Phase generated carrier interrogator and associated phase generated carrier interrogation method

Abstract

A phase generated carrier interrogator comprising a complex multi-carrier synthesizer operable to generate at least a laser modulation carrier signal for modulating a laser radiation source; a power monitor branch operable to measure a power metric of the laser radiation source; a wavelength reference branch comprising a wavelength reference for wavelength stabilization of the laser radiation source, a wavelength reference detector for measuring said wavelength reference to obtain a wavelength reference signal; and a signal processing module. The signal processing module is operable to: estimate a monitor power-carrier delay describing a propagation delay of the power monitor signal in the laser power monitor signal path; estimate, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrier delay, the wavelength reference power-carrier delay describing a propagation delay of the wavelength reference signal in the wavelength reference branch.

Description

PHASE GENERATED CARRIER INTERROGATOR AND ASSOCIATED PHASE GENERATED CARRIER INTERROGATION METHOD CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The application claims priority of EP application 23157591.1 which was filed on 20 February, 2023 and which is incorporated herein in its entirety by reference. FIELD

[0002] The present invention relates to interferometry, and in particular to methods and apparatuses for phase generated carrier (PGC) interferometer interrogation. BACKGROUND

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

[0004] As semiconductor manufacturing processes continue to advance, the dimensions of circuit elements have continually been reduced while the amount of functional elements, such as transistors, per device has been steadily increasing over decades, following a trend commonly referred to as ‘Moore’s law’. To keep up with Moore’s law the semiconductor industry is chasing technologies that enable to create increasingly smaller features. To project a pattern on a substrate a lithographic apparatus may use electromagnetic radiation. The wavelength of this radiation determines the minimum size of features which are patterned on the substrate. Typical wavelengths currently in use are 365 nm (i-line), 248 nm, 193 nm and 13.5 nm. A lithographic apparatus, which uses extreme ultraviolet (EUV) radiation, having a wavelength within a range of 4 nm to 20 nm, for example 6.7 nm or 13.5 nm, may be used to form smaller features on a substrate than a lithographic apparatus which uses, for example, radiation with a wavelength of 193 nm.

[0005] Within a lithographic apparatus, it is important to accurately measure the position of certain components of the lithographic apparatus. Such components may comprise inter alia one or more wafer (or substrate) stages, the reticle (or mask) stage and / or one or more optical components (e.g., mirrors within a projection system of the lithographic apparatus. To do this, an interferometer may be used, and in particular a multi-axis interferometer. A multi-axis interferometer may comprise multiple individual interferometers, one per axis, wherein each axis relates to position measurement of a different degree of freedom. In an embodiment, the interferometer may comprise a 6-axis interferometer for position measurement in 6 degrees of freedom: three mutually perpendicular spatial axes, conventionally referred to as the x-axis, y-axis and z axis, and rotations around each of these axes Rx, Ry, Rz.

[0006] It is known to stabilize a wavelength of a laser radiation source of such a phase generated carrier interferometer interrogator using a wavelength reference such as a gas cell reference.

[0007] It would be desirable inter alia to improve on methods of using such a wavelength reference. SUMMARY

[0008] In a first aspect of the invention, there is provided a phase generated carrier interrogator, comprising: a complex multi-carrier synthesizer being operable to generate at least a laser modulation carrier signal for modulating a laser radiation source; a laser power monitor branch comprising a power monitor detector and being operable to measure a power metric of the laser radiation source to obtain a power monitor signal; a wavelength reference branch comprising a wavelength reference for wavelength stabilization and / or control of the laser radiation source, and a wavelength reference detector for measuring said wavelength reference to obtain a wavelength reference signal; and a signal processing module being operable to: estimate a monitor power-carrier delay describing a propagation delay of the power monitor signal in the laser power monitor signal path; and estimate, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrier delay, the wavelength reference power-carrier delay describing a propagation delay of the wavelength reference signal in the wavelength reference branch.

[0009] In a second aspect of the invention, there is provided a method of estimating at least a wavelength reference excess delay relating to a wavelength reference, the method comprising: measuring a power metric of a laser radiation source to obtain a power monitor signal; stabilizing and / or controlling the wavelength of the laser radiation source using a wavelength reference signal from a wavelength reference; estimating a monitor power-carrier delay describing a propagation delay of the power monitor signal in a laser power monitor signal path; and estimating, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrier delay, the wavelength reference power- carrier delay describing a propagation delay of the wavelength reference signal in a wavelength reference branch. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which: - Figure 1 depicts a schematic overview of a lithographic apparatus; - Figure 2 is a flow diagram conceptually illustrating a signal processing method in PGC interferometry according to a known method; - Figure 3 is a flow diagram conceptually illustrating a signal processing method for determining a carrier delay in the context of PGC interferometry according to a first embodiment;- Figure 4 is a flow diagram conceptually illustrating a signal processing method for determining a power monitor carrier delay in in the context of PGC interferometry according to a second embodiment; - Figure 5 is a flow diagram conceptually illustrating a signal processing method for calibrating a wavelength reference excess carrier delay in the context of PGC interferometry according to a third embodiment; - Figure 6 is a flow diagram conceptually illustrating a signal processing method for determining a laser pseudo delay, and using each of the delays calibrated in Figures 4-6 for laser wavelength stabilization, in in the context of PGC interferometry according to a fourth embodiment; and - Figure 7 is a flow diagram conceptually illustrating a signal processing method for calibrating an interferometer excess carrier delay, and using each of the delays calibrated in Figures 4-7 for interferometer signal processing in the context of PGC interferometry according to a fifth embodiment. DETAILED DESCRIPTION

[0011] In the present document, the terms “radiation” and “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range of about 5- 100 nm). Interferometers described herein may use radiation having a wavelength greater than 400 or greater than 500nm. More specifically, by way of specific examples, the interferometer radiation wavelength may be 633nm or 1530nm.

[0012] The term “reticle”, “mask” or “patterning device” as employed in this text may be broadly interpreted as referring to a generic patterning device that can be used to endow an incoming radiation beam with a patterned cross-section, corresponding to a pattern that is to be created in a target portion of the substrate. The term “light valve” can also be used in this context. Besides the classic mask (transmissive or reflective, binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include a programmable mirror array and a programmable LCD array.

[0013] Figure 1 schematically depicts a lithographic apparatus LA. The lithographic apparatus LA includes an illumination system (also referred to as illuminator) IL configured to condition a radiation beam B (e.g., UV radiation, DUV radiation or EUV radiation), a mask support (e.g., a mask table) MT constructed to support a patterning device (e.g., a mask) MA and connected to a first positioner PM configured to accurately position the patterning device MA in accordance with certain parameters, a substrate support (e.g., a wafer table) WT constructed to hold a substrate (e.g., a resist coated wafer) W and connected to a second positioner PW configured to accurately position the substrate support in accordance with certain parameters, and a projection system (e.g., a refractive projection lens system) PS configured to project a pattern imparted to the radiation beam B by patterning device MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.

[0014] 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 opticalcomponents, such as refractive, reflective, magnetic, electromagnetic, electrostatic, and / or other types of optical components, or any combination thereof, for directing, shaping, and / or controlling radiation. The illuminator IL may be used to condition the radiation beam B to have a desired spatial and angular intensity distribution in its cross section at a plane of the patterning device MA.

[0015] The term “projection system” PS used herein should be broadly interpreted as encompassing various types of projection system, including refractive, reflective, catadioptric, anamorphic, magnetic, electromagnetic and / or electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, and / or for other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term “projection lens” herein may be considered as synonymous with the more general term “projection system” PS.

[0016] The lithographic apparatus LA may be of a type wherein at least a portion of the substrate may be covered by a liquid having a relatively high refractive index, e.g., water, so as to fill a space between the projection system PS and the substrate W – which is also referred to as immersion lithography. More information on immersion techniques is given in US6952253, which is incorporated herein by reference.

