Phase-generated carrier interrogator and associated phase-generated carrier interrogation method

Abstract

A phase-generated carrier interrogator is disclosed, comprising: one or more interferometers, each interferometer being operable to receive modulated radiation and to generate a respective interferometer signal comprising a phase of interest caused by a measured object, each said interferometer signal further comprises a periodic phase modulation caused by said modulation radiation, where the periodic phase modulation comprises an amplitude described by a respective modulation index of each interferometer; at least one detector operable to detect each interferometer signal; and a signal processing module. The signal processing module is operable to: estimate a respective modulation index for each interferometer, respectively, obtaining a respective estimated modulation index for each interferometer; the interferometer signal is demodulated using the respective estimated modulation index for each interferometer to estimate a respective phase of interest from each interferometer signal to obtain a respective phase of interest estimate for each interferometer.

Description

[0001] Cross-reference to related applications

[0002] This application claims priority to EP Application No. 22211743.4 filed on December 6, 2022 and EP Application No. 22213974.3 filed on December 15, 2022, and the entire contents of the above EP applications are incorporated herein by reference. Technical field

[0003] The present invention relates to interferometry techniques and, in particular, to methods and apparatus for phase generated carrier (PGC) interferometer interrogation. The present invention also relates to a lithographic apparatus and a projection system for an optical lithographic system including a position measurement system. Background art

[0004] A lithographic apparatus is a machine for applying a desired pattern onto a substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). For example, a lithographic apparatus can project a pattern of a patterning device (e.g., a mask), often also referred to as a "design layout" or "design", onto a layer of radiation-sensitive material (resist) provided on a substrate (e.g., a wafer).

[0005] With the continuous progress of semiconductor manufacturing processes, the dimensions of circuit elements have been continuously reduced, while the number of functional elements (such as transistors) per device has steadily increased over several decades, a trend commonly known as "Moore's Law". To follow Moore's Law, the semiconductor industry is pursuing technologies capable of fabricating smaller features. To project a pattern onto a substrate, a lithographic apparatus can use electromagnetic radiation. The wavelength of this radiation determines the minimum size of the features patterned on the substrate. Currently, typical wavelengths used are 365 nm (i-line), 248 nm, 193 nm, and 13.5 nm. Compared to lithographic apparatuses using radiation with a wavelength of, for example, 193 nm, a lithographic apparatus using extreme ultraviolet (EUV) radiation with a wavelength in the range of 4 nm to 20 nm (e.g., 6.7 nm or 13.5 nm) can be used to form smaller features on a substrate.

[0006] In a lithographic apparatus, it is important to accurately measure the position of certain components of the lithographic apparatus. These components may include one or more wafer (or substrate) tables, a mask table, and / or one or more optical components (e.g., mirrors within the projection system of the lithographic apparatus). For this purpose, interferometers can be used, in particular multi-axis interferometers. A multi-axis interferometer can include a plurality of independent interferometers, one for each axis, where each axis is associated with a position measurement of a different degree of freedom. In an embodiment, the interferometer can include a six-axis interferometer for position measurement of six degrees of freedom: three mutually perpendicular spatial axes (commonly referred to as the x-axis, y-axis, and z-axis), and rotations Rx, Ry, Rz about each of these axes.

[0007] There is a desire to improve such interferometers, in particular multi-axis interferometers. Summary of the Invention

[0008] In a first aspect of the present invention, there is provided a phase-generated carrier interrogator, the phase-generated carrier interrogator comprising: one or more interferometers, each of the one or more interferometers being operable to receive modulated radiation and generate a corresponding interferometer signal, the interferometer signal including a phase of interest caused by the object under test, each interferometer signal further including a periodic phase modulation caused by the modulated radiation, wherein the periodic phase modulation includes an amplitude described by a respective modulation index of the one or more interferometers; at least one detector operable to detect each interferometer signal; and a signal processing module operable to: estimate the modulation index for each of the one or more interferometers separately to obtain a respective estimated modulation index for each of the one or more interferometers; and use the respective estimated modulation index of each interferometer to demodulate each interferometer signal in the interferometer signals to estimate a corresponding phase of interest according to each interferometer signal in the interferometer signals, thereby obtaining a respective estimated value of the phase of interest for each interferometer.

[0009] In a second aspect of the present invention, there is provided a method of processing one or more interferometer signals to determine an estimated value of a phase of interest caused by a target under test, each of the one or more interferometer signals being obtained from a respective interferometer of one or more interferometers; the method comprising: modulating radiation for each of the one or more interferometers to apply a periodic phase modulation having an amplitude described by a respective modulation index of each interferometer; detecting a respective interferometer signal for each of the one or more interferometers, each interferometer signal including, in addition to the periodic phase modulation, a phase of interest caused by the target under test; estimating a modulation index for each of the one or more interferometers separately to obtain a respective estimated modulation index of each of the one or more interferometers; and demodulating each interferometer signal of the interferometer signals using the respective estimated modulation index of each interferometer to estimate a respective phase of interest from each interferometer signal, thereby obtaining a respective estimated value of the phase of interest of each interferometer. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Embodiments of the present invention will now be described, by way of example only, with reference to the accompanying drawings, in which:

[0011] - Figure 1 is a schematic diagram depicting a lithographic apparatus;

[0012] - Figure 2 is a flow chart conceptually showing a signal processing method in PGC interferometry according to a known method;

[0013] - Figure 3 is a flow chart in a first level of abstraction conceptually showing a signal processing method in PGC interferometry according to an embodiment;

[0014] - Figure 4 is conceptually showing Figure 3 a flow chart in a lower second level of abstraction of the signal processing method shown in; and

[0015] - Figure 5 is conceptually showing Figure 3 a flow chart in a lowest third level of abstraction of the signal processing method shown in. DETAILED DESCRIPTION

[0016] In this document, the terms "radiation" and "beam" are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g., having wavelengths of 365 nm, 248 nm, 193 nm, 157 nm, or 126 nm) and EUV (extreme ultraviolet radiation, e.g., having wavelengths in the range of about 5 nm to 100 nm). The interferometers described herein may use radiation having a wavelength greater than 400 nm or greater than 500 nm. More specifically, by way of specific example, the interferometer radiation wavelength may be 633 nm or 1530 nm.

[0017] The terms "reticle", "mask", or "patterning device" as used herein may be broadly understood to refer to a general patterning device that can be used to endow an incident radiation beam with a patterned cross-section corresponding to a pattern to be formed in a target portion of a substrate. The term "light valve" may also be used in this context. Examples of other such patterning devices include programmable mirror arrays and programmable LCD arrays in addition to classical masks (transmission or reflection, binary, phase-shifting, hybrid, etc.).

[0018] Figure 1 A lithographic apparatus LA is schematically depicted. The lithographic apparatus LA includes: an illumination system (also referred to as an illuminator) IL configured to condition a radiation beam B (e.g., ultraviolet radiation, deep ultraviolet radiation, or extreme ultraviolet radiation); a reticle support (e.g., a reticle stage) MT constructed to support a patterning device (e.g., a reticle) MA and connected to a first positioning device PM configured to accurately position the patterning device MA according to certain parameters; a substrate support (e.g., a wafer stage) WT constructed to hold a substrate (e.g., a wafer coated with resist) W and connected to a second positioning device PW configured to accurately position the substrate support according to certain parameters; and a projection system (e.g., a refractive projection lens system) PS configured to project the pattern imparted to the radiation beam B by the patterning device MA onto a target portion C (e.g., including one or more dies) of the substrate W.

