Title of Invention: Method and apparatus for inspection and metrology Abstract: A method and apparatus for optical metrology is disclosed. There is disclosed, for example, a method involving a radiation intensity distribution for a target measured using an optical component at a gap from the target, the method including calculating a correction factor for the variation of radiation intensity of the radiation intensity distribution as a function of variation of the distance of the gap. Detailed Description: Part general_description_paragraphs: This application claims the benefit of priority of European Patent Application No. 15157799.6, filed on Mar. 5, 2015, which is incorporated herein in its entirety by reference. Part FIELD: The present description relates to a method and apparatus for correction of error in measured radiation distribution captured from a metrology target. Part BACKGROUND: A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g., including part of, one, or several dies) on a substrate (e.g., a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at one time, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate. In order to monitor the lithographic process, the patterned substrate is inspected and one or more parameters of the patterned substrate are measured. The one or more parameters may include, for example, the overlay error between successive layers formed in or on the patterned substrate and/or critical linewidth of developed photosensitive resist. This measurement may be performed on a target of the product substrate itself and/or on a dedicated metrology target provided on the substrate. There are various techniques for making measurements of the microscopic structures formed in lithographic processes, including the use of a scanning electron microscope and/or various specialized tools. A fast and non-invasive form of specialized inspection tool is a scatterometer in which a beam of radiation is directed onto a target on the surface of the substrate and properties of the scattered or reflected beam are measured. By comparing one or more properties of the beam before and after it has been reflected or scattered by the substrate, one or more properties of the substrate can be determined. Two main types of scatterometer are known. A spectroscopic scatterometer directs a broadband radiation beam onto the substrate and measures the spectrum (intensity as a function of wavelength) of the radiation scattered into a particular narrow angular range. An angularly resolved scatterometer uses a monochromatic radiation beam and measures the intensity of the scattered radiation as a function of angle. A particular application of scatterometry is in the measurement of feature asymmetry within a periodic target. This can be used as a measure of overlay error, for example, but other applications are also known. In an angle resolved scatterometer, asymmetry can be measured by comparing opposite parts of the diffraction spectrum (for example, comparing the −1st and +1st orders in the diffraction spectrum of a periodic grating). This can be done simply in angle-resolved scatterometry, as is described for example in U.S. patent application publication US2006-066855. Part SUMMARY: With reduction of the physical dimensions in lithographic processing, there is demand to, for example, increase measurement accuracy and/or reduce the space occupied by targets dedicated to metrology. Image based scatterometry measurements have been devised to allow the use of smaller targets, by taking separate images of the target using −1st and +1st order radiation in turn. Examples of this image based technique are described in published U.S. patent application publication nos. US2011-0027704, US2011-0043791 and US2012-0044470, which are incorporated herein in their entirety by reference. Demand for further reduction in target size and for improved accuracy continues, however, and existing techniques suffer from various constraints that make it difficult to maintain accuracy and/or reduce the size of the targets. Another way to improve on inspection and measurement techniques is to use a solid immersion lens (SIL) as the optical element nearest the substrate surface. The extreme proximity of the SIL with the substrate surface (e.g., target surface) results in a very high effective numerical aperture (NA) larger than 1. Using an incoherent or coherent radiation source with this SIL allows a very small target to be inspected. To take advantage of the increasing numerical aperture, the gap between the SIL and the substrate should be set to a desired value. For example, the gap may be within the range of 10-50 nm to have the SIL in effective optical contact with the substrate. An example optical gap measuring method and apparatus can involve detecting cross components of polarization in the high numerical aperture element. The cross polarized signal is then recorded by a detector and can be used as an input parameter into a gap control process. In another example, the gap may be controlled by reference to reflected laser radiation intensity. As will be appreciated, other methods and apparatus may be used to arrive at a signal representative of the gap (e.g., representative of its size or of its variation from a nominal size). Irrespective of any detecting method, the gap between the SIL (or other component) and the substrate (or other surface) should be established, and maintained at, a desired gap distance or distance range, typically by an associated actuator and control system. This is because the measurement data (e.g., intensity data, image, etc.) derived from the radiation redirected by the target, and obtained using the SIL (or other component), depends on the gap, and so any parameter of interest (e.g., height of a part of the target pattern, width of a part of the target pattern, thickness of one or more various layers of the target pattern, etc.) will depend on the gap distance because the radiation redirected from the substrate is evanescently coupled to the SIL. But, regardless of the