[0017] The lithographic apparatus LA may also be of a type having two or more substrate supports WT (also named “dual stage”). In such “multiple stage” machine, the substrate supports WT may be used in parallel, and / or steps in preparation of a subsequent exposure of the substrate W may be carried out on the substrate W located on one of the substrate support WT while another substrate W on the other substrate support WT is being used for exposing a pattern on the other substrate W.

[0018] In addition to the substrate support WT, the lithographic apparatus LA may comprise a measurement stage. The measurement stage is arranged to hold a sensor and / or a cleaning device. The sensor may be arranged to measure a property of the projection system PS or a property of the radiation beam B. The measurement stage may hold multiple sensors. The cleaning device may be arranged to clean part of the lithographic apparatus, for example a part of the projection system PS or a part of a system that provides the immersion liquid. The measurement stage may move beneath the projection system PS when the substrate support WT is away from the projection system PS.

[0019] In operation, the radiation beam B is incident on the patterning device, e.g. mask, MA which is held on the mask support MT, and is patterned by the pattern (design layout) present on patterning device MA. Having traversed the mask MA, the radiation beam B passes through the projection system PS, which focuses the beam onto a target portion C of the substrate W. With the aid of the second positioner PW and a position measurement system IF, the substrate support WT can be moved accurately, e.g., so as to position different target portions C in the path of the radiation beam B at a focused and aligned position. Similarly, the first positioner PM and possibly another position sensor (which is not explicitly depicted in Figure 1) may be used to accurately position the patterning device MA with respect to the path of the radiation beam B. Patterning device MA and substrate W may be aligned using mask alignment marks M1, M2 and substrate alignment marks P1, P2. Although thesubstrate alignment marks P1, P2 as illustrated occupy dedicated target portions, they may be located in spaces between target portions. Substrate alignment marks P1, P2 are known as scribe-lane alignment marks when these are located between the target portions C.

[0020] To clarify the invention, a Cartesian coordinate system is used. The Cartesian coordinate system has three axes, i.e., an x-axis, a y-axis and a z-axis. Each of the three axes is orthogonal to the other two axes. A rotation around the x-axis is referred to as an Rx-rotation. A rotation around the y-axis is referred to as an Ry-rotation. A rotation around about the z-axis is referred to as an Rz-rotation. The x- axis and the y-axis define a horizontal plane, whereas the z-axis is in a vertical direction. The Cartesian coordinate system is not limiting the invention and is used for clarification only. Instead, another coordinate system, such as a cylindrical coordinate system, may be used to clarify the invention. The orientation of the Cartesian coordinate system may be different, for example, such that the z-axis has a component along the horizontal plane.

[0021] The position measurement system PMS may comprise any type of sensor that is suitable to determine a position of the substrate support WT. The position measurement system PMS may comprise any type of sensor that is suitable to determine a position of the mask support MT. The sensor may be an optical sensor such as an interferometer or an encoder. The position measurement system PMS may comprise a combined system of an interferometer and an encoder. The sensor may be another type of sensor, such as a magnetic sensor, a capacitive sensor or an inductive sensor. The position measurement system PMS may determine the position relative to a reference, for example the metrology frame MF or the projection system PS. The position measurement system PMS may determine the position of the substrate table WT and / or the mask support MT by measuring the position or by measuring a time derivative of the position, such as velocity or acceleration.

[0022] The position measurement system PMS may comprise an interferometer system. An interferometer system is known from, for example, United States patent US6,020,964, filed on July 13, 1998, hereby incorporated by reference. The interferometer system may comprise a beam splitter, a mirror, a reference mirror and a sensor. A beam of radiation is split by the beam splitter into a reference beam and a measurement beam. The measurement beam propagates to the mirror and is reflected by the mirror back to the beam splitter. The reference beam propagates to the reference mirror and is reflected by the reference mirror back to the beam splitter. 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 a phase or a frequency of the combined radiation beam. The sensor generates a signal based on the phase or the frequency. The signal is representative of a displacement of the mirror. In an embodiment, the mirror is connected to the substrate support WT. The reference mirror may 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 an additional optical component instead of the beam splitter.

[0023] In lithographic processes, it is desirable to make frequently measurements of the structures created, e.g., for process control and verification. Tools to make such measurement are typically called metrology tools MT. Different types of metrology tools MT for making such measurements are known, including scanning electron microscopes or various forms of scatterometer metrology tools MT. Scatterometers are versatile instruments which allow measurements of the parameters of a lithographic process by having a sensor in the pupil or a conjugate plane with the pupil of the objective of the scatterometer, measurements usually referred as pupil based measurements, or by having the sensor in the image plane or a plane conjugate with the image plane, in which case the measurements are usually referred as image or field based measurements. Such scatterometers and the associated measurement techniques are further described in patent applications US20100328655, US2011102753A1, US20120044470A, US20110249244, US20110026032 or EP1,628,164A, incorporated herein by reference in their entirety.

[0024] It is proposed that interferometry for the position measurement system of either a lithographic apparatus (scanner) or a metrology apparatus may use phase generated carrier (PGC) interferometer interrogation. PGC interferometry requires less expensive interferometer hardware than conventional homodyne or heterodyne interrogation techniques, but requires more complicated signal processing.

[0025] A radiation source or laser source such as a wavelength-modulated laser diode may be used in combination with an unbalanced interferometer (having a nonzero optical path difference OPD), to create a periodic phase modulation or sinusoidal modulation ^ ∙ sin^^^^^∙ ^^ of the interferometer phase, in addition to the phase of interest ^^^^ induced by the measurand. The modulation index ^ of this modulation can be described by ^ = 2^ ∙^^^^ ∙ ^̂^^^, where ^̂^^^is the laser frequency excursion amplitude and c is the speed of light.

[0026] The wavelength modulation may be created in a number of different ways, of which two alternative methods will be described: 1) the emission wavelength of the laser may be changed (this is the lowest cost option) or 2) the laser may be operated at constant emission frequency and the periodic wavelength modulation created by an additional (electro-optic) phase modulator component downstream from the laser but common to all interferometers. This latter method has an advantage over the first method in that it avoids modulation of the laser output power, which is a source of inaccuracy.

[0027] Figure 2 is a flow diagram conceptually illustrating the signal processing in PGC interferometry. A multi-carrier synthesizer MCS generates the sinusoidal modulation sin^^^^^∙ ^^ for a laser diode LD of the interferometer. This sinusoidal modulation is combined with (e.g., multiplied by) the modulation amplitude or current excitation amplitude ^^̂^^and further combined with (e.g., summed with) the bias current BC. This laser current excitation amplitude ^^̂^^is related to the laser frequency excursion amplitude ^̂^^^by: ^̂^^^ = ^ ∙ ^̂^^^where η is the laser’s current to frequency sensitivity.

[0028] The optical power (interferometer signal ^^^^^^^) detected by the photodetector PD of the interferometer can then be written as (ignoring delays for now): ^^^^^^^ = ^ + ! ∙ cos^^^^^ + ^ ∙ sin^^$%& ∙ ^^^where B is a background power and A is the amplitude of the power of the interference fringes. The ratio A / B is the interference contrast and satisfies 0 ≤ ! / ^ ≤ 1. The phase modulation amplitude m will be referred to as the modulation index from hereon.

[0029] This sinusoidal phase modulation serves to generate a series of demodulation carrier frequencies or phase-generated carrier frequencies in the detected output power of the interferometer as harmonics of the modulation frequency ^$%&. These phase-generated carriers may take the form of: sin+^2, − 1^ ∙ ^^^^ ∙ ^., cos^2, ∙ ^^^^ ∙ ^^where k represents to n.