[0019] 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 for guiding, shaping, and / or controlling the radiation, such as refractive, reflective, magnetic, electromagnetic, electrostatic, and / or other types of optical components, or any combination thereof. The illuminator IL may be used to condition the radiation beam B to have a desired spatial and angular intensity distribution in a cross-section of the radiation beam B in the plane of the patterning device MA.

[0020] The term "projection system" PS as used herein should be broadly understood to encompass various types of projection systems, including refractive, reflective, catadioptric, anamorphic, magnetic, electromagnetic, and / or electrostatic optical systems, or any combination thereof, as applicable for the exposure radiation used and / or other factors, such as the use of an immersion liquid or the use of a vacuum. The term "projection lens" as used herein may be considered synonymous with the more general term "projection system" PS.

[0021] The lithographic apparatus LA may be of a type in which at least a portion of the substrate is covered by a liquid having a relatively high refractive index (e.g., water), thereby filling the space between the projection system PS and the substrate W - this is also known as immersion lithography. More information on immersion techniques is given in US6952253, which is incorporated herein by reference.

[0022] The lithographic apparatus LA may also be of a type having two or more substrate supports WT (also known as "dual stage"). In such a "multi-stage" machine, the substrate supports WT may be used in parallel, and / or preparatory steps for subsequent exposure of a substrate W may be carried out on a substrate W located on one of the substrate supports WT, while a pattern on another substrate W is being exposed using another substrate W located on another substrate support WT.

[0023] In addition to the substrate support WT, the lithographic apparatus LA may also include a measurement stage. The measurement stage is arranged to hold sensors and / or cleaning devices. The sensors may be arranged to measure properties of the projection system PS or the radiation beam B. The measurement stage may hold multiple sensors. The cleaning device may be arranged to clean a part of the lithographic apparatus, such as a part of the projection system PS or a part of the system providing the immersion liquid. The measurement stage may move under the projection system PS when the substrate support WT is moved away from the projection system PS.

[0024] In operation, the radiation beam B is incident on a patterning device (e.g., a mask) MA, which is held on a mask support MT and patterned by a pattern (design layout) present on the patterning device MA. The radiation beam B passes through the mask MA and then through the projection system PS, which focuses the beam onto a target portion C of the substrate W. By means of a second positioner PW and a position measurement system IF, the substrate support WT can be accurately moved, for example to position different target portions C at the focusing and alignment positions on the path of the radiation beam B. Similarly, a first positioner PM and possibly another position sensor ( Figure 1Another position sensor (not explicitly depicted in the figure) can be used to precisely position the patterning device MA relative to the path of the radiation beam B. Mask alignment marks M1, M2 and substrate alignment marks P1, P2 can be used to align the patterning device MA and the substrate W. Although as shown in the figure, the substrate alignment marks P1, P2 occupy dedicated target portions, the substrate alignment marks P1, P2 can also be located in the space between the target portions. When the substrate alignment marks P1, P2 are located between the target portions C, the substrate alignment marks P1, P2 are referred to as scribe alignment marks.

[0025] To clarify the present invention, a Cartesian coordinate system is used. The Cartesian coordinate system has three axes, namely the x-axis, the y-axis and the z-axis. Each of these three axes is orthogonal to the other two axes. A rotation about the x-axis is referred to as an Rx rotation. A rotation about the y-axis is referred to as an Ry rotation. A rotation about the z-axis is referred to as an Rz rotation. The x-axis and the y-axis define a horizontal plane, while the z-axis is in the vertical direction. The Cartesian coordinate system does not limit the present invention, but is only used to clarify the present invention. Alternatively, other coordinate systems (e.g., cylindrical coordinate systems) can also be used to clarify the present invention. The orientation of the Cartesian coordinate system can be different, for example such that the z-axis has a component along the horizontal plane.

[0026] The position measurement system PMS can include any type of sensor suitable for determining the position of the substrate support WT. The position measurement system PMS can include any type of sensor suitable for determining the position of the mask support MT. The sensor can be an optical sensor, such as an interferometer or an encoder. The position measurement system PMS can include a combined system of an interferometer and an encoder. The sensor can be another type of sensor, such as a magnetic sensor, a capacitive sensor or an inductive sensor. The position measurement system PMS can determine the position relative to a reference (e.g., a metrology frame MF or a projection system PS). The position measurement system PMS can determine the position of the substrate table WT and / or the mask support MT by measuring the position or the time derivative of the position (such as velocity or acceleration).

[0027] The position measurement system PMS may include an encoder system. For example, an encoder system can be learned from U.S. Patent Application US2007 / 0058173A1 filed on September 7, 2006, which is incorporated herein by reference. The encoder system includes an encoder head, a grating, and a sensor. The encoder system can receive a main radiation beam and a secondary radiation beam. Both the main radiation beam and the secondary radiation beam originate from the same radiation beam, i.e., the original radiation beam. At least one of the main radiation beam and the secondary radiation beam is generated by diffracting the original radiation beam using the grating. If both the main radiation beam and the secondary radiation beam are generated by diffracting the original radiation beam using the grating, the main radiation beam needs to have a different diffraction order from the secondary radiation beam. Different diffraction orders are, for example, the +1 order, -1 order, +2 order, and -2 order. The encoder system optically combines the main radiation beam and the secondary radiation beam into a combined radiation beam. The sensor in the encoder head determines the phase or phase difference of the combined radiation beam. The sensor generates a signal based on the phase or phase difference. The signal represents the position of the encoder head relative to the grating. One of the encoder head and the grating can be arranged on the substrate structure WT. The other of the encoder head and the grating can be arranged on the metrology frame MF or the base frame BF. For example, multiple encoder heads are arranged on the metrology frame MF, and the grating is arranged on the top surface of the substrate support WT. In another example, the grating is arranged on the bottom surface of the substrate support WT, and the encoder head is arranged below the substrate support WT.

[0028] The position measurement system PMS may include an interferometer system. The interferometer system can be learned, for example, from U.S. Patent No. 6,020,964 filed on July 13, 1998, which is incorporated herein by reference. The interferometer system may include a beam splitter, a mirror, a reference mirror, and a sensor. The radiation beam is split by the beam splitter into a reference beam and a measurement beam. The measurement beam propagates to the mirror and is reflected back to the beam splitter by the mirror. The reference beam propagates to the reference mirror and is reflected back to the beam splitter by the reference mirror. At the beam splitter, the measurement beam and the reference beam are combined into a combined radiation beam. The combined radiation beam is incident on the sensor. The sensor determines the phase or frequency of the combined radiation beam. The sensor generates a signal based on the phase or frequency. The signal represents the displacement of the mirror. In an embodiment, the mirror is connected to the substrate support WT. The reference mirror can be connected to the metrology frame MF. In an embodiment, the measurement beam and the reference beam are combined into a combined radiation beam by additional optical components instead of the beam splitter.

[0029] The interferometry method for the position measurement system proposed can use the phase generated carrier (PGC) interferometer interrogation technique. Compared with the traditional homodyne or heterodyne interrogation technique, the PGC interferometer requires less expensive interferometer hardware but more complex signal processing.