control mechanism used to establish, and maintain, the desired gap, a residual dynamic error is generally present in the gap, i.e., there is a dynamic error in the gap distance from the desired or expected gap distance. And, it has been discovered that even a small variation in the gap distance in the absolute sense may cause unacceptably large error in the determination of one or more parameters of interest derived from the measurement data. Accordingly, it is desired to provide, for example, one or more methods and apparatus to correct measurement data obtained using a SIL (or other component) for a residual error in the gap between the SIL and the substrate, and/or derive a parameter of interest using corrected measurement data. In an aspect, there is provided a method to correct radiation intensity associated with a target for reconstruction of one or more parameters of interest relating to the target. In an aspect, there is provided a method involving a radiation intensity distribution for a target measured using an optical component at a gap from the target, the method comprising: calculating a correction factor for the variation of radiation intensity of the radiation intensity distribution as a function of variation of the distance of the gap. In an aspect, there is provided a method comprising: for a given target structure, calculating a second-order derivative tensor for dependence of radiation intensity on a gap between the target structure and an optical element; determining a statistical variance of a gap variation distribution over a measurement period; and determining a variation in radiation intensity for the target structure based on the statistical variance and the second-order derivative tensor. In an aspect, there is provided a method comprising: for a given target structure, calculating a second-order derivative tensor for dependence of radiation intensity on a gap between the target structure and an optical element; evaluating statistical variance of a gap variation distribution for a measurement period of the target, the gap variation distribution being based on a measured gap signal; and evaluating a mean radiation intensity variation across the target structure as a function of variation of the gap based on the second-order derivative tensor and the statistical variance of the gap variation distribution. Part DETAILED DESCRIPTION: Before describing embodiments in detail, it is instructive to present an example environment in which embodiments may be implemented. FIG. 1 schematically depicts a lithographic apparatus LA. The apparatus comprises: an illumination system (illuminator) IL configured to condition a radiation beam B (e.g. UV radiation or DUV radiation). a support structure (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 in accordance with certain parameters; a substrate table (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 in accordance with certain parameters; and a projection system (e.g. a refractive projection lens system) PL 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, the projection system supported on a reference frame (RF). The illumination system may include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic or other types of optical components, or any combination thereof, for directing, shaping, or controlling radiation. The support structure supports the patterning device in a manner that depends on the orientation of the patterning device, the design of the lithographic apparatus, and other conditions, such as for example whether or not the patterning device is held in a vacuum environment. The support structure can use mechanical, vacuum, electrostatic or other clamping techniques to hold the patterning device. The support structure may be a frame or a table, for example, which may be fixed or movable as required. The support structure may ensure that the patterning device is at a desired position, for example with respect to the projection system. Any use of the terms “reticle” or “mask” herein may be considered synonymous with the more general term “patterning device.” The term “patterning device” used herein should be broadly interpreted as referring to any device that can be used to impart a radiation beam with a pattern in its cross-section such as to create a pattern in a target portion of the substrate. It should be noted that the pattern imparted to the radiation beam may not exactly correspond to the desired pattern in the target portion of the substrate, for example if the pattern includes phase-shifting features or so called assist features. Generally, the pattern imparted to the radiation beam will correspond to a particular functional layer in a device being created in the target portion, such as an integrated circuit. The patterning device may be transmissive or reflective. Examples of patterning devices include masks, programmable mirror arrays, and programmable LCD panels. Masks are well known in lithography, and include mask types such as binary, alternating phase-shift, and attenuated phase-shift, as well as various hybrid mask types. An example of a programmable mirror array employs a matrix arrangement of small mirrors, each of which can be individually tilted so as to reflect an incoming radiation beam in different directions. The tilted mirrors impart a pattern in a radiation beam, which is reflected by the mirror matrix. The term “projection system” used herein should be broadly interpreted as encompassing any type of projection system, including refractive, reflective, catadioptric, magnetic, electromagnetic and electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, 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”. As here depicted, the apparatus is of a transmissive type (e.g. employing a transmissive mask). Alternatively, the apparatus may be of a reflective type (e.g. employing a programmable mirror array of a type as referred to above, or employing a reflective mask). The lithographic apparatus may be of a type having two (dual stage) or more tables (e.g., two or more