[0030] The amplitudes of odd-order carrier frequencies (carriers) are proportional to the sine of the measurand-induced phase ^^^^, and the amplitudes of even-order carrier frequencies (carriers) are proportional to the cosine of the measurand-induced phase ^^^^ . Each carrier of order n is also attenuated by the modulation index dependent factor 01^^^, which is a Bessel function of the first kind of order n (where 02^^^is a DC component of series described by the bracketed terms).Therefore the optical power S(t) may be described as: 7 ^^^^^^^ = ^ + ! ∙ cos^^^^∙302^^^ + 24056^^^ ∙ cos^2, ∙ ^$%&∙ ^^:

[0031] ^^^^ ) is not directly accessible. Instead, this measurand-induced phase ^^^^ (the phase of interest) may be estimated by a dynamic state estimator SE, also known as the process model (e.g., which models the dynamic state of the measurand; for example this may be the instantaneous position of a body such as a stage). This estimation process may calculate a phase prediction ^<^^^ (estimated phase of interest value), which can be compared to the true measurand-induced phase ^^^^ comprised in the signal ^^^^. The phase residual ^<^^^ − ^^^^ from this comparison can be used to correct the dynamic state predicted by theprocess model SE. As such, the dynamic state estimator may determine a new state comprising said estimated phase prediction ^<^^ from a previous state and from said phase residual ^<^^^ − ^^^^ The corrected state is the output of the system. The resulting estimation loop is actually a (digital) phase- locked loop (PLL), which in turn is a Kalman filter.

[0032] By proper processing of the signal ^^^^(e.g., within a signal processing module), a complex phasor ! ∙ ?@+ A^B^;AC ^B^ .(phase residual phasor) can be constructed which has the desired phase residual ^<^^^ − ^^^^ as its argument. The required signal processing involves three conceptual steps. In a first step, the interferometer signal ^^^^ is multiplied by a unity-magnitude complex conjugate phasor ?;@AC ^B^having an argument which is the negative phase prediction −^<^^^. A first look-up table or look-up table LUT1 can be used to construct this phasor. In a second step, respective demodulate components D1^^^or demodulates are formed respectively for each carrier to be taken into account (i.e., at least one of even-order and one of odd-order). Synchronous demodulation involves multiplication by the respective carrier cos^2, ∙ ^^^^∙ ^^, -sin+^2, − 1^∙ ^^^^∙ ^., followed by low-pass filtering LPF. These two steps together result in the following even-order and odd-order demodulates: D56 = 〈^^^^^^^ ∙ ?;@AC ^B^ 1∙ cos^2, ∙ ^$%& ∙ ^^〉 ≈ ! ∙ @+ A^B^;AC ^B^ . ;@+ A^B^HAC ^B^ .2 〈? + ? 〉 ∙ 056^^^〈 〉 or pass two low-pass filtered phasors: a first phasor comprising the desired argument ^<^^^ − ^^^^ and an additional phasor having the argument −+^<^^^ + ^^^^.. In the situation where the process model is tracking the desired phase ^^^^ phase residual ^<^^^− ^^^ will be small and the^corresponding first phasor ! ∙ ?@+ A^B^;AC ^B^ .will continuously be near or at the real axis. However, the additional phasor ! ∙ ?Aat a rate of twice the rate of change of the measurand- induced phase ^^^^, which can be substantial. Thus, depending on the rate of change (e.g., velocity) this additional phasor will experience attenuation by the lowpass filtering action, while the desired first phasor will not.

[0033] In the third processing step, even-order and odd-order demodulates are combined, by a properly weighted summation, in order to remove the additional phasor. In the simplest case, where one even- order demodulate is combined with one odd-order demodulate, the proper weighting coefficients are the reciprocal Bessel functions:056;9^^^∙ D56+ I ∙ 056;9;9^^^∙ D56;9≈ ! ∙ ?@+ A^B^;AC ^B^ .

[0034] Note are identical andtherefore the

[0035] The above three processing steps result in a phase-residual first phasor ! ∙ ?@+ A^B^;AC ^B^ .that, provided that the estimate ^<^^^ is accurate such that the phase residual ^<^^^ −zero, is on or near the real axis. Magnitude and phase residual can now be separated by inputting this phasor into a Cartesian-to-Polar coordinate transformation unit CtP, which may be conveniently implemented in a FPGA (or in software) as a CORDIC algorithm, for example. For notational convenience the time- dependence of the phase residual ^<^^^ − ^^^^ will now be dropped, since it is quasi-static.

[0036] In order to correctly detect the various (odd & even-order) carriers in the photodetector signal, it is important that the demodulation carriers are phase-aligned to the respective carrier signals in the photodetector signal. A carrier phase deviation reduces the strength of the demodulates (cosine dependency) determined from the demodulation carriers. Therefore, in extreme cases (e.g., an absolute phase deviation of a quarter cycle or more), a demodulate may become zero or obtain the wrong sign, which will render the interrogator inoperative. For very small phase deviations the strength reduction is a second-order effect.

[0037] The wavelength modulation frequency of the laser diode LD is usually in the order of a few (tens of) MHz. This may be accomplished by current modulation of a distributed-feedback (DFB) laserdiode, which results in modulation of the junction temperature and subsequently the emission frequency^^^^, according to:^^^^ = ^2 + ^̂^^^ ∙ sin+^$%& ∙ ^ + J^^$%&^.where J^^$%&^ is a due to the thermal process inthe laser junction and ^̂^^^is the laser frequency excursion amplitude around a mean optical frequency^2.

[0038] However, the temperature sensitivity of the wavelength also affects the mean wavelength, which leads to drift of the measurand-induced phase (phase of interest), if not counteracted. An accurate PGC interrogator therefore should have the laser diode stabilized against a wavelength reference. The wavelength reference can be a gas cell, for example, offering multiple candidate absorption lines with accurately-known (e.g., traceable to NIST standard) center wavelengths. For the purpose of the description herein, the gas cell optical transmission characteristic around one such line center will be approximated by a Lorentzian profile according to: 1− KK^^^ = 1 − $L1^− ^^5where ^^is the center frequency and ^NON^is the half-width half-maximum linewidth, while K$L1represents the transmission minimum at the line center.

[0039] The laser mean optical frequency can be locked to the center of an absorption line by continuously adjusting the laser bias current in response to a suitable control residual. In principle the fundamental (modulation) frequency component, or other odd frequency component in the gas cell output power can be used as a control residual (any odd frequency component may be used, although the fundamental is the most sensitive). In the hypothetical absence of laser power modulation, this component will vanish when the mean wavelength is at the line center, because that is a point of even symmetry. This mechanism is a variant of the Pound-Drever-Hall (PDH) stabilization scheme. As an example, assuming the optical frequency to remain sufficiently close to the line center, the Lorentzian transmission characteristic can be further approximated by: ^− ^5 K^^^ ≈ K$L1 + ^1 − K$L1^ ∙ Q^^R

[0040] Substituting the time- K+^^^^ ≈ K + ^1 − K ^ ∙^^2 − ^^^5 + 12 ^̂^5^^+ 2^1 − K ^ ∙^^2 − ^^^ ∙

[0041] The fundamental frequency component 2^1 − K$L1^ ∙ ^ST;SU^∙VSWXUSY\ZY[ ∙ sin+^$%& ∙ ^ +J^^ ^ vanishes when ^2 = ^^ , which illustrates that used as aresidual to stabilize the mean optical frequency ^2to the line center ^^.

[0042] In practice, however, the laser current modulation will also lead to power modulation at the fundamental modulation frequency. In other words the laser power ]^^^ can be written as: ]^^^ = ]2 − ]^^^^ ∙ sin^^$%& ∙ ^^where ]^^^^is the laser power excursion amplitude around mean power ]2.

[0043] The optical power ]_`^^^at the output of the gas cell is therefore of the form: ]_`^^^= K+^^^^. ∙ ]^^^

[0044] The laser power modulation at the fundamental frequency will add two extra components to the gas cell’s output power at the fundamental frequency. One of these extra component is due to mixing with the DC component in T(t); the other is due to mixing with the second harmonic component in T(t). The resulting components at DC, fundamental frequency and 2nd harmonic frequency are given by: ]^ ^\ b \_`^^^ ≈aKS$L1+ ^1 − K$L1^ ∙T;SU H\VSWXUSY\ZY c∙ ]2+ −two components at cause mean optical frequency ^2to deviate from the line center ^^. The most dominant undesired term is the one that is proportional to K$L1. The worst-case optical frequency error can be coarsely estimated by equating the amplitude of the desired term to that of the dominant undesired term: 2^1 − K$L1^ ∙ ]2 ∙|^2 − ^^| ∙ ^̂^^^^5 = ^1 − K$L1^ ∙ ]^^^^ ∙K$L11− K ⇒

[0046] This a gas narrow ^NON^) and deep (small K$L1) absorption lines and additionally a high laser power (large ]2). A laser with small ratio]^^^^ / ^̂^^^ is also beneficial.