[0030] A radiation source or a laser source (such as a wavelength-modulated laser diode) can be used in combination with an unbalanced interferometer (having a non-zero optical path difference OPD), which, in addition to the phase of interest caused by the object under test, also generates a periodic or sinusoidal phase modulation of the interferometer phase. Outside of this, a periodic phase modulation or a sinusoidal modulation of the interferometer phase is generated. The modulation index of this modulation can be described by where is the laser frequency offset amplitude and c is the speed of light.

[0031] Wavelength modulation can be achieved in a number of different ways. Two alternative methods will be described herein: 1) The emission wavelength of the laser can be changed (this is the least costly option); or 2) The laser can operate at a constant emission frequency, and a periodic wavelength modulation can be generated by an additional (electro-optic) phase modulator component located downstream of the laser and shared by all interferometers. Since the latter method avoids modulating the output power of the laser, which is a source of inaccuracy, the latter method has an advantage over the first method.

[0032] Figure 2 is a flow chart conceptually showing the signal processing in PGC interferometry. The multi-carrier synthesizer generates a sinusoidal modulation for the laser diode LD of the interferometer This sinusoidal modulation is combined with the modulation amplitude or the current excitation amplitude (for example, multiplied by this modulation amplitude or current excitation amplitude ), and further combined with the bias current BC (for example, added to the bias current BC). The laser current excitation amplitude is related to the laser frequency offset amplitude by the following relationship:

[0033]

[0034] where η is the current-to-frequency sensitivity of the laser.

[0035] The optical power (interferometer signal) detected by the photodetector PD of the interferometer can then be written as:

[0036]

[0037] where B is the background power and A is the amplitude of the power of the interference fringes. The ratio A / B is the interference contrast and satisfies . Hereinafter, the phase modulation amplitude m is referred to as the modulation index.

[0038] This sinusoidal phase modulation is used to generate a series of demodulated carrier frequencies or phase-generated carrier frequencies in the detected output power of an interferometer as harmonics of the modulation frequency. These phase-generated carriers can take the following forms:

[0039]

[0040] where k represents an integer between 1 and infinity to define the order n.

[0041] The amplitude of the odd-order carrier frequencies (carriers) is proportional to the sine of the phase caused by the object under test, and the amplitude of the even-order carrier frequencies (carriers) is proportional to the cosine of the phase caused by the object under test. Each nth-order carrier is also attenuated by a modulation-index-dependent factor which is the nth-order Bessel function of the first kind (where is the DC component of the Fourier series described by the term in parentheses). Thus, the optical power S(t) can be described as:

[0042] .

[0043] The phase caused by the object under test that varies with time (and its (multiple) time derivatives) cannot be directly obtained. Instead, the phase caused by the object under test (the phase of interest) can be estimated by a dynamic state estimator SE, which is also referred to as a process model (e.g., the process model models the dynamic state of the object under test; for example, this can be the instantaneous position of an object such as a stage). This estimation process can calculate a phase prediction (the estimated value of the phase of interest), and the phase prediction can be compared with the true phase caused by the object under test included in the signal . According to the phase residual of this comparison, it can be used to correct the dynamic state predicted by the process model SE. Thus, the dynamic state estimator can determine a new state, which includes the estimated phase prediction according to the previous state and the phase residual . The corrected state is the output of the system. The resulting estimation loop is actually a (digital) phase-locked loop (PLL), and this phase-locked loop is in turn a Kalman filter.

[0044] By appropriately processing the signal (e.g., within a signal processing module), a complex phasor (the first phasor) can be constructed, which has a desired phase residual As its argument. The desired signal processing involves three conceptual steps. In the first step, the interferometer signal is multiplied by a complex conjugate phasor of unit magnitude , which has an argument that is the negative phase prediction . This phasor can be constructed using a first look-up table or a sine / cosine look-up table LUT1. In the second step, for each carrier to be considered (i.e., at least one even-order carrier and at least one odd-order carrier), the corresponding demodulation component or demodulation result is separately formed. Synchronous demodulation involves multiplying by the corresponding carrier 、- and then performing low-pass filtering LPF. These two steps together produce the following even-order demodulation result and odd-order demodulation result:

[0045]

[0046] where, V-shaped represents time averaging or low-pass filtering. Each demodulation includes two low-pass filtering phasors: a first phasor including the desired argument and an additional phasor having an argument . In the case where the process model closely tracks the desired phase , the phase residual will be very small, and the corresponding first phasor will remain close to or on the real axis. However, the additional phasor will rotate at a rate that is twice the rate of change of the phase caused by the object under test, which may be large. Therefore, depending on the rate of change (e.g., speed), this additional phasor will experience attenuation due to the low-pass filtering effect, while the desired first phasor will not.

[0047] In the third processing step, the even-order and odd-order demodulation results are combined by appropriately weighted summation in order to eliminate the additional phasor. In the simplest case, i.e., one even-order demodulation result is combined with one odd-order demodulation result, the appropriate weighting coefficient is the reciprocal Bessel function:

[0048] .

[0049] Note that the digital low-pass filter LPF acting on two of these demodulation results is the same, and thus the second phasors of each of these demodulation results will cancel each other out.

[0050] Assume that the estimate is accurate such that the phase residual Approaching zero, the above three processing steps generate a first phasor of the phase residual , the first phasor of the phase residual lies on or near the real axis. Now, the magnitude and phase residual can be separated by inputting this phasor into a Cartesian-to-polar transformation unit CtP, which can be conveniently implemented in an FPGA (or software) as, for example, a CORDIC algorithm. For convenience of counting, since the phase residual is quasi-static, the time dependence of the phase residual will now be reduced.

[0051] As can be seen from the above description, the Bessel function applied in the above third processing step depends on the modulation index m. As already mentioned, the modulation index m is determined by the product of the optical frequency (or wavelength) offset amplitude and the optical path difference OPD; that is, according to:

[0052]

[0053] where c is the speed of light.

[0054] Most PGC interrogators adopt the following method: where the modulation index m is controlled by adjusting the optical frequency offset amplitude (for example, by the laser current amplitude ), so that the modulation index m moves towards a convenient operating point. A convenient operating point can be, for example, a point where the Bessel function corresponding to a specific carrier becomes zero, while the amplitudes of the odd / even pairs of other carriers reach near the maximum intensity. At such a predetermined operating point, the ratio of the amplitudes of the members of the odd / even pair is known a priori (i.e., the ratio of the corresponding Bessel functions). This a priori knowledge can be used to equalize these amplitudes and construct a phasor A with a constant magnitude.

[0055] This method is well applicable to a single interferometer and laser arrangement, or well applicable to the case where each interferometer of the interrogator has a corresponding dedicated laser. For example, by individually adjusting each laser and the optical frequency offset amplitude of each interferometer, the modulation index of each interferometer can be individually controlled for the corresponding dedicated laser. However, for a multi-axis interrogator such as may be used as part of a position measurement system within a lithography apparatus, this may be impractical in terms of cost and / or volume. Since each axis essentially includes a corresponding different interferometer, implementing this method may require each axis to be equipped with a laser; for example, for the six axes of a six-degree-of-freedom position measurement system, six lasers are required, with each axis corresponding to one laser.

