repeating a. to c. until a predetermined criterion is satisfied, and using the matrix to allocate reflective elements that direct radiation to different locations in the illumination mode.27. The illumination system of clause 26, wherein if the new matrix provides a worse cost function or merit function than the previous matrix, then the processor is configured to retain the new matrix if a number generated using a random number generator satisfies a predetermined test.28. The illumination system of clause 26 or clause 27, wherein the merit function or cost function takes into account one or more of the following: a size of a radiation coherence cell at the array of reflective elements, a shape of a radiation coherence cell at the array of reflective elements, distance between locations in the pupil plane which receive radiation from neighbouring mirrors or mirrors within the same coherence cell, and a comparison of the resulting illumination mode with a desired illumination mode.29. The illumination system of any of clauses 26 to 28, wherein the predetermined criterion is one or more of: the cost function falling below a threshold, the cost function falling below a threshold normalized with respect to the cost function of the initial matrix, the merit function moving above a threshold, the merit function moving above a threshold normalized with respect to the merit function of the initial matrix.30. The illumination system of any of clauses 26 to 29, wherein the predetermined criterion is one or more of: a predetermined number of iterations of a. to c. have taken place, the illumination mode corresponds with a desired illumination mode to within a desired accuracy, a predetermined processing time has elapsed, a new matrix has been rejected less than a threshold number of times.

[0105] 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.

Claims

CLAIMS1. A method of forming an illumination mode using an illumination system (IL) comprising an array (33,50) of individually controllable reflective elements (33a-e) arranged to direct portions of a radiation beam to desired locations in a pupil plane (PP); wherein the method comprises allocating different reflective elements of the array to direct radiation to different locations in the illumination mode by:determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency;applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; andusing the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.

2. The method of claim 1, wherein the illumination mode comprises a plurality of spatially separated illuminated areas, and wherein the different locations to which radiation is directed correspond with the spatially separated illuminated areas.

3. The method of claim 2, wherein the spatially separated illuminated areas of the illumination mode are poles (pole 1-4).

4. The method of any preceding claim, wherein the power spectral density function includes a low frequency cutoff.

5. The method of claim 4, wherein a coherence size is used to determine the low frequency cutoff of the power spectral density function.

6. The method of claim 4 or claim 5, wherein the power spectral density function includes an additional and different low frequency cutoff used for reflective element allocation in an orthogonal direction on the array of reflective elements.

7. The method of any preceding claim, wherein the power spectral density function includes a high frequency cutoff.

8. The method of any preceding claim, wherein transforming the power spectral density function into the spatial domain signal is performed using an inverse Fourier transform.

9. The method of any preceding claim, wherein the different phases applied to the frequency components are random phases.

10. A method of exposing a substrate using a lithographic apparatus, the method comprising: providing a beam of radiation using an illumination system;using the method of claim 1 to form an illumination mode using the beam of radiation; directing the beam of radiation with the illumination mode onto a patterning device to impart the radiation beam with a pattern in its cross-section; andprojecting the patterned radiation beam onto a target portion of a substrate.

11. An illumination system comprising an array (33,50) of individually controllable reflective elements (33a-e) arranged to direct portions of a radiation beam to desired locations in a pupil plane (PP), and further comprising a processor, wherein the processor is configured to allocate different reflective elements of the array to direct radiation to different locations in an illumination mode by:determining a desired power spectral density function of radiation directed by the array of reflective elements to the illumination mode, the desired power spectral density function including a cutoff frequency;applying different phases to frequency components of the power spectral density function; transforming the power spectral density function into a spatial domain signal that varies as a function of positions of the reflective elements; andusing the spatial domain signal to allocate reflective elements that direct radiation to different locations in the illumination mode.

12. The illumination system of claim 11, wherein the illumination mode comprises a plurality of spatially separated illuminated areas, and wherein the different locations to which radiation is directed correspond with the spatially separated illuminated areas.

13. The illumination system of claim 11 or claim 12, wherein the power spectral density function includes a low frequency cutoff.

14. The illumination system of claim 13, wherein a coherence size is used to determine the low frequency cutoff of the power spectral density function.

15. The illumination system of claim 13 or claim 14, wherein the power spectral density function includes an additional and different low frequency cutoff used for reflective element allocation in an orthogonal direction on the array of reflective elements.

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