[0047] However, to achieve accurate wavelength stabilization, the power modulation factor in the photodetector signal should be compensated. This can be done by detecting the instantaneous laser power (a power metric) using, for example, a fiber splitter to branch off a fixed fraction of the laser power onto a dedicated power monitor detector (photodiode or photodetector). The digitized instantaneous signal from the gas cell detector can be power-normalized by dividing it by the digitized instantaneous signal from the power monitor detector. The digitized interferometer photodetector signals from the (multiple) interferometer axes can also be power-normalized in this way.

[0048] For the power normalization to be accurate, the power modulation factors in all of the digitized photodetector signals should be phase-aligned to the power modulation term in the normalizing monitorsignal. This will generally not be the case without compensation, because gas cell and interferometer axes are interconnected by fibers of lengths which differ from that of the fiber going to the power monitor photodetector; these buffer delays are equal and / or common to all axes and do not affect the phase alignment to the normalizing monitor signal. The (digitized) power monitor signal ^$%1obtained from the power monitor photodetector may be described by: ^$%1^^^= ^$%1 ∙ ]^^ − r$%1^= ^$%1 ∙ +]2 − ]^^^^ ∙ sin+^$%& ∙^^ − r$%1^..where r$%1power- signal path and ^$%1is the power efficiency of the monitor. Likewise, the gas cell signal ^_`^^^ from the gas cell photodetector is delayed by a gas cell power-carrier delay r_`of the gas cell signal in the gas cell signal path, and may be described by: ^_`^^^= ^_` ∙ ]_`^^ − r_`^= ^_` ∙ +]2 − ]^^^^ ∙ sin+^$%& ∙^^ − r_`^.. ∙ K+^^^ − r_`^.to power gas be delayed by the difference of the monitor power-carrier delay and gas cell power-carrier delay, hereafter described as the gas cell excess delay r_`,^^= r_`− r$%1, i.e. the gas cell signal may be normalized by: ^$%1+^ − r_`,^^. = ^$%1 ∙ +]2 − ]^^^^ ∙ sin+^$%& ∙ ^^ − r_`^..

[0050] In the ideal case, a power- of the form: ^_`^^^=^_`^∙ K+^^^ − r_`^.^ +^ − r

[0051] One problem gas excess delay r_`,^^. Once the gas cell excess delay estimate r̃_`,^^. has been determined, the monitor signal may be delayed by this estimate prior to using it for normalization of the gas cell signal. This delaying of the monitor signal may be achieved, for example, using a variable fractional delay filter (frequency transfer function), since this estimate will in general not be an integer number of sampling intervals.

[0052] Another issue is that the normalized gas cell signal depends on the delayed optical frequency signal:^^^ − r_`^ = ^2 + ^̂^^^ ∙ sin+^$%& ∙ ^^ − r_`^ + J^^$%&^.

[0053] As concept a and may be definedas comprising the sum of gas cell power-carrier delay r_` and a laser pseudo delay rt^^^$%&^ =−J^^$%&^ / ^$%& resultant from the (possibly drifting) frequency response of the wavelength-modulating thermal process in the laser.

[0054] Thus, to ensure that the fundamental frequency component in the (properly) normalized gas cell signal will be detected by means of synchronous demodulation, with adequate sensitivity and correct sign, the demodulation carrier should be brought to, and maintained in, phase-alignment to the (vanishing) fundamental frequency component. In other words the demodulation carrier may ideally be phase-shifted over: −^$%&∙ +r_`+ rt^^^$%&^.

[0055] Laser it results in harmonic mixing in the gas cell signal,leads to mixing even carriers in the interferometer signal, which inturn results in cyclic errors in the phase estimate. Taking into account the laser phase retardationJ^^$%&^, the interferometer signal ^^^^^^^ takes the form:]^^^^^^^^ = a1 − ^^^∙ sin+^$%& ∙ ^^ − r^^^^.cwhere ^^^^ r^^^a power-carrier delay describing the unknown and / or time-varying propagation delay of the radiation as it travels from the laser, along the two beam paths comprising the interferometer, to the interferometer photodetector. More on this delay (which optionally may be tracked in embodiments) will be described later. In other embodiments, it may be ignored.

[0056] This can be rewritten, using the Jacobi-Anger expansion (a special Fourier series expansion), as: ^^^^^^^ö ÷ø

[0057] The terms proportional to the power modulation amplitude can be written in a form that clearly shows their distinct frequency components: The power modulation terms in ^^^^^^^= −]^^^^∙ ^ ∙ sin+^$%&∙ ^^ − r^^^^.−]^^^^∙ ! ö ÷÷ø^ ∙ ^$%&, power neighboring carriers at^^ ± 1^∙ ^$%&(recognizable by the indices of the involved Bessel functions). The carrier phase of each undesired component is determined by the laser phase retardation J^^$%&^ and by the sum and difference of the Bessel functions of the neighboring desired carriers. Odd-order spurious carriers are proportional to cos ^^^^, while even-order spurious carriers are proportional tosin ^^^^.

[0059] The disclosed method relates to the calibrating and / or monitoring of one or more of the abovementioned delays. In particular, the method may relate to the determination of a wavelength reference excess delay (hereafter a gas cell excess delay) describing a difference of a wavelength reference power-carrier delay (hereafter a gas cell power-carrier delay) and a monitor power-carrier delay, the gas cell power-carrier delay describing a propagation delay of a wavelength reference signal (hereafter a gas cell signal) in a wavelength reference (hereafter a gas cell) signal path and the monitor power-carrier delay describing a propagation delay of a power monitor signal in a power monitor signal path. The wavelength reference excess delay can then be used to delay the power monitor signal, to obtain a first delayed power monitor signal that may be used to power-normalize the gas cell signal.

[0060] Such a method may comprise determining and continually monitoring / tracking the monitor power-carrier delay r$%1, e.g., based on the determination and nullifying of a monitor power-carrier delay residual r̃$%1-r$%1, where r̃$%1is the estimated monitor power-carrier delay (generally a tilde will signify an estimate herein). The monitor power-carrier delay residual r̃$%1- r$%1may be determined from a demodulated fundamental component (n=1) in the monitor photodetector signal (ideally the power monitor signal does not contain any other harmonic components).

[0061] A (e.g., one time) calibration of the gas cell excess delay r_`,^^in the gas cell optical path, relative to the monitor power-carrier delay r$%1, may then be performed, where r_`,^^= r_`− r$%1. Such a method may comprise nullifying a gas cell excess delay residual r̃_`,^^-r_`,^^, where r̃_`,^^isthe gas cell excess delay estimate, when the mean laser wavelength is in a region between absorption lines of the gas cell reference. The monitor signal may be delayed by the estimate of the gas cell excess delay r̃_`,^^, prior to applying it to normalize the gas cell signal. Alternatively a reciprocal of the monitor signal may be delayed by the gas cell excess delay r̃_`,^^, prior to multiplying it by the gas cell signal to obtain the normalized gas cell signal. This requires only a single division stage for the monitor signal rather than a dedicated division stage for each photodetector signal to be normalized.

[0062] The method may also comprise determining and / or continuously monitoring an estimate of a laser pseudo-delay rt^^^$%&^= −J^^$%&^ / ^$%&, while locked to an absorption line of the gas cell reference, the laser pseudo-delay resulting from a (possibly drifting) frequency response of the wavelength-modulating thermal process in the laser. This laser pseudo-delay rt^^^$%&^may be determined from a laser pseudo-delay residual r̃t^^^$%&^− rt^^^$%&^as constructed from even- order harmonic components of the gas cell signal.