[0056] To reduce cost and / or volume, it is desirable to use a single laser (or at least fewer lasers than interferometers) for each interferometer, so that a multi-axis interferometric device (e.g., a multi-axis PGC interferometer interrogator) can be implemented using only a single laser. When one laser is used for multiple interferometers, the existing methods of controlling the modulation index to achieve the above signal processing will no longer be feasible. This is because the OPD of each interferometer will typically be different, and thus for a single optical frequency offset amplitude setting (laser current amplitude setting), the corresponding modulation indices will also be different. This makes it impossible to construct phasors with constant magnitudes for all axes simultaneously. For small deviations in the modulation index, this will translate into periodic errors, i.e., a decrease in measurement accuracy. For larger deviations, singularities, loss of lock, and / or sign reversals of the motion of the object under test may occur.

[0057] This document discloses a method in which the demodulation of a measurement signal is performed using an estimated demodulation index value based on the relative amplitudes of N≥2 even-order carrier frequencies and the relative amplitudes of N≥2 odd-order carrier frequencies. Although this method can be used for a single interferometer as an alternative to the existing methods of controlling the modulation index to a set value, the main application will be to facilitate the use of a single laser for multiple interferometers (e.g., as part of a multi-axis interrogator). This method will enable the construction of phasors with constant magnitudes for multiple interferometers and thus for multiple measurement axes when only a single laser source is used for multiple interferometers.

[0058] The basic concept is to allow each measurement axis to have a corresponding modulation index that is allowed to vary individually over time as the optical path difference (OPD) of its axis changes (due to changes in the object under test), rather than controlling the modulation index to a predetermined value. Then, each of these modulation indices is estimated, and the estimated values for each axis are used to demodulate the corresponding measurement signals from that axis.

[0059] Accordingly, a phase generated carrier interrogator and an associated signal processing method are disclosed. The phase generated carrier interrogator includes: one or more interferometers, each of the one or more interferometers being operable to receive (e.g., frequency) modulated radiation and generate a corresponding interferometer signal, the interferometer signal including a phase of interest caused by the object under test, each of the interferometer signals further including a periodic phase modulation caused by the modulated radiation, wherein the periodic phase modulation includes an amplitude described by a corresponding modulation index of each of the one or more interferometers; at least one detector operable to detect each of the interferometer signals; and a signal processing module operable to: respectively estimate the modulation index for each of the one or more interferometers to obtain a corresponding estimated modulation index for each of the one or more interferometers; and use the corresponding estimated modulation index of each interferometer to demodulate each of the interferometer signals in the interferometer signals to estimate a corresponding phase of interest according to each of the interferometer signals in the interferometer signals, thereby obtaining a corresponding estimated value of the phase of interest for each of the interferometers.

[0060] Figure 3 is a flow chart at a higher level of abstraction showing such a method for a single axis. It is recommended to independently estimate the modulation index m of each axis using the relative amplitudes of N≥2 even-order demodulated carriers and the relative amplitudes of N≥2 odd-order demodulated carriers. If the features of this method are the same as the features of the Figure 2 shown method, then there is no need to repeat the description of these features. Then, the estimated modulation index represented as can be used to define a first set of weighting coefficients (e.g., via a first weighted coefficient look-up table LUT2) , hoping that the first set of weighting coefficients can equalize the corresponding carrier amplitudes, thereby obtaining a first phasor with a constant amplitude . Through the Cartesian to polar coordinate transformation unit CtP, the phase residual can be extracted from this constant-amplitude phasor without periodic error , and the phase residual can be used to correct the dynamic state predicted by the process model SE.

[0061] In an embodiment, the first weighted combination or sum of demodulation may include:

[0062]

[0063] wherein each demodulation component or demodulation result includes each corresponding demodulated carrier of interest and the measured interferometer signal and a unit amplitude complex conjugate phasor The low - pass filtered product, as described previously. In this embodiment, now (at least) four demodulation components or demodulation results will be required, namely, two even demodulation results and two odd demodulation results, instead of two demodulation results (i.e., N = 2). Although more than four demodulation results can be used, the cost is the complexity of implementation.

[0064] In order for this weighted addition to result in the desired complex phasor , the following three conditions should be satisfied:

[0065]

[0066] The first condition stipulates that the modulation index estimate should be accurate. How to achieve this will be described later. The other two conditions impose a single constraint on the set of first - order weighted coefficients of even order and the set of first - order weighted coefficients of odd order. Assuming there are N≥2 coefficients in each set, this provides N - 1≥1 degrees of freedom for choosing coefficients in each of the two sets. This degree of freedom can be used to minimize the noise amplification caused by the weighted addition. The following will give an example for the case where the noise spectral density of the signal is white noise. In this case, the demodulation results include independent additive noise with equal variances. This means that the root - mean - square (rms) noise in the weighted sum can be calculated as the rms noise in any one of the demodulation results, and multiplied by the corresponding factor; for example:

[0067] ,

[0068] These factors can be identified as the lengths of the corresponding vectors:

[0069] .

[0070] Therefore, for the minimum vectors and the minimum noise amplification is achieved and the above - mentioned constraint conditions are satisfied. These constraint conditions can in turn be interpreted as the dot products of the corresponding vectors of Bessel functions of even order and Bessel functions of odd order:

[0071]

[0072] Therefore, in searching for the minimum vectors and , the minimum vectors and will form unit dot products with the corresponding vectors and In other words, the vectors and should be chosen to be orthogonal to the vectors and are parallel and should be normalized to their squared lengths:

[0073] .

[0074] Thus, the noise-optimal coefficients can be found by and :

[0075] .

[0076] These weighting coefficients can be pre-computed for an expected range of the estimated modulation index and stored in a look-up table LUT2 (e.g., stored in an FPGA embodying the algorithm of the proposed method). If it is known that the noise in the detected signal has a specific non-white spectral density, the calculation of the above coefficients can be modified by first applying a noise whitening virtual gain to each demodulation result in the demodulation result. The calculation of the G coefficient after applying the noise whitening virtual gain to each demodulation result in the demodulation result is conceptually the same. Then, the stored coefficient is determined as the product of the corresponding noise whitening gain and the corresponding G coefficient, and the corresponding G coefficient is calculated after applying the noise whitening gain to the demodulation result.

[0077] Figure 4 The proposed method at the second level of abstraction is described. Returning to the first condition for accurate phasor construction, i.e., the modulation estimation should be accurate , it is proposed that this can be achieved by iteratively updating (refining) the modulation index estimate . This iterative update of the modulation index estimate can be based on the modulation index residual , which approximates the true error and is composed of a second weighted sum of the demodulation results . This is inspired by the concept that both the ratio of different even-order Bessel functions and the ratio of different odd-order Bessel functions contain information about the modulation index . Given a reasonable initial estimate (e.g., according to existing knowledge), the residual can be constructed based on the known (first-order) derivative of the Bessel function at the estimate . The inventors have determined that the next second weighted combination or weighted summation produces a phasor (second phasor) that is linearly proportional to the modulation index residual ,

[0078] .