substrate tables WTa, WTb, two or more patterning device tables, a substrate table WTa and a table WTb below the projection system without a substrate that is dedicated to, for example, facilitating measurement, and/or cleaning, etc.). In such “multiple stage” machines the additional tables may be used in parallel, or preparatory steps may be carried out on one or more tables while one or more other tables are being used for exposure. For example, alignment measurements using an alignment sensor AS and/or level (height, tilt, etc.) measurements using a level sensor LS may be made. The lithographic apparatus may also 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 and the substrate. An immersion liquid may also be applied to other spaces in the lithographic apparatus, for example, between the patterning device and the projection system. Immersion techniques are well known in the art for increasing the numerical aperture of projection systems. The term “immersion” as used herein does not mean that a structure, such as a substrate, must be submerged in liquid, but rather only means that liquid is located between the projection system and the substrate during exposure. Referring to FIG. 1, the illuminator IL receives a radiation beam from a radiation source SO. The source and the lithographic apparatus may be separate entities, for example when the source is an excimer laser. In such cases, the source is not considered to form part of the lithographic apparatus and the radiation beam is passed from the source SO to the illuminator IL with the aid of a beam delivery system BD comprising, for example, suitable directing mirrors and/or a beam expander. In other cases the source may be an integral part of the lithographic apparatus, for example when the source is a mercury lamp. The source SO and the illuminator IL, together with the beam delivery system BD if required, may be referred to as a radiation system. The illuminator IL may comprise an adjuster AD configured to adjust the angular intensity distribution of the radiation beam. Generally, at least the outer and/or inner radial extent (commonly referred to as σ-outer and σ-inner, respectively) of the intensity distribution in a pupil plane of the illuminator can be adjusted. In addition, the illuminator IL may comprise various other components, such as an integrator IN and a condenser CO. The illuminator may be used to condition the radiation beam, to have a desired uniformity and intensity distribution in its cross-section. The radiation beam B is incident on the patterning device (e.g., mask) MA, which is held on the support structure (e.g., mask table) MT, and is patterned by the patterning device. Having traversed the patterning device MA, the radiation beam B passes through the projection system PL, which focuses the beam onto a target portion C of the substrate W. With the aid of the second positioner PW and position sensor IF (e.g. an interferometric device, linear encoder, 2-D encoder or capacitive sensor), the substrate table WT can be moved accurately, e.g. so as to position different target portions C in the path of the radiation beam B. Similarly, the first positioner PM and another position sensor (which is not explicitly depicted in FIG. 1) can be used to accurately position the patterning device MA with respect to the path of the radiation beam B, e.g. after mechanical retrieval from a mask library, or during a scan. In general, movement of the support structure MT may be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which form part of the first positioner PM. Similarly, movement of the substrate table WT may be realized using a long-stroke module and a short-stroke module, which form part of the second positioner PW. In the case of a stepper (as opposed to a scanner) the support structure MT may be connected to a short-stroke actuator only, or may be fixed. Patterning device MA and substrate W may be aligned using patterning device alignment marks M1, M2 and substrate alignment marks P1, P2. Although the substrate alignment marks as illustrated occupy dedicated target portions, they may be located in spaces between target portions (these are known as scribe-lane alignment marks). Similarly, in situations in which more than one die is provided on the patterning device MA, the patterning device alignment marks may be located between the dies. The depicted apparatus could be used in at least one of the following modes: 1. In step mode, the support structure MT and the substrate table WT are kept essentially stationary, while an entire pattern imparted to the radiation beam is projected onto a target portion C at one time (i.e. a single static exposure). The substrate table WT is then shifted in the X and/or Y direction so that a different target portion C can be exposed. In step mode, the maximum size of the exposure field limits the size of the target portion C imaged in a single static exposure. 2. In scan mode, the support structure MT and the substrate table WT are scanned synchronously while a pattern imparted to the radiation beam is projected onto a target portion C (i.e. a single dynamic exposure). The velocity and direction of the substrate table WT relative to the support structure MT may be determined by the (de-) magnification and image reversal characteristics of the projection system PL. In scan mode, the maximum size of the exposure field limits the width (in the non-scanning direction) of the target portion in a single dynamic exposure, whereas the length of the scanning motion determines the height (in the scanning direction) of the target portion. 