[0063] A wavelength reference wavelength-carrier delay (hereafter gas cell wavelength-carrier delay)rt^^^$%&^ + r_` can then be continually predicted as a sum of the laser pseudo-delay rt^^^$%&^estimate, monitor power-carrier delay r$%1and gas cell excess delay r_`,^^, while locked to an absorption line. This gas cell wavelength-carrier delay can be used to adjust a demodulation carrier phase or demodulation component phase so as to align its phase with the fundamental component (or other odd component if used) of the gas cell signal; i.e., the phase of demodulation carriers or demodulation components used to demodulate the (power-normalized) gas cell signal when locked to an absorption line of the gas cell. As such, the effective gas cell carrier delay can be used to rotate a complex demodulate of the fundamental frequency component of the (normalized) gas cell signal onto the real axis, thereby locking a mean optical laser frequency to the center of an absorption line; e.g., by continuously adjusting the laser bias current based on the resultant demodulated wavelength reference signal (e.g., so as to nullify the normalized and phase-aligned fundamental frequency component).

[0064] The method may further comprise determining or calibrating an interferometer excess delayr^^^,^^, defined as the difference of an interferometer power-carrier delay r^^^ and the monitor power-carrier delay r$%1. This interferometer excess delay r^^^,^^may be used to delay the monitor signal or a reciprocal thereof (to obtain a second delayed (e.g., reciprocal) power monitor signal) prior to using the (e.g., reciprocal) monitor signal to power-normalize a respective interferometer signal for each interferometer.

[0065] An interferometer wavelength-carrier delay r may be predicted (e.g., per interferometer) from the sum of: the interferometer excess delay r^^^,^^, the monitor power-carrier delay r$%1and the laser pseudo-delay rt^^^$%&^; e.g., r = r^^^,^^+ r$%1+ rt^^^$%&^ . This interferometer wavelength- carrier delay prediction may be used to adjust respective partial demodulation carrier phases or demodulation component phases so as to phase-align the partial demodulation components with respective carrier signals in the interferometer signal.

[0066] In summary, the delay-quantities of which estimates may be maintained and / or calibrated are:the monitor power-carrier delay r$%1 , the gas cell excess delay r_`,^^ , the laser pseudo-delayrt^^^$%&^ and (for each individual interferometer axis) the interferometer excess delay r^^^,^^. Themonitor power-carrier delay r$%1and the laser pseudo delay rt^^^$%&^ are associated to active components and may therefore be drifting; as such it is proposed that their estimates may be updated continuously. The excess delay quantities represent optical propagation delays of passive components and their estimates, and as such should not require frequent updating, but only a (e.g., one-time) calibration.

[0067] The excess delay estimates (of the gas cell and, where appropriate, the interferometer axes) may be used to delay the (e.g., reciprocal) monitor signal prior to using it to normalize the respective photodetector signals. Each of the carrier delay predictions is an absolute (rather than relative) delay that is used to rotate the complex demodulates, resulting from demodulation by complex phasors, to the real axis of the complex plane. The real part of the rotated complex demodulate then becomes the signed amplitude of the corresponding component in the photodetector signal.

[0068] Figure 3 is a flow diagram illustrating the basic principle of how at least some of these delay estimates may be refined by determination of a corresponding delay residual. Each of these delay residuals may be extracted as a phase residual from a complex demodulation stage. In Figure 3 and the corresponding description, a generic delay r will be described which may be any of the delays determined herein. The estimate of this delay will be denoted by r̃, and the residual therefore denoted as r̃ − r. Where features of this method are the same as that illustrated by Figure 2, these features will not necessarily be described again.

[0069] In a complex demodulation stage, a real delayed signal component (nth harmonic) of the form! ∙ cos+^ ∙ ^$%& ∙ ^^ − r^. is multiplied by a synchronous complex demodulation carrier (e.g.,generated by complex multi-carrier synthesizer CMCS) of the form ?;@∙1∙hijk∙B, which gives a product: ! ∙ cos+^ ∙ ^$%&∙ ^^ − r^. ∙ ?;@∙1∙hijk∙B=1 1 ^ ^2 ! ∙ ?;@∙1∙hijk∙^+ 2 ! ∙ ?;@∙1∙hijk∙ 5B;^two components. pass LPF, hereafter by chevrons 〈 〉, rejects the (rapidly revolving) second component while leaving the (static) first component as the useful / desired demodulate: 〈! ∙ cos+^ ∙ ^$%&∙ ^^ − r^.∙ ?;@∙1∙hijk∙B〉 =12 ! ∙ ?;@∙1∙hijk∙^

[0070] This phasor makes an angle −^ ∙ ^$%&∙ r with the real axis of the complex plane. A carrier delay prediction r̃ may be applied to this phasor in such a way as to rotate the demodulate to the real axis. This may comprise multiplication with a phasor ?@∙1∙hijk∙^^, such that the phasor product yields: ?@∙1∙hijk∙^^∙ 〈! ∙ cos+^ ∙ ^$%&∙ ^^ − r^. ∙ ?;@∙1∙hijk∙B〉 =12 ! ∙ ?@∙1∙hijk∙^^^; ^^

[0071] argument − r , scaled by the modulation angular frequency ^$%&and by the harmonic order n. Thus the delay residual can be conveniently extracted by feeding the phasor into a CORDIC unit CtP (Cartesian to Polar coordinate transformation), to obtain the argument, and multiplying this argument by ^^ ∙ ^$%&^;9. The delay residual r̃ − r may then be used to refine the delay estimate r̃ maintained by DT. Therefinement is a scaled-down version of the negated residual. The scaling factor L<1 determines the bandwidth of the delay tracker.

[0072] The range of the delay residual and therefore the convergence range of the delay estimator amounts to + / - half a cycle of the nth harmonic component ^$%&, i.e. |r̃ − r| <95 ∙ ^^ ∙ ^$%&^;9.

[0073] This approach is very implementation-friendly: the themselves are not(individually) phase-shifted and can therefore be distributed to many (monitor, gas cell, multiple interferometer axes), while the rotation of the demodulates happens at a much lower sampling rate, allowing these (e.g., FPGA-based) resources to be shared (pipelined) among these channels, at the expense of only a few FPGA clock cycles additional latency.

[0074] Figure 4 is a flow diagram showing the basic arrangement of Figure 3, in the context of determining and monitoring a monitor carrier delay r$%1using a monitor carrier delay tracker MDT. A (delayed) monitor signal ^$%1^^^ is obtained, via monitor analog to digital converter Mon ADC and buffer Bf, having been measured by a monitor photodetector (not shown). Each of the monitor ADC, the buffer and the signal carriers may contribute to the unknown delay of the monitor signal ^$%1^^^. During normal operation, the monitor signal ^$%1^^^comprises only a (large) DC component and small component at the fundamental modulation frequency. The fundamental component is due to power modulation only. To obtain a monitor carrier delay residual r̃$%1− r$%1, it is proposed to demodulate the fundamental component (n=1) in the monitor photodetector signal.

[0075] From the viewpoint of delay residual construction, it is unhelpful that the fundamental frequency component is very small compared to the DC component. In the demodulation stage the largeDC component will first be frequency-shifted to the negative (fundamental) modulation frequency−^$%&, and then attenuated by the low-pass filter LPF.

[0076] In an embodiment, the low-pass filter in the demodulation stage may comprise a symmetric FIR decimation filter (e.g., comprising 512 taps), having a frequency response which exhibits a numberof notches at a regular frequency-pitch. To obtain sufficient rejection (of the frequency-shifted DC component) the modulation frequency ^$%&(as well as its low-order harmonics) may be carefully chosen to coincide with the notch frequencies of the decimation filter. This can be accomplished in the digital synthesizer hardware.