[0079] By determining a second set of weighting coefficients that satisfy the equation , the modulation index residual can be extracted by multiplying by the complex conjugate of the squared amplitude normalized to the position residual phasor (including its sign). The construction of the position residual phasor has been described as:

[0080]

[0081] By substituting the expressions for demodulation, the H-weighted addition can be rewritten as:

[0082]

[0083] Using the estimate to approximate the true but inaccessible modulation index m, and according to the estimation error of satisfies:

[0084]

[0085] Then the Bessel function can be linearized using the first derivative around :

[0086]

[0087]

[0088] where the superscript symbol denotes the derivative.

[0089] Since this expression must satisfy the range of values of , and any value of the phase residual , this expression decomposes into the following set of four requirements:

[0090]

[0091] In a manner similar to the G coefficients, two H vectors can be defined according to:

[0092]

[0093] And a set of four equations can be written as two sets of independent dot product equations:

[0094]

[0095] ​For the N = 2 embodiment, then the two equations in each group completely define the two components of the corresponding H vector. The first equation of each set defines the (2D) direction of the corresponding H vector as perpendicular to the corresponding J vector. The second equation sets the length (and sign) of the H vector. Thus, in the case of N = 2, the individual coefficients can be calculated according to the following formula:

[0096]

[0097] The Bessel function derivative can be expressed in terms of higher-order or lower-order Bessel functions using a recurrence relation:

[0098] ,

[0099] This gives the final expression:

[0100]

[0101] These H coefficients can be pre-computed for the expected range of and stored in a lookup table (e.g., stored in a second weighted coefficient lookup table, such as implemented in an FPGA including the algorithm embodying the proposed method). This second weighted coefficient lookup table can be the same table as the first weighted coefficient lookup table, such as LUT2, etc. The above proves that it is indeed possible to calculate H coefficients that satisfy all the constraints. Thus, the modulation index residual can be constructed over a range of modulation indices . However, it can be understood that there may be some limitations to this range. For both lower and higher modulation indices, one or more of the H coefficients (and even the G coefficients) may increase too much, such that this results in extreme noise amplification; these cases should be avoided. By considering higher-order PGC (i.e., N > 2), the available range can be extended on the higher modulation index side. However, the cost is a higher degree of FPGA resource usage and higher requirements for analog bandwidth and sampling rate. When N > 2, the H vector includes more components, which provides freedom for noise optimization, as has already been demonstrated by the G coefficients.

[0102] Then, a scaled-down version of the signed modulation index residual can be used to update the modulation index estimate. This update improves the accuracy of the modulation index estimate and ensures that the modulation index estimate will track the actual modulation index The change over time is caused by an OPD that changes over time. This can be done separately for different interferometers (e.g., different axes of a multi-axis interrogator), such that the respective modulation indices of multiple measurement axes can be tracked simultaneously and independently. This allows these axes to perform simultaneous measurements using a single laser diode without periodic errors.

[0103] The proposed modulation index tracking actually does not require a reasonable estimate of the phase caused by the object under test. Therefore, modulation index tracking can be started before "locking" the main measurement value. This greatly simplifies the initialization procedure required for the interrogator to operate precisely.

[0104] Modulation index and phase caused by the object under test is proportional, where the proportionality factor is the wavelength offset amplitude of the laser radiation with respect to the center wavelength of the (quasi-static) ratio (i.e., :

[0105]

[0106] Therefore, the modulation index estimator can be implemented as a second dynamic state estimator (not shown in the figure) similar to the dynamic state estimator SE of the process model, so as to be able to accurately track the OPD changes caused by the object under test at higher speeds / accelerations / acceleration change rates. This means that the method uses two dynamic state estimators, which track substantially the same (but differently scaled) OPD changes and both have similar bandwidths. This method is within the scope of the present disclosure, and thus the method can include using a (second) dynamic state estimator to determine the modulation index estimate , to determine the G coefficient and the H coefficient.

[0107] Figure 5 Shows the proposed alternative method at a lower level of abstraction, where the modulation index estimate is predicted based on the phase estimate caused by the object under test output from the dynamic state estimator SE or the process model (e.g., the phase estimate indicates the instantaneous position of the component being tracked, at which the interrogator is part of a position measurement system). Since this requires much less repetitive work, it is more efficient than providing a second dynamic state estimator.

[0108] This embodiment includes defining two additional estimation variables: one estimation variable is the aforementioned ratio , denoted as , and the other estimation variable is the estimated phase caused by the object under test Static “fringe count error” (FCE) in The estimated values of these two estimated variables are essentially quasi-static variables. Compared with the rapidly changing modulation index, estimating these quasi-static variables can be achieved with a lower bandwidth, so the noise averaging effect is better. Predicted modulation index And the phase estimate caused by the object under test The relationship between them can be defined as:

[0109]

[0110] According to the definition, the estimated values of the wavelength ratios of all axes of the multi-axis system Are the same. The estimated value of the wavelength ratio Can be preset, calibrated, or continuously estimated. According to the following first-order update equation, the estimated value of the fringe count error of each axis Can be corrected by the modulation index residual As follows:

[0111]

[0112] Where the coefficient Is the bandwidth parameter, and l is the sampling instant index.

[0113] The proposed signal processing method (e.g., the algorithm and associated look-up table embodying the method) can be executed in software or firmware (e.g., in a programmable firmware component such as a field programmable gate array FPGA).

[0114] Also disclosed is a phase generated carrier interferometer interrogator including at least one interferometer, which can operate to perform any of the methods disclosed herein to process (e.g., demodulate) the detected interferometer signal. The phase generated carrier interferometer interrogator can include, for example, a multi-interferometer or multi-axis phase generated carrier interferometer interrogator that uses a common laser (e.g., a laser diode) for each interferometer / axis.

[0115] There is also disclosed a position measurement system for measuring the position of a component of a machine, the machine being a lithographic apparatus such as a scanner including at least one interferometer, and operable to perform any of the methods disclosed herein for processing (e.g., demodulating) detected interferometer signals. The position measurement system may include a (e.g., multi-axis) phase generated carrier interferometer interrogator as described above. The component may include, for example, a stage of a lithographic apparatus (e.g., a wafer stage or a mask stage), or any other component, such as a projection component (e.g., a steering mirror or a beam delivery mirror) of a projection system of a lithographic apparatus (e.g., an EUV lithographic apparatus). The position measurement system may be operable to measure any two or more axes, each of the two or more axes being associated with a respective degree of freedom of the component being measured. In a specific example, the position measurement system may be operable to measure six axes, namely three spatial axes x, y, z and rotations Rx, Ry, Rz about each of these axes.

[0116] There is also disclosed a lithographic apparatus including at least one such position measurement system for measuring the position of a component of the lithographic apparatus.

[0117] Although specific reference may be made herein to the use of a lithographic apparatus in the manufacture of integrated circuits, it should be understood that the lithographic apparatus described herein may have other applications. Other possible applications include the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat panel displays, liquid crystal displays (LCDs), thin film magnetic heads, and the like.

[0118] Although embodiments of the invention may be specifically referred to herein in the context of a lithographic apparatus, embodiments of the invention may also be used in other apparatuses. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus for measuring or processing an object such as a wafer (or other substrate) or a mask (or other patterning device). Such apparatuses may generally be referred to as lithographic tools. Such lithographic tools may use vacuum conditions or ambient (non-vacuum) conditions.

[0119] Although embodiments of the invention may have been specifically referred to above in the context of optical lithography, it will be appreciated that, where the context allows, the invention is not limited to optical lithography techniques and may be used in other applications, such as imprint lithography.