3. In another mode, the support structure MT is kept essentially stationary holding a programmable patterning device, and the substrate table WT is moved or scanned while a pattern imparted to the radiation beam is projected onto a target portion C. In this mode, generally a pulsed radiation source is employed and the programmable patterning device is updated as required after each movement of the substrate table WT or in between successive radiation pulses during a scan. This mode of operation can be readily applied to maskless lithography that utilizes programmable patterning device, such as a programmable mirror array of a type as referred to above. Combinations and/or variations on the above described modes of use or entirely different modes of use may also be employed. As shown in FIG. 2, the lithographic apparatus LA may form part of a lithographic cell LC, also sometimes referred to a lithocell or cluster, which also includes apparatuses to perform pre- and post-exposure processes on a substrate. Conventionally these include one or more spin coaters SC to deposit one or more resist layers, one or more developers DE to develop exposed resist, one or more chill plates CH and/or one or more bake plates BK. A substrate handler, or robot, RO picks up one or more substrates from input/output port I/O1, I/O2, moves them between the different process apparatuses and delivers them to the loading bay LB of the lithographic apparatus. These apparatuses, which are often collectively referred to as the track, are under the control of a track control unit TCU which is itself controlled by the supervisory control system SCS, which also controls the lithographic apparatus via lithography control unit LACU. Thus, the different apparatuses can be operated to maximize throughput and processing efficiency. In order that a substrate that is exposed by the lithographic apparatus is exposed correctly and consistently, it is desirable to inspect an exposed substrate to measure one or more properties such as overlay error between subsequent layers, line thickness, critical dimension (CD), etc. Accordingly a manufacturing facility in which lithocell LC is located also typically includes a metrology system MET which receives some or all of the substrates W that have been processed in the lithocell. The metrology system MET may be part of the lithocell LC, for example it may be part of the lithographic apparatus LA. Metrology results may be provided directly or indirectly to the supervisory control system SCS. If an error is detected, an adjustment may be made to exposure of a subsequent substrate (especially if the inspection can be done soon and fast enough that one or more other substrates of the batch are still to be exposed) and/or to subsequent exposure of the exposed substrate. Also, an already exposed substrate may be stripped and reworked to improve yield, or discarded, thereby avoiding performing further processing on a substrate known to be faulty. In a case where only some target portions of a substrate are faulty, further exposures may be performed only on those target portions which are good. Within a metrology system MET, an inspection apparatus is used to determine one or more properties of the substrate, and in particular, how one or more properties of different substrates vary or different layers of the same substrate vary from layer to layer. The inspection apparatus may be integrated into the lithographic apparatus LA or the lithocell LC or may be a stand-alone device. To enable rapid measurement, it is desirable that the inspection apparatus measure one or more properties in the exposed resist layer immediately after the exposure. However, the latent image in the resist has a low contrast—there is only a very small difference in refractive index between the parts of the resist which have been exposed to radiation and those which have not—and not all inspection apparatus have sufficient sensitivity to make useful measurements of the latent image. Therefore measurements may be taken after the post-exposure bake step (PEB) which is customarily the first step carried out on an exposed substrate and increases the contrast between exposed and unexposed parts of the resist. At this stage, the image in the resist may be referred to as semi-latent. It is also possible to make measurements of the developed resist image—at which point either the exposed or unexposed parts of the resist have been removed—or after a pattern transfer step such as etching. The latter possibility limits the possibilities for rework of a faulty substrate but may still provide useful information. FIG. 3 depicts an example inspection apparatus (e.g., a scatterometer). It comprises a broadband (white light) radiation projector 2 which projects radiation onto a substrate W. The reflected radiation is passed to a spectrometer detector 4, which measures a spectrum 10 (intensity as a function of wavelength) of the specular reflected radiation, as shown, e.g., in the graph in the lower left. From this data, the structure or profile giving rise to the detected spectrum may be reconstructed by processor PU, e.g. by Rigorous Coupled Wave Analysis and non-linear regression or by comparison with a library of simulated spectra as shown at the bottom right of FIG. 3. In general, for the reconstruction the general form of the structure is known and some parameters are assumed from knowledge of the process by which the structure was made, leaving only a few parameters of the structure to be determined from the measured data. Such an inspection apparatus may be configured as a normal-incidence inspection apparatus or an oblique-incidence inspection apparatus. Another inspection apparatus that may be used is shown in FIG. 4. In this device, the radiation emitted by radiation source 2 is collimated using lens system 12 and transmitted through interference filter 13 and polarizer 17, reflected by partially reflecting surface 16 and is focused into a spot S on substrate W via an objective lens 15, which has a high numerical aperture (NA), desirably at least 0.9 or at least 0.95. An immersion inspection apparatus (using a relatively high refractive index fluid such as water) may even have a numerical aperture over 1. As in the lithographic apparatus LA, one or more substrate tables may be provided to hold the substrate W during measurement operations. The substrate tables may be similar or identical in form to the substrate tables WTa, WTb of FIG. 1. In an example where the