[0077] As mentioned in the foregoing, the complex demodulate is rotated by a monitor power-carrier delay phasor ?@∙hijk∙^^ijn(in each of the figures, a generic r is used in the argument for each respective phasor, the actual delay is clear in context) having an argument proportional to the monitor power- carrier delay prediction, so as to bring the demodulate to or near the real axis (i.e., to bring into phase- alignment with the fundamental frequency component). The result is a monitor power-carrier delay residual phasor proportional to ?@∙hijk∙^^^ijn;^ijn^, having an argument that is proportional to the monitor power-carrier

[0078] Figure 5 is flow diagram of a proposed method for performing a calibration of gas cell excess delay r_`,^^= r_`− r$%1. This calibration may be performed as a one-time calibration. The method and drawing builds upon and uses the monitor carrier delay prediction r̃$%1described in relation to Figure 4. Such a method may yield a gas cell excess delay estimate r̃_`,^^.

[0079] This calibration requires the laser wavelength to be kept somewhere in the relatively wide regime between two neighboring absorption lines of the wavelength reference. In such a regime the gas cell transmission has very little wavelength sensitivity. Therefore the optical frequency modulation is not a factor and the modulation of the gas cell output power is determined entirely from the input power modulation at the fundamental modulation frequency. Maintaining the mean laser wavelength in a region between absorption lines makes the gas cell carrier delay estimation incompatible with normal operation. Hence this delay cannot be tracked continuously. Fortunately, continuous tracking is not required, as this excess delay in only due to the excess optical path length in the gas cell channel, which does not significantly vary with environmental influences. A one-time calibration at startup is therefore sufficient.

[0080] A calibration wavelength reference signal or calibration gas cell signal ^_`^^^^ is obtained, via gas cell analog to digital converter GC ADC and buffer Bf (and is delayed as a consequence), having been measured by a gas cell photodetector (not shown) when the mean laser wavelength is in a region between absorption lines. To obtain a gas cell carrier delay residual r̃_`− r_`, it is proposed to demodulate the fundamental frequency component (n=1) in the gas cell photodetector signal ^_`^^^ (although any odd-order frequency component may be used). While this calibration is in progress, the power normalization of the gas cell signal may be temporarily suspended. The complex demodulate is rotated by a phasor prediction ?@∙hijk∙+^^ijnH^^^^,WX.to bring it near the real axis. The result is a phasor proportional to: ?@∙hijk∙+^^^^,WX;^^^,WXH^^ijn;^ijn., having an argument which is proportional to the sum of gas cell and monitor power-carrier delay residual r̃$%1− r$.%1Since the monitor power-carrier delay residual r̃$%1− r$%1is already being controlled to zero by themonitor power-carrier delay tracker MDT (as already described), substantially only the gas cell excess delay residual r̃_`,^^− r_`,^^remains. A gas cell excess delay tracker / estimator GCDE acts to controlthis residual to zero (in this calibration phase only), so as to calibrate the gas cell excess delay estimater̃_`,^^.

[0081] Figure 6 is a flow diagram which describes how the aforementioned delays may be used to perform: (1) delay-compensated gas cell power normalization using a fractionally delayed FDF reciprocal monitor signal, (2) laser pseudo-delay tracking based on the second harmonic component in the signal from the locked gas cell, and (3) laser mean wavelength control based on the delay- compensated fundamental frequency component of the gas cell signal.

[0082] The gas cell excess delay tracker and its residual determination loop is no longer part of the scheme since it is not available in this operating mode; instead the calibrated gas cell excess delay estimate r̃_`,^^is used as a static value. Once the gas cell excess delay r_`,^^has been calibrated, thegas cell power normalization can be resumed (gas cell power-carrier delay prediction r̃_` = r̃_`,^^ +r̃$%1, will now be accurate for the first time, as the gas cell excess delay r_`,^^ is now accuratelyestimated) and lock on an absorption line can be achieved (e.g., by slowly scanning the laser temperature). Successful lock may be detected from a sufficiently strong second harmonic amplitude in combination with a vanishing fundamental frequency component.

[0083] When locked to the center of an absorption line, the normalized gas cell signal no longer has a fundamental frequency (or other odd-order harmonic) component. Hence it is proposed that the laser pseudo-delay residual r̃t^^^$%&^ − rt^^^$%&^ is constructed from even-order (e.g., second order) harmonic components. For the small (relative to the absorption linewidth) frequency excursion amplitudes ^̂^^^that are of interest, the second-order harmonic dominates. This second-order harmonicis demodulated by a complex demodulation stage, which involves multiplication by a phasor?;@∙5∙hijk∙B (e.g., as generated by complex multi-carrier synthesizer CMCS), followed by low-passfiltering LPF, e.g., by a FIR decimation filter identical to that used in the fundamental-frequency complex demodulation stages.

[0084] Rotating the 2nd-order complex demodulate by a phasor prediction: ?@∙5∙hijk∙M^^ijnH^^^^,WXH^^^^^hijk^Pshould bring it near the real axis, this phasor having an argument proportional to the predicted gas cell wavelength-carrier delay r̃_`+ r̃t^^^$%&^ (noting that r_`= r_`,^^+ r$%1). The result is a phasor proportional to: ?@∙5∙hijk∙+^^^^^hijk^;^^^^hijk^H^^^^,WX;^^^,WXH^^ijn;^ijn.having an argument which is proportional to the sum of laser pseudo-delay residual r̃t^^^$%&^ −rt^^^$%&^, gas cell excess delay residual r̃_`,^^− r_`,^^and monitor power-carrier delay residualr̃$%1 − r$%1. Since the monitor power-carrier delay residual is being controlled to zero by monitorcarrier delay tracker MDT and the gas cell excess delay residual has already been calibrated to zero, only the laser pseudo-delay residual r̃t^^^$%&^− rt^^^$%&^remains. This residual may be controlled to zero via a laser pseudo-delay residual tracker / estimator LDpD, so as to determine a laser pseudo-delay estimate r̃t^^^$%&^.

[0085] Using this approach, it is possible to correctly rotate the fundamental frequency component of the normalized gas cell signal, even when it is controlled to zero in order to lock the mean laser frequency to the center of an absorption line. The monitor power-carrier delay estimate, the gas cell excess delay estimate and the laser pseudo-delay estimate may be combined to construct a (predicted) gas cell wavelength-carrier delay phasor ?@∙hijk∙+^^^^^hijk^H^^^^,WXH^^ijn.based on the gas cell wavelength-carrier delay estimate; i.e., the combination of the gas cell power-carrier delay estimate and the estimated laser pseudo-delay. The complex demodulate at the fundamental frequency can therefore be rotated by this gas cell wavelength-carrier delay phasor ?@∙hijk∙+^^^^^hijk^H^^^^,WXH^^ijn.. This stabilizes the sensitivity of the fundamental frequency complex demodulate component to deviations of the mean wavelength from the absorption line center, allowing it to be used as a residual for the control of the mean wavelength, e.g., via laser bias control LBC so as to control the laser bias current BC.

[0086] Figure 7 is a flow diagram describing a further embodiment, in which the measured and delayed interferometer signal is now considered. As mentioned above, a cause of carrier phase deviation is the unknown and / or time-varying propagation delay of the radiation as it travels from the laser, along the two beam paths comprising the interferometer, to the photodetector.

[0087] The delayed interferometer photodetector signal ^^^^^^^ can be modeled as: ^^^^^^^ = ^ + ! ∙ cos+^^^^ + ^ ∙ sin +^$%& ∙ ^^ − r^..where the delay r an to the gas cellwavelength-carrier delay above in that it comprises the sum of interferometer power-carrier delayr^^^ and the laser pseudo delay rt^^^$%&^. Also, analogously to the gas cell excess delay, theinterferometer excess delay r^^^,^^may be defined as the difference of the interferometer power-carrier delay r^^^and the monitor power-carrier delay r$%1; i.e., r^^^,^^= r^^^− r$%1. As such, the interferometer wavelength-carrier delay r may be determined (e.g., per interferometer) from the sum of: the interferometer excess delay r^^^,^^, the monitor power-carrier delay r$%1and the laser pseudo- delay rt^^^$%&^; e.g., r = r^^^,^^+ r$%1+ rt^^^$%&^. This interferometer wavelength-carrier delay describes a propagation delay of the carrier signal used for modulating the laser radiation wavelength.