[0120] Where context permits, embodiments of the present invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the present 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, acoustic, or other forms of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.). Additionally, firmware, software, routines, instructions may be described herein as performing certain operations. However, it should be understood that such descriptions are for convenience only and that such actions are actually performed by computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc., and may cause actuators or other devices to interact with the physical world when performing such actions.

[0121] Although specific embodiments of the present invention have been described above, it will be understood that the present invention may be implemented in ways different from those described. The above description is intended to be illustrative, not restrictive. Thus, those skilled in the art will understand that the present invention as described may be modified without departing from the scope of the claims set forth below. Other aspects of the present invention are described in the numbered aspects below.

[0122] 1. A phase-generated carrier interrogator, comprising:

[0123] One or more interferometers, each of the one or more interferometers being operable to receive modulated radiation and generate a corresponding interferometer signal, the interferometer signal including a phase of interest caused by the object under test, each of the interferometer signals further including a periodic phase modulation caused by the modulated radiation, wherein the periodic phase modulation includes an amplitude described by a respective modulation index of each of the one or more interferometers;

[0124] At least one detector, the at least one detector being operable to detect each of the interferometer signals; and

[0125] A signal processing module, the signal processing module being operable to:

[0126] Estimate the modulation index for each of the one or more interferometers separately to obtain a respective estimated modulation index for each of the one or more interferometers; and

[0127] Demodulate each interferometer signal in the interferometer signals using a respective estimated modulation index of each of the interferometers to estimate a respective phase of interest based on each interferometer signal in the interferometer signals, thereby obtaining a respective phase of interest estimate for each of the interferometers.

[0128] 2. The phase generation carrier interrogator according to aspect 1, wherein the signal processing module is operable to determine, for each of the one or more interferometers, a respective demodulation component of each of at least four demodulation carrier frequencies, the at least four demodulation carrier frequencies including at least two even-order demodulation carrier frequencies and at least two odd-order demodulation carrier frequencies, the demodulation carrier frequencies including harmonics of a modulation frequency of the periodic phase modulation.

[0129] 3. The phase generation carrier interrogator according to aspect 2, wherein the signal processing module is operable to determine, for each of the one or more interferometers, each demodulation component as a combination of: each respective demodulation carrier frequency, the interferometer signal, and a unit magnitude complex conjugate phasor having an argument that is a negative phase of interest estimate.

[0130] 4. The phase generation carrier interrogator according to aspect 3, wherein the signal processing module is operable to determine the unit magnitude complex conjugate phasor based on the phase of interest estimate.

[0131] 5. The phase generation carrier interrogator according to any one of aspects 2 to 4, wherein the signal processing module is operable to, for each of the one or more interferometers, use a respective estimated modulation index of the interferometer to define a first set of weighting coefficients for the demodulation components, the first weighting coefficients being defined such that a first weighted combination of the demodulation components weighted by the first weighting coefficients produces a first phasor having an argument that includes a phase residual describing an error in the phase of interest estimate.

[0132] 6. The phase generation carrier interrogator according to aspect 5, wherein the signal processing module is further operable to determine the phase residual based on the first phasor.

[0133] 7. The phase generation carrier interrogator according to aspect 6, wherein the signal processing module further includes a dynamic state estimator operable to determine a new state, the new state including the phase of interest estimate from a previous state and from the phase residual.

[0134] 8. A phase-generated carrier interrogator according to aspect 5, 6, or 7, wherein the signal processing module includes a first weighting coefficient look-up table, and the first set of weighting coefficients is selected based on the first weighting coefficient look-up table according to each respective estimated modulation index.

[0135] 9. A phase-generated carrier interrogator according to any one of aspects 5 to 8, wherein the first set of weighting coefficients includes respective even-order first weighting coefficient vectors for weighting each of the at least two even-order demodulation components, and respective odd-order first weighting coefficient vectors for weighting each of the at least two odd-order demodulation components, and

[0136] wherein the signal processing module is operable to, for each of the one or more interferometers:

[0137] determine each of the even-order first weighting coefficient vectors to be parallel to a respective normalized vector of an even-order Bessel function of the same order and corresponding to the estimated modulation index; and

[0138] determine each of the odd-order first weighting coefficient vectors to be parallel to a respective normalized vector of an odd-order Bessel function of the same order and corresponding to the estimated modulation index.

[0139] 10. A phase-generated carrier interrogator according to any one of aspects 5 to 9, wherein the signal processing module is operable to estimate each modulation index based on the relative amplitudes of the two even-order demodulation components and at least two odd-order carrier demodulation components.

[0140] 11. A phase-generated carrier interrogator according to any one of aspects 5 to 10, wherein the signal processing module is operable to estimate the modulation index for each of the one or more interferometers separately by iteratively updating the modulation index estimate based on the modulation index residual.

[0141] 12. A phase-generated carrier interrogator according to aspect 11, wherein the signal processing module is operable to estimate the modulation index residual based on a second weighted combination of the demodulation components weighted by a second set of weighting coefficients to produce a second phasor that is linearly proportional to the modulation index residual.

[0142] 13. A phase-generated carrier interrogator according to aspect 12, wherein the signal processing module is operable to extract the modulation index residual by combining the second phasor with the normalized conjugate of the first phasor.

[0143] 14. The phase-generated carrier interrogator according to aspect 12 or 13, wherein the second set of weighting coefficients includes respective even-order second weighting coefficient vectors for weighting each of the at least two even-order demodulation components, and respective odd-order second weighting coefficient vectors for weighting each of the at least two odd-order demodulation components; and wherein the second set of weighting coefficients is defined such that:

[0144] The sum of the dot products of each of the even-order second weighting coefficient vectors and the corresponding even-order Bessel function vectors of the estimated modulation index is zero;

[0145] The sum of the dot products of each of the odd-order second weighting coefficient vectors and the corresponding odd-order Bessel function vectors of the estimated modulation index is zero;

[0146] The sum of the dot products of each of the even-order second weighting coefficient vectors and the derivatives of the corresponding even-order Bessel function vectors of the estimated modulation index is -1; and

[0147] The sum of the dot products of each of the odd-order second weighting coefficient vectors and the derivatives of the corresponding odd-order Bessel function vectors of the estimated modulation index is -1.

[0148] 15. The phase-generated carrier interrogator according to aspect 12, 13 or 14, wherein the signal processing module includes a second weighting coefficient look-up table, and the second set of weighting coefficients is selected based on each respective estimated modulation index according to the second weighting coefficient look-up table.

[0149] 16. The phase-generated carrier interrogator according to any one of aspects 11 to 15, wherein the signal processing module is operable to determine the estimated modulation index according to an updated fringe count error estimate among the phase estimates of interest, the updated fringe count error estimate having been updated by the modulation index residual.

[0150] 17. The phase-generated carrier interrogator according to aspect 16, wherein the signal processing module is operable to determine the estimated modulation index according to the updated fringe count error estimate and an estimated ratio of the wavelength offset amplitude to the center wavelength of the modulated radiation.

[0151] 18. The phase-generated carrier interrogator according to any of the preceding aspects, the phase-generated carrier interrogator including a programmable firmware component programmed to implement the signal processing module.

[0152] 19. A phase-generated carrier interrogator according to any of the foregoing aspects, the phase-generated carrier interrogator further comprising a plurality of said interferometers.