inspection apparatus is integrated with the lithographic apparatus, they may even be the same substrate table. Coarse and fine positioners may be provided to a second positioner PW configured to accurately position the substrate in relation to a measurement optical system. Various sensors and actuators are provided for example to acquire the position of a target of interest, and to bring it into position under the objective lens 15. Typically many measurements will be made on targets at different locations across the substrate W. The substrate support can be moved in X and Y directions to acquire different targets, and in the Z direction to obtain a desired location of the target relative to the focus of the optical system. It is convenient to think and describe operations as if the objective lens is being brought to different locations relative to the substrate, when, for example, in practice the optical system may remain substantially stationary (typically in the X and Y directions, but perhaps also in the Z direction) and only the substrate moves. Provided the relative position of the substrate and the optical system is correct, it does not matter in principle which one of those is moving in the real world, or if both are moving, or a combination of a part of the optical system is moving (e.g., in the Z and/or tilt direction) with the remainder of the optical system being stationary and the substrate is moving (e.g., in the X and Y directions, but also optionally in the Z and/or tilt direction). The radiation redirected by the substrate W then passes through partially reflecting surface 16 into a detector 18 in order to have the spectrum detected. The detector may be located in a back-projected pupil plane 11, which is at the focal length of the lens system 15, however the pupil plane may instead be re-imaged with auxiliary optics (not shown) onto the detector. The pupil plane is the plane in which the radial position of radiation defines the angle of incidence and the angular position defines azimuth angle of the radiation. The detector may be a two-dimensional detector so that a two-dimensional angular scatter spectrum of a substrate target 30 can be measured. The detector 18 may be, for example, an array of CCD or CMOS sensors, and may use an integration time of, for example, 40 milliseconds per frame. A reference beam may be used, for example, to measure the intensity of the incident radiation. To do this, when the radiation beam is incident on the partially reflecting surface 16 part of it is transmitted through the partially reflecting surface 16 as a reference beam towards a reference mirror 14. The reference beam is then projected onto a different part of the same detector 18 or alternatively on to a different detector (not shown). One or more interference filters 13 is available to select a wavelength of interest in the range of, say, 405-790 nm or even lower, such as 200-300 nm. The interference filter may be tunable rather than comprising a set of different filters. A grating could be used instead of an interference filter. An aperture stop or spatial light modulator (not shown) may be provided in the illumination path to control the range of angle of incidence of radiation on the target. The detector 18 may measure the intensity of redirected radiation at a single wavelength (or narrow wavelength range), the intensity separately at multiple wavelengths or integrated over a wavelength range. Furthermore, the detector may separately measure the intensity of transverse magnetic- and transverse electric-polarized radiation and/or the phase difference between the transverse magnetic- and transverse electric-polarized radiation. The target 30 on substrate W may be a 1-D grating, which is printed such that after development, the bars are formed of solid resist lines. The target 30 may be a 2-D grating, which is printed such that after development, the grating is formed of solid resist pillars or vias in the resist. The bars, pillars or vias may be etched into the substrate. The pattern (e.g., of bars, pillars or vias) is sensitive to chromatic aberration in the lithographic projection apparatus, particularly the projection system PS, and illumination symmetry and the presence of such aberration will manifest in a variation in the printed grating. Accordingly, the measured data of the printed grating is used to reconstruct the grating. One or more parameters of the 1-D grating, such as line width and/or shape, or one or more parameters of the 2-D grating, such as pillar or via width or length or shape, may be input to the reconstruction process, performed by processor PU, from knowledge of the printing step and/or other inspection processes. In addition to measurement of a parameter by reconstruction, angle resolved scatterometry is useful in the measurement of asymmetry of features in product and/or resist patterns. A particular application of asymmetry measurement is for the measurement of overlay, where the target 30 comprises one set of periodic features superimposed on another. The concepts of asymmetry measurement using the instrument of FIG. 3 or FIG. 4 are described, for example, in U.S. patent application publication US2006-066855, which is incorporated herein in its entirety. Simply stated, while the positions of the diffraction orders in the diffraction spectrum of the target are determined only by the periodicity of the target, asymmetry in the diffraction spectrum is indicative of asymmetry in the individual features which make up the target. In the instrument of FIG. 4, where detector 18 may be an image sensor, such asymmetry in the diffraction orders appears directly as asymmetry in the pupil image recorded by detector 18. This asymmetry can be measured by digital image processing in unit PU, and calibrated against known values of overlay. FIG. 5 illustrates a plan view of a typical target 30, and the extent of illumination spot S in the apparatus of FIG. 4. To obtain a diffraction spectrum that is free of interference from surrounding structures, the target 30, in