[0088] The corresponding expansion of the interferometer signal into phase generated carriers is: 7 ^^^^^^^ = ^ + ! ∙ cos ^^^^ ∙ 302^^^ + 24056^^^ ∙ cos +2, ∙ ^$%&∙ ^^ − r^.:

[0089] In excess may In an embodiment, as the interferometer excess delay estimate r̃^^^,^^is largely static, it may be estimated in a (e.g., one-time) calibration step, and the calibrated value used in combination with the (e.g., continuously tracked) monitor power-carrier delay and laser pseudo-delay to determine the interferometer wavelength carrier delay.

[0090] Once calibrated, this interferometer excess delay estimate may be combined with the monitor power-carrier delay estimate and the laser pseudo-delay estimate to determine an interferometer wavelength-carrier delay prediction. This interferometer wavelength-carrier delay prediction can then be used to correct an estimated phase of interest value determined from the respective interferometer signal, e.g., by rotating partial demodulation components of the interferometer signal by a respective angle determined from said interferometer wavelength-carrier delay prediction, to place the partial demodulation components onto the real axis (i.e., to phase-align the partial demodulation components with respective carrier signals in the interferometer signal).

[0091] The interferometer excess delay estimate r̃^^^,^^may also be used to delay the (e.g., reciprocal) monitor signal prior to using the (e.g., reciprocal) monitor signal to power-normalize the interferometer signal.

[0092] In an embodiment, the interferometer excess delay estimate may be determined based on the construction of an interferometer wavelength-carrier delay residual ^r = ^ r̃ − r^ obtained from a weighted combination of the imaginary parts of the rotated partial demodulation components: ?@∙5∙hijk∙M+^^^^^hijk^;^^^^hijk^.H+^^^^[,WX;^^^[,WX.H^^^ijn;^ijn^Pwhich has an argument which is proportional to the sum of laser pseudo-delay residual r̃t^^^$%&^ −rt^^^$%&^, interferometer excess delay residual r̃^^^,^^ − r^^^,^^ and monitor power-carrier delayresidual r̃$%1− r$%1. Since the monitor power- delay residual is being controlled to zero bymonitor power-carrier delay tracker MDT and the laser pseudo-delay residual r̃t^^^$%&^ −rt^^^$%&^ is being controlled to zero via the laser pseudo-delay residual tracker / estimator LDpD, onlythe interferometer excess delay residual r̃^^^,^^− r^^^,^^remains; this interferometer excess delayresidual may be nullified by an interferometer excess delay tracker IFM EDT to determine interferometer excess delay r̃^^^,^^

[0093] In this embodiment, the (real) carriers in the photodetector signal may be demodulated using complex demodulation carriers or complex demodulation phasors ?;@56∙hijk∙B, −I ∙ ?;@^56;9^∙hijk∙Brather than by real demodulation carriers (as per the conventional example of Figure 2), which have to be phase-shifted. This results in a two complex partial demodulation components or demodulates (rather than a single real demodulate) per carrier. These two complex partial demodulates are denoted by DRn and DIn, where partial demodulate DRn may comprise a (e.g., low-pass filtered) combination or product of the complex demodulation phasors with the detected signal ^^^^^^^and the real part of the complex conjugate phasor prediction ?;@AC ^B^, and partial demodulate DRn may comprise a (e.g., low-pass filtered) combination or product ofdemodulation phasors with the detected signal ^^^^^^^ and the imaginary part of the complex conjugate phasor prediction ?;@AC ^B^. The complexityfrom the complex conjugate phasor prediction is temporarilyseparating the complex conjugate phasor prediction into its constituent two real expressions. A new complexity is now introduced for the purpose of describing the operation of demodulating the detected signal ^^^^^^^ by the complex demodulation phasors ?;@56∙hijk∙B, −I ∙ ?;@^56;9^∙hijk∙B. The partial demodulates of even-order and odd-order are thus formed according to: D^56 = 〈^^^^^^^ ∙ Re^?;@AC ^B^^ ∙ ?;@56∙hijk∙B〉

[0094] τ- ^^^^^^^ yields: D^56≈ ! ∙12 Re^〈?@+ A^B^;AC ^B^ .+ ?;@+ A^B^HAC ^B^ .〉^ ∙ ?;@56∙hijk∙^∙ 056^^^

[0095] This shows that these partial demodulates are oriented in the complex plane at respective angles−2, ∙ ^$%& ∙ r and −^2, − 1^ ∙ ^$%& ∙ r with respect to the real axis. It is proposed herein to rotatethese partial demodulates over respective angles 2, ∙ ^$%&∙ r̃ and^2, − 1^∙ ^$%&∙ r̃ onto the realaxis by multiplication with the respective delay correction phasors ?@56∙hijk∙^^and ?@^56;9^∙hijk∙^^, both of which may be constructed from the interferometer wavelength-carrier delay prediction r̃ (as determined from the sum of the interferometer excess delay estimate r̃^^^,^^, the monitor power-carrier delay estimate r̃$%1and the laser pseudo-delay estimate r̃t^^^$%&^. The rotation operation and its result are given by: D^56∙ ?@56∙hijk∙^^≈ ! ∙95 Re^〈?@+ A^B^;AC ^B^ .+ ?;@+ A^B^HAC ^B^ .〉^ ∙ ?;@56∙hijk∙^^;^^^∙ 056^^^accurate, demodulates will be on the real axis. Hence the full demodulates D1(including their complexity) may be obtained by combining the real parts of the rotated partial demodulates. This may be achieved, for example, in the following manner: D56 = Re^D^56 ∙ ?@56∙hijk∙^^^ + I ∙ Re^D^56 ∙ ?@56∙hijk∙^^ ^^

[0097] This to D56= ! ∙95 ∙ 056^^^ ∙ 〈?@+ A^B^;AC ^B^ .+ ?;@+ A^B^HAC ^B^ .〉 ∙ cos+2, ∙ ^$%&∙ ^r − r̃^.to except that the strength reduction factors are now functions of the interferometer wavelength-carrier delay residual rather than the true interferometer wavelength-carrier delay. Hence, provided that an accurate interferometer wavelength-carrier delay prediction r̃ is maintained, the strength reduction can be eliminated, avoiding operation failure and cyclic errors due to the delay.

[0098] When the interferometer wavelength-carrier delay prediction is accurate the rotated partial demodulates will be on the real axis of the complex plane. Hence non-zero imaginary parts of the rotated partial demodulates are an indication of an inaccurate delay prediction.