[0153] 20. A phase-generated carrier interrogator according to aspect 19, the phase-generated carrier interrogator further comprising a common radiation source for generating radiation to be modulated to obtain the modulated radiation for each of the plurality of said interferometers.

[0154] 21. A multi-axis position measurement system comprising a phase-generated carrier interrogator according to aspect 19 or 20, wherein each of the plurality of interferometers is for measuring a respective axis of the multi-axis position measurement system.

[0155] 22. A multi-axis position measurement system according to aspect 21, wherein the multi-axis position measurement system comprises at least three axes for measuring three rigid body position coordinates related to the object under test.

[0156] 23. A multi-axis position measurement system according to aspect 21, wherein the multi-axis position measurement system comprises at least six axes for measuring three rigid body position coordinates and three rigid body attitude coordinates related to the object under test.

[0157] 24. A lithographic apparatus comprising a multi-axis position measurement system according to any one of aspects 21 to 23, wherein the multi-axis position measurement system is operable to measure the position of a component of the lithographic apparatus.

[0158] 25. A projection system for an optical lithography system, comprising a multi-axis position measurement system according to any one of aspects 21 to 23, wherein the multi-axis position measurement system is operable to measure the position of a component of the projection system.

[0159] 26. A method of processing one or more interferometer signals to determine an estimated phase of interest caused by an object under test, each of the one or more interferometer signals being obtained from a respective interferometer of one or more interferometers; the method comprising:

[0160] modulating radiation for each of the one or more interferometers to apply a periodic phase modulation having an amplitude described by a respective modulation index of each interferometer;

[0161] detecting a respective interferometer signal for each of the one or more interferometers, each interferometer signal comprising, in addition to the periodic phase modulation, a phase of interest caused by the object under test;

[0162] Estimate the modulation index for each of the one or more interferometers separately to obtain a corresponding estimated modulation index for each of the one or more interferometers; and

[0163] Demodulate each interferometer signal in the interferometer signals using the corresponding estimated modulation index of each interferometer to estimate a corresponding phase of interest according to each interferometer signal in the interferometer signals, thereby obtaining a corresponding estimated value of the phase of interest for each interferometer.

[0164] 27. The method according to aspect 26, including, for each of the one or more interferometers: determining a corresponding demodulation component for each of at least four demodulation carrier frequencies, the at least four demodulation carrier frequencies including at least two even-order demodulation carrier frequencies and at least two odd-order carrier demodulation frequencies, the demodulation carrier frequencies including harmonics of the modulation frequency of the periodic phase modulation.

[0165] 28. The method according to aspect 27, including, for each of the one or more interferometers: determining each demodulation component as a combination of: each corresponding demodulation carrier frequency, the interferometer signal, and a unit magnitude complex conjugate phasor having an argument that is the negative of the estimated value of the phase of interest.

[0166] 29. The method according to aspect 28, including determining the unit magnitude complex conjugate phasor according to the estimated value of the phase of interest.

[0167] 30. The method according to any one of aspects 27 to 29, including, for each of the one or more interferometers: using the corresponding estimated modulation index of the interferometer to define a first set of weighting coefficients for the demodulation components, the first weighting coefficients being defined such that a first weighted combination of the demodulation components weighted by the first weighting coefficients produces a first phasor having an argument that includes a phase residual describing an error in the estimated value of the phase of interest.

[0168] 31. The method according to aspect 28, wherein the signal processing module is further operable to determine the phase residual according to the first phasor; and to determine the estimated value of the phase of interest according to the phase residual.

[0169] 32. The method according to aspect 30 or 31, including selecting the first set of weighting coefficients based on each corresponding estimated modulation index according to the first weighting coefficient look-up table.

[0170] 33. The method according to any one of aspects 30 to 32, wherein the first set of weighting coefficients includes respective first even-order weighting coefficients for weighting each of the at least two even-order demodulation components, and respective first odd-order weighting coefficients for weighting each of the at least two odd-order demodulation components, and the method further includes, for each of the one or more interferometers:

[0171] Determining each of the first even-order weighting coefficient vectors to be parallel to a respective normalized vector of an even-order Bessel function of the same order and corresponding to the estimated modulation index; and

[0172] Determining each of the first odd-order weighting coefficient vectors to be parallel to a respective normalized vector of an odd-order Bessel function of the same order and corresponding to the estimated modulation index.

[0173] 34. The method according to any one of aspects 30 to 33, including estimating each modulation index based on the relative amplitudes of the two even-order demodulation components and at least two odd-order carrier demodulation components.

[0174] 35. The method according to any one of aspects 30 to 34, including separately estimating the modulation index for each of the one or more interferometers by iteratively updating the modulation index estimate based on the modulation index residual.

[0175] 36. The method according to aspect 35, including estimating the modulation index residual based on a second weighted combination of the demodulation components weighted by a second set of weighting coefficients to produce a second phasor that is linearly proportional to the modulation index residual.

[0176] 37. The method according to aspect 36, including extracting the modulation index residual by combining the second phasor with the normalized conjugate of the first phasor.

[0177] 38. The method according to aspect 36 or 37, wherein the second set of weighting coefficients includes respective second even-order weighting coefficients for weighting each of the at least two even-order demodulation components, and respective second odd-order weighting coefficients for weighting each of the at least two odd-order demodulation components; and wherein the second set of weighting coefficients is defined such that:

[0178] The dot product of the vectors of all the second even-order weighting coefficients and the vectors of the corresponding even-order Bessel functions is zero;

[0179] The dot product of all the vectors of the odd-order second weighting coefficients and the vectors of the corresponding odd-order Bessel functions is zero;

[0180] The dot product of all the vectors of the even-order second weighting coefficients and the derivatives of the vectors of the corresponding even-order Bessel functions is -1; and

[0181] The dot product of all the vectors of the odd-order second weighting coefficients and the derivatives of the vectors of the corresponding odd-order Bessel functions is -1.

[0182] 39. The method according to aspect 36, 37 or 38, including selecting the set of second weighting coefficients based on each corresponding estimated modulation index according to a second weighting coefficient look-up table.

[0183] 40. The method according to any one of aspects 35 to 39, including determining the estimated modulation index according to an updated fringe count error estimate value among the phase estimates of interest, the updated fringe count error estimate value having been updated by the modulation index residual.

[0184] 41. The method according to aspect 40, including determining the estimated modulation index according to the updated fringe count error estimate value and an estimated ratio of the wavelength offset amplitude to the center wavelength of the modulated radiation.

[0185] 42. The method according to any one of aspects 26 to 41, wherein the method includes processing corresponding interferometer signals from each of a plurality of the interferometers, the plurality of the interferometers using a common radiation source.

[0186] 43. The method according to any one of aspects 26 to 42, including determining a corresponding position value describing the position coordinates of a component of a machine for each phase estimate of interest.

[0187] 44. The method according to aspect 43, wherein the machine includes a lithographic apparatus.

[0188] 45. A computer program, including computer-readable instructions capable of operating to execute the method according to any one of aspects 26 to 44.

[0189] 46. A programmable firmware component, the programmable firmware component being programmed to implement the method according to any one of aspects 26 to 44.