an embodiment, is a periodic structure (e.g., grating) larger than the width (e.g., diameter) of the illumination spot S. The width of spot S may be smaller than the width and length of the target. The target in other words is ‘underfilled’ by the illumination, and the diffraction signal is free from interference by product features and the like outside the target itself. The illumination arrangement 2, 12, 13, 17 may be configured to provide illumination of a uniform intensity across a pupil plane of objective 15. Alternatively, by, e.g., including an aperture in the illumination path, illumination may be restricted to on axis or off axis directions. FIG. 6 schematically depicts an example process of the determination of one or more parameters of interest of a target pattern based on measurement data obtained using metrology. Radiation detected by detector 18 provides a measured radiation distribution 108 for target 30′. This measured radiation distribution 108 contains information to enable derivation of a parameter of interest such as the overlay error between successive layers formed in or on the substrate and/or critical dimension of, e.g., developed photosensitive resist. FIG. 7 depicts an example portion of a target (such as target 30, 30′) and example layers of various materials making up, and associated with, the target. For example, the target may comprise a layer of silicon dioxide (SiO2) 710 overlying the bare silicon substrate 700. Overlying layer 710 may be a layer of silicon nitride (Si3N4) 720, which may form a grating feature, having a layer of TEOS (tetraethyl orthosilicate) 730 overlying it. Overlying layer 730 is a further layer of silicon nitride (Si3N4) 740, which may form a further grating feature (e.g., a grating feature for measuring overlay). FIG. 7 further depicts various parameters of the target that help to define the target, e.g., the distance H1 representing the thickness of layer 710, the distance H2 representing the thickness of layer 720, the distance H3 representing the thickness of layer 730, the distance H4 representing the thickness of layer 740, top critical dimension TCD of layer 740, and bottom critical dimension BCD of layer 740. These various (and other) parameters influence the radiation distribution obtained at detector 18. For a given target 30′, a radiation distribution 208 can be computed/simulated from a parameterized model 206 of the pattern for target 30′ using, for example, a numerical Maxwell solver 210. The parameterized model 206 may include one or more of the parameters identified in FIG. 7 and/or other parameters such as refractive index of one or more of the layers, a sidewall angle of one or more layers, etc. The initial values of the parameters may be those expected for the target being measured. The measured radiation distribution 108 is then compared at 212 to the computed radiation distribution 208 to determine the difference between the two. If there is a difference, the values of one or more of the parameters of the parameterized model 206 may be varied, a new computed radiation distribution 208 calculated and compared against the measured radiation distribution 108 until there is sufficient match between the measured radiation distribution 108 and the computed radiation distribution 208. At that point, the values of the parameters of the parameterized model 206 provide a good or best match of the geometry of the actual target 30′. One of those parameters of the parameterized model (e.g., CD) may be used by the user for evaluating the lithographic process. Additionally or alternatively, a parameter of interest may be derived from one or more of the values of the parameterized model. But, there is demand to reduce the space occupied by metrology targets. For example, there is a desire to reduce the width of ‘scribe lanes’ between target portions C on the substrate, where metrology targets have conventionally been located. Additionally or alternatively, there is a desire, for example, to include metrology targets within the device patterns themselves, to allow more accurate monitoring and correction of variations in parameters such as CD and/or overlay. To this end, alternative methods of diffraction based metrology have been devised more recently. For example, in image-based metrology, two images of the target are made, each using different selected orders of the diffraction spectrum. Comparing the two images, one can obtain asymmetry information. By selecting parts of the images, one can separate the target signal from its surroundings. The targets can be made smaller, and need not be square, so that several can be included within the same illumination spot. Examples of this technique are described in U.S. patent application publications US2011-0027704, US2011-0043791, and US2012-0044470. In addition to or alternatively to reducing the space occupied by metrology targets, there is demand to improve the nature of the measurements themselves, such as their accuracy. For example, there is a desire to, for example, obtain higher sensitivity of measurement. Additionally or alternatively, there is a desire to, for example, obtain better decoupling between various parameters in the reconstruction described above. For example, it is desired to obtain better values for each of the specific parameters of interest, by reducing or eliminating the effect of measurements associated with one parameter of interest influencing another parameter of interest. As the demand for size reduction and/or accuracy continues, existing techniques may meet some technical limitations. For example, some methods desire to capture at least the ±1st diffraction orders. Taking into account the numerical aperture of the objective 15, this constrains the pitch (L) of a periodic structure of the target. To improve sensitivity and/or to reduce target size, one can consider using shorter wavelengths λ. Further, the target cannot be too small otherwise it will not have enough features to be considered as a periodic structure. Consequently, overlay, as an