Claims

1. A phase-generated carrier interrogator, comprising: One or more interferometers, each of the one or more interferometers being operable to receive modulated radiation and generate a corresponding interferometer signal, the interferometer signal including a phase of interest caused by the object under test, each of the interferometer signals further including a periodic phase modulation caused by the modulated radiation, wherein the periodic phase modulation includes an amplitude described by a corresponding modulation index of each of the one or more interferometers; At least one detector, the at least one detector being operable to detect each of the interferometer signals; and A signal processing module, the signal processing module being operable to: Estimate the modulation index for each of the one or more interferometers separately to obtain a corresponding estimated modulation index for each of the one or more interferometers; and Demodulate each of the interferometer signals in the interferometer signals using the corresponding estimated modulation index of each interferometer to estimate a corresponding phase of interest according to each of the interferometer signals in the interferometer signals, thereby obtaining a corresponding estimated value of the phase of interest for each of the interferometers.

2. The phase-generated carrier interrogator according to claim 1, wherein The signal processing module is operable to determine, for each of the one or more interferometers, a corresponding demodulation component of each of at least four demodulation carrier frequencies, the at least four demodulation carrier frequencies including at least two even-order demodulation carrier frequencies and at least two odd-order demodulation carrier frequencies, the demodulation carrier frequencies including harmonics of the modulation frequency of the periodic phase modulation.

3. The phase generation carrier interrogator according to claim 2, wherein, The signal processing module is operable to determine, for each of the one or more interferometers, each demodulation component as a combination of: each corresponding demodulation carrier frequency, the interferometer signal, and a unit magnitude complex conjugate phasor having an argument that is a negative estimated value of the phase of interest.

4. The phase-generated carrier interrogator according to claim 3, wherein, The signal processing module is operable to determine the unit magnitude complex conjugate phasor based on the estimated value of the phase of interest, and / or Wherein the signal processing module is operable to, for each of the one or more interferometers, use the corresponding estimated modulation index of the interferometer to define a first set of weighting coefficients for the demodulation components, the first weighting coefficients being defined such that a first weighted combination of the demodulation components weighted by the first weighting coefficients produces a first phasor, the first phasor having an argument including a phase residual describing an error in the estimated value of the phase of interest.

5. The phase-generated carrier interrogator according to claim 4, wherein, The signal processing module is further operable to determine the phase residual based on the first phasor, and / or Wherein the signal processing module includes a first weighting coefficient look-up table, and the first set of weighting coefficients is selected based on the first weighting coefficient look-up table based on each corresponding estimated modulation index, and / or Wherein the signal processing module is operable to estimate each modulation index based on the relative amplitudes of the two even-order demodulation components and at least two odd-order carrier demodulation components, and / or Wherein, the signal processing module is operable to estimate a modulation index for each of the one or more interferometers respectively by iteratively updating a modulation index estimate based on a modulation index residual, and / or Wherein, the signal processing module is operable to estimate the modulation index residual based on a second weighted combination of the demodulated components weighted by a second set of weighting coefficients to produce a second phasor linearly proportional to the modulation index residual.

6. The phase-generated carrier interrogator according to claim 5, wherein, The signal processing module further includes a dynamic state estimator operable to determine a new state, the new state including the estimated phase of interest from a previous state and from the phase residual, and / or Wherein, the signal processing module is operable to extract the modulation index residual by combining the second phasor with a normalized conjugate of the first phasor, and / or Wherein, the signal processing module is operable to determine the estimated modulation index according to an updated fringe count error estimate among the estimated phases of interest, the updated fringe count error estimate having been updated by the modulation index residual.

7. The phase generation carrier interrogator according to any one of claims 4 to 6, wherein, The first set of weighting coefficients includes respective even-order first weighting coefficient vectors for weighting each of the at least two even-order demodulated components, and respective odd-order first weighting coefficient vectors for weighting each of the at least two odd-order demodulated components, and Wherein, the signal processing module is operable, for each of the one or more interferometers: Determine each of the even-order first weighting coefficient vectors to be parallel to a respective normalized vector of an even-order Bessel function of the same order and corresponding to the estimated modulation index; and Determine each of the odd-order first weighting coefficient vectors to be parallel to a respective normalized vector of an odd-order Bessel function of the same order and corresponding to the estimated modulation index, and / or Wherein, the second set of weighting coefficients includes respective even-order second weighting coefficient vectors for weighting each of the at least two even-order demodulated components, and respective odd-order second weighting coefficient vectors for weighting each of the at least two odd-order demodulated components; and wherein, the second set of weighting coefficients is defined such that: The sum of the dot products of each of the even-order second weighting coefficient vectors with a corresponding even-order Bessel function vector of the estimated modulation index is zero; The sum of the dot products of each of the odd-order second weighting coefficient vectors with a corresponding odd-order Bessel function vector of the estimated modulation index is zero; The sum of the dot products of each of the even-order second weighting coefficient vectors with the derivative of a corresponding even-order Bessel function vector of the estimated modulation index is -1; and The sum of the dot products of each of the odd-order second weighting coefficient vectors with the derivative of a corresponding odd-order Bessel function vector of the estimated modulation index is -1.

8. The phase-generated carrier interrogator according to claim 5, 6 or 7, wherein The signal processing module includes a second weighting coefficient look-up table, and the second set of weighting coefficients is selected based on each respective estimated modulation index from the second weighting coefficient look-up table, and / or wherein the signal processing module is operable to determine the estimated modulation index based on the updated fringe count error estimate and the estimated ratio of the wavelength offset amplitude to the central wavelength of the modulated radiation.

9. A phase generated carrier interrogator as claimed in any preceding claim, the phase generated carrier interrogator including a programmable firmware component programmed to implement the signal processing module, and / or the phase generated carrier interrogator further includes a plurality of the interferometers, preferably further including a common radiation source for generating the radiation to be modulated to obtain the modulated radiation for each of the plurality of interferometers.

10. A multi-axis position measurement system, comprising the phase-generated carrier interrogator as described in claim 9, wherein, Each of the plurality of interferometers is for measuring a respective axis of the multi-axis position measurement system.

11. A lithographic apparatus, comprising a multi-axis position measurement system as claimed in claim 10, wherein, The multi-axis position measurement system is operable to measure the position of a component of the lithographic apparatus.

12. A projection system for an optical lithography system, comprising the multi-axis position measurement system as claimed in claim 11, wherein, The multi-axis position measurement system is operable to measure the position of a component of the projection system.

13. A method of processing one or more interferometer signals to determine an estimated value of a phase of interest caused by an object under test, each of the one or more interferometer signals being obtained from a respective interferometer of one or more interferometers; the method comprising: modulating radiation for each of the one or more interferometers to apply a periodic phase modulation having an amplitude described by a respective modulation index of each interferometer; detecting a respective interferometer signal for each of the one or more interferometers, each interferometer signal including in addition to the periodic phase modulation a phase of interest caused by the object under test; estimating the modulation index for each of the one or more interferometers separately to obtain a respective estimated modulation index for each of the one or more interferometers; and demodulating each of the interferometer signals using the respective estimated modulation index of each interferometer to estimate a respective phase of interest based on each of the interferometer signals, thereby obtaining a respective estimated value of the phase of interest for each of the interferometers.

14. A computer program comprising computer-readable instructions operable to execute the method as claimed in claim 13.

15. A programmable firmware component programmed to execute the method as claimed in claim 13.

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