example, is measured using periodic structures features (e.g., lines) having dimensions far bigger than those of the product (e.g., device) layout, making overlay measurement less reliable. Ideally the feature line and pitch should have similar dimensions to the product features. FIG. 8 shows an inspection apparatus in which improvement of the nature of the measurements themselves (e.g., accuracy) and/or reduction of target size may be realized. In FIG. 8, a spot S′ (which may be smaller than convention if, for example, a smaller target is desired) can be applied to a target 30′ (which may be smaller than convention, e.g., features of smaller pitch, if, for example, a smaller target is desired). Like reference numerals refer to like components throughout the figures. Comparing the apparatus of FIG. 8 with that of FIG. 4, a first difference is the provision of an additional lens element 60 close to the target 30′. This additional lens is a miniature solid immersion lens (SIL), with a width (e.g., diameter) only on the order of a millimeter, for example in the range of 1 mm to 5 mm, for example about 2 mm. The SIL comprises, in an example, a hemisphere of material that receives rays of radiation at substantially normal incidence to its surface. In an embodiment, the SIL may be a different shape such as a super-hemisphere. In an embodiment, the SIL is made up of a material of refractive index n, such as glass, fused quartz, a combination of materials, etc. Within the SIL material, the numerical aperture (NA) of the original rays is multiplied by n. The received rays come to focus at about the center of the hemisphere or super-hemisphere and form a spot that is smaller by a factor of n compared to what would have been in the absence of the SIL. For example, a typical glass hemisphere having n=2 will reduce the width of the focused spot by a factor of 2. Immersion of optical elements in liquid has been used to increase resolution in microscopy and photolithography. The solid immersion lens may achieve similar gains without the inconvenience/problems of liquid immersion. However, to ensure that the increased NA does indeed increase the resolution of the system, the bottom of the SIL must either be in contact with the target 30 or positioned extremely closely to it. This restricts its practical applications. A so-called micro-SIL may also be used. The width (e.g., diameter) of such a SIL is many times smaller, for example about 2 microns in width instead of about 2 millimeters. In an example where SIL 60 in the FIG. 8 apparatus is a micro-SIL, it may have a width (e.g., diameter) less than or equal to 10 μm, potentially less than or equal to 5 μm. Whether a miniature or micro-SIL 60 is used, it can be attached to a movable support so that controlling the alignment and proximity to the substrate is much simpler than in the case of a lens with bigger width. For example, the SIL 60 in FIG. 8 is mounted to a frame 62. In an embodiment, frame 62 is movable. An actuator may be provided to move frame 62. In an embodiment, the frame 62 supports the objective 15. Accordingly, in an embodiment, the frame 62 may move both the objective 15 and the SIL 60 together. In an embodiment, the actuator for the frame 62 may be configured to move the frame 62 (and the SIL 60) in substantially the Z direction. In an embodiment, the actuator for the frame 62 may be configured to move the frame 62 (and the SIL 60) around the X axis and/or Y axis. In an embodiment, the SIL 60 is in relative fixed position relative to the frame 62. This may be referred to a single stage arrangement, where the objective 15 and SIL 60 are fixed relative to each and are moved by the actuator of frame 62. In such a case, a benefit may be that the SIL can be mechanically positioned in the focus of the objective. As noted above, the SIL 60 in FIG. 8 is mounted to a frame 62, which in an embodiment supports objective 15. Of course, the SIL 60 may be mounted on a separate frame from that supporting objective 15. In an embodiment, the SIL 60 is connected to a frame (e.g., frame 62) via an arm 64 and actuator 66. Actuator 66 may be, for example, piezoelectric in operation or voice coil actuated. The arrangement where the SIL 60 has an actuator to cause relative movement between a movable objective 15 and the SIL 60 may be referred to as a dual stage arrangement. In a dual stage, certain functionalities may be separated. For example, the (relatively large) objective stage comprises the relatively heavy objective and can have relatively large motion range. In an embodiment, the objective stage may move only substantially in the Z-direction (substantially normal to the surface). Further, it can have a certain bandwidth (e.g., ˜100 Hz) sufficient for relatively long displacement ranges, but perhaps not sufficient (e.g., too low) for suppression of small surface disturbances. The (relatively small) SIL stage comprises the relatively light SIL and can have a relatively small motion range. In an embodiment, the SIL stage may move in at least 3 degrees of freedom, e.g., in the Z-direction and around the X-axis and/or the Y-axis, to position the SIL substantially parallel to the surface. Further, it can have a certain bandwidth (e.g., sufficiently high) to suppress small surface disturbances (e.g., ˜1-5 nm). The SIL stage may not have a mechanical range sufficient to cover the desired full travel range. So, the SIL stage can be used to position the SIL at about 10-50 nm above the surface, while the objective stage can position the objective at focus with respect to the surface. Actuator 66 may operate in combination with one or more other actuators positioning the objective as a whole in relation to the target. In relation to the coarse and fine positioners mentioned above, for example, the actuator 66 may be regarded as an ultra-fine positioner. The servo control loops of these different positioners can be integrated with one another. The components 62, 64 and 66, together with the substrate table and positioners (mentioned above but not shown in FIG. 8), form a support apparatus for positioning the SIL and the target T in close proximity to one another. AS noted above, in principle, SIL 60 could be mounted rigidly to the frame 62, and/or may be of larger width. The separate arm and actuator allows easier control of the very small gap, as discussed in more detail below. Inclusion of the SIL 60 opens the possibility of focusing to a much smaller spot S′. The SIL works by capturing the near-field radiation from the target, and to this end it is positioned substantially closer than one wavelength (λ) of radiation from the target structure, generally closer than a half wavelength, for example around λ/20. The closer the distance, the stronger will be the coupling of near-field signals into the instrument. The gap between the SIL 60 and target 30′ may therefore be less than 100 nm, for example between 10 nm and 50 nm. Because the NA of the inspection apparatus is effectively increased, the sensitivity and parameter de-correlation is enhanced such that the pitch of the target periodic structure may be reduced closer to product dimensions. In examples where a micro-SIL would be used, incoherent radiation of the type conventionally used in, for example, a scatterometer cannot be focused to a micron-sized spot as small as the micro-SIL. Accordingly, in such an embodiment or in an embodiment using a macro-SIL (i.e., one larger than a micro-SIL) the radiation source 2 may be changed to a coherent source. Therefore a laser source 70 is coupled to illumination optics 12, etc. via an optical fiber 72. The limit on the spot size on the substrate is set by the numerical aperture of the focusing lens system and the laser wavelength. As an additional benefit of using spatially coherent radiation, the instrument with laser radiation source 70 can be used to perform different types of scatterometry or measurement. For example, coherent Fourier scatterometry (CFS) may be used to measure the target. As highlighted above, a small gap should be maintained between the SIL and the target. As also highlighted above, known techniques for controlling the gap have limitations, particularly when a variety of different target structures and materials are to be inspected. But, even if the control for gap is improved, a residual dynamic error of the control system will be difficult, if not impossible, to eliminate. That is, regardless of the control mechanism, the actual value of a gap (e.g., a value in the range of 10-50 nm, e.g., 20, 25, 30, or 35 nm) may vary because of various factors such as vibrations in the apparatus, movement around the apparatus, vibration of the floor on which the apparatus sits, variability in Van der Waals forces, and so forth. FIG. 9 shows an example deviation from the nominal value of the gap versus time for the acquisition of measurement data using a SIL at a gap in the range of 10-50 nm. That is, the variable plotted in FIG. 9 is the change from a desired or expected gap value, rather than a measurement of the absolute gap distance, and this data is for a SIL controlled to be at the certain nominal value. As can be seen in FIG. 9, while the absolute value of the variability can be relatively small, the variation is quite significant and moreover, some values may be relatively high compared to the nominal value. Accordingly, it may be incorrect to assume when deriving one or more parameters of interest from a radiation distribution measured using a SIL that the gap assumes a single value over the acquisition time and that the error is not significant enough. So, assuming a single value for the gap distance over an acquisition period of measurement data may decrease precision of the one or more reconstructed parameters of interest. Accordingly, it is desirable to apply a correction to the measurement data, and/or one or more parameters of interest derived from the measurement data, to account for the variability in gap distance. So, to make this kind of correction, there should be a measure of the gap distance variability. The gap variation may be determined/measured using any available gap related signal. In an embodiment, the control signal used to control the gap may be used as a proxy for the gap variation. Such a signal may be sufficient to measure the gap variation distribution in real time for a particular data collection. Further, as known in the art, a gap error signal (GES) may be used and obtained/measured using any of the techniques known in the art, now or hereafter. Further, to enable correction of a particular set of measurement data, information about (e.g., values of) the gap variation related to the measurement data should be available. For example, there may be provided information about (e.g., values of) the gap variation for at least part, if not all, of the acquisition time of the measurement data. For example, there may be provided information about (e.g., values of) the gap variation that is obtained prior to a time of obtaining all or part of the measurement data, e.g., dynamic behavior obtained without actually measuring it during acquisition. FIGS. 10A-10C show graphs for distribution of gap variation over three different time intervals (2 ms, 10 ms, and 60 ms, respectively) for two example sets of data collections. The distributions depict the probability in the y-axis for a particular variation in gap in the x-axis during the particular acquisition time. As can be seen, the gap variation distribution may differ for different data collections. The gap variation distribution, thus, depends not only on the dynamic behavior of the SIL at and during the particular acquisition time, but also on the acquisition time over which the behavior is measured. Accordingly, the measurement of the same target under the same general conditions
{'bibliographic_information': "{'country': 'US', 'doc-number': '15053968', 'kind': 'B2', 'date': '20160225', 'extension': '16', 'invention_title': 'Method and apparatus for inspection and metrology'}", 'source_file': './grant/2017/ipg171226.xml', 'source_data': 'USPTO-Grant'}