[0099] A method for constructing an interferometer wavelength-carrier delay residual ^r =^r̃ − r^from the imaginary parts of the rotated partial demodulates will now be described. The imaginary parts may be combined similarly to how the real parts were combined above when determining the full demodulates D1, e.g.: Im^D^56 ∙ ?@56∙hijk∙^^^ + I ∙ Im^D^56 ∙ ?@56∙hijk∙^^^are to residual r̃ - r, scaled by the modulation frequency ^$%&and the PGC order 2k, respectively 2k-1. For small values of the interferometer wavelength-carrier delay residual, these sine functions can be approximated by their respective arguments. Furthermore these combinations contain the complex phasor factor 〈?@+A;AC .− ?;@+AHAC .〉, which they have in common with the demodulates D2kand D2k-1. Hence from these combinations it is possible to construct (e.g., by a weighted addition) a phasor of the form^r̃ − r^∙ ^^^^∙ ! ∙ ?@+A;AC ., which in turn allows the (scaled) interferometer wavelength-carrier delay obtained by multiplying that phasor by the conjugate (& normalized)C / N phase-residual phasor according to: ;@+A;AC .^r̃ − r^ ∙ ^^^^=M^r̃ − r^ ∙ ^^^^∙ ∙ @+A;AC .P∙^! ∙ ?^

[0100] ^ − r^ = ^r, a phasor of the required form may be obtained by constructing a first weighted combination or first weighted addition of the imaginary parts of the rotated partial demodulates according to: ^r̃ − r^ ∙ ^^^^ ∙ ! ∙ ?@+A;AC .≈where a set of first weighting coefficients ^1^^ ^, dependent on the modulation index estimate ^  , may be defined. Requirements on the first weighting coefficients ^1^^  ^ can be found by substituting the evaluated form of the combinations and replacing the sine functions therein by their arguments. This gives: ^r̃ − r^ ∙ ^^^^ ∙ ! ∙ ?@+A;AC .≈

[0101] Since this has to hold for all possible values of the phase-residual ^<− ^, then the following should be met: ^  = ^¡ 4 ^56^^^ ∙ 2, ∙ 056^^^ = 1 1

[0102] The latter two of on the even-order first weighting coefficients and the odd-order first weighting coefficients. For practical configurations, where N ≥ 2 even-order and N ≥ 2 odd-order PGCs are taken into account, this leaves at least one degree of freedom in choosing the coefficients. This freedom can be used to optimize the noise performance of the interferometer wavelength-carrier delay residual construction. For the case where the noise in the photodetector signal S(t) is white, the optimal noise performance is obtained when the vectors: ^5^^^¢^£^1^^^ = 3 ⋮ : :are parallel to the respective vectors:2 ∙ 05^^^ 09^^^3 ⋮ , 3 ⋮

[0103] This is the case are according to: 2, ∙ 056^^^ ^2, − 1^ ∙ 056; ^^^^ ^^^ ^ ^ 956 =5^56;9 ^ =

[0104] In an K ^^¨§^ can for a modulation index setpoint msp (and stored as design constants on the FPGA) . As such, the selection of the first weighting coefficients K may be based on the setpoint value for the modulation index ^¨§, e.g., the value to which the modulation index is controlled and therefore is assumed to be. In such an embodiment, reference to the modulation index in the above equations should be read as the setpoint value for the modulation index ^¨§. The lookup table LUT2 may also comprise the Bessel functions01^^^ (or second weighting coefficients ©1^^^ in the embodiment below).

[0105] In another embodiment, rather than controlling the modulation index to a predetermined value, each measurement axis may have a respective modulation index which is allowed to vary individually over time as the OPD of its axis varies (due to changing measurand). Each of these modulation indices are then estimated, with the estimated value for each axis used to demodulate a respective measured signal from that axis. In such an embodiment, reference to the modulation index in the above equations may be read as the estimated modulation index ^  .

[0106] The estimated modulation index, denoted by ^  , can then be used to define (e.g., via a second weighting coefficients look-up table, which may be the same as first weighting coefficients look-up table e.g., LUT2) a set of second weighting coefficients ©9^^  ^, ©5^^  ^, ©ª^^ ^, ©o^^ ^ which are expected to equalize the corresponding demodulation carrier amplitudes in such a way as to obtain a constant-magnitude phase residual first phasor ! ∙ ?@+ A;AC ., and therefore the phase residual ^<− ^.

[0107] In an embodiment, a second weighted or summation of demodulates may comprise:% ( G m % j ⋅ J m + % ( φ − φ ) NN2k( )2k( )G 2k− 1(m)⋅ J 2k− 1(m) )⋅ e

[0108] An example for the case where the noise spectral density of the signal ^^^^ is white, the demodulates D1comprise independent additive noises of equal variance. This means that the root-mean-square (rms) noise in the weighted sum can be calculated as the rms noise in any of the demodulates, amplified by the respective factors; e.g.: N N ^ G22k ( m%) , ^ G22k −1( m%) ,

[0109] These factors can be recognized as the lengths of the respective vectors: ^ G 2 ( m % ) ^ ^ G 1 ( m % ) ^ ^ ^ ^ ^ ^ ^ ^ .

[0110] Therefore, minimal noise amplification is achieved for the smallest vectors «^£^1and «%&&that satisfy the constraints given above. These constraints in turn can be interpreted as dot-products with the respective vectors of even-order and odd-order Bessel functions: ^ J 2 ( m % ) ^ ^ J 1 ( m % ) ^ ^ ^ ^ ^ ^ ^ ^

[0111] Thus the smallest vectors «^£^1and «%&&are sought which will make a unity dot product with the respective vectors ¬^£^1and ¬%&&. In other words, the vectors «^£^1and «%&&should be chosen parallel to the vectors ¬^£^1and ¬%&&, and should be normalized to their squared length: J m % J ( m % ) G m % = even( )% odd even,G odd m =2.

[0112] Therefore noise-optimal coefficients ©56^^ ^, ©56;9^^ ^may be found from: J 2 k m % J 2 k − 1 m % G 2k(m %)=N( ), G 2k− 1(m %)=N( )) .

[0113] These weighting coefficients can be pre-calculated for an expected range of the estimated modulation index and stored in lookup tables LUT2 (e.g., in an FPGA comprising the algorithm embodying the proposed method). If the noise in the detected signal ^^^^ is known to have a particular non-white spectral density, the above calculation of the coefficients can be modified by first applying noise-whitening virtual gains to each of the demodulates, after which the calculation of the G coefficients is conceptually the same. The stored coefficients are then determined as the products of the respective noise-whitening gains and the corresponding G coefficients, calculated after applying the noise whitening gains to the demodulates.

[0114] As already mentioned, the interferometer excess delay estimate may be used to delay the (e.g., reciprocal) monitor signal (e.g., using a fractional delay filter) in order to perform accurate instantaneous power normalization. When the interferometer excess delay calibration first commences, the power normalization will therefore not yet be accurate. This will thus also impact the accuracy of the wavelength-carrier delay residual construction, since each PGC demodulate will experience some ‘crosstalk’ from its neighbors. This effect may result in a (small) reduction of the convergence range. But as the interferometer excess delay estimate converges, this crosstalk will get increasingly smaller, leaving no impact on the final accuracy of the interferometer excess delay.

[0115] The monitor power-carrier delay is the only absolute delay among the delay estimates that are maintained. The other delay estimates are relative delays. Hence the monitor delay is by far the greatest among the four delay contributions described above. It includes all the delays encountered by the fundamental frequency, beginning at the carrier synthesizer (e.g., on the FPGA), a laser signal FIFO buffer, the laser DAC, laser driver, laser, optical fiber path, photodetector and amplifier, ADC, a photodetector signal FIFO buffer and ending at the demodulator stage (e.g., back on the FPGA). This delay estimate serves as a common base to the delays seen in the other photodetector channels. This is a) because the number of sample delays in the digital signal processing stages of the different photodetector signals can easily be made identical, b) because the path from synthesizer to fiber splitter is common to all channels, c) the path from fiber splitter branch to the monitor photodetector is the shortest amongst all the optical channels.

[0116] To ensure rapid and reliable convergence of the delay estimators at startup of the system, the initial delay estimates may be preset to suitable nominal values (e.g., as known from design parameters and experience). However, depending on the initial uncertainty of the various delay items and on the used modulation frequency, the convergence range of the delay estimator may not be sufficiently large. The initial delay uncertainty should not exceed ± half the cycle duration of the used harmonic frequency. To extend the convergence range, the system may use a (much) lower modulation frequency at startup. Furthermore, as the laser pseudo-delay is not a pure delay, but rather a frequency-dependent delay, various intermediate modulation frequencies may also be used, in order to ensure a reliable reconvergence of the pseudo-delay estimate at the next higher modulation frequency.

[0117] Furthermore, because this laser pseudo delay is a (strong) function of the modulation frequency it has a large initial uncertainty. The convergence range (allowable initial delay uncertainty) of the present method amounts to + / - half a period of the second harmonic frequency. This large guaranteed convergence range means that the interferometer excess delay calibration can be done anywhere within the operating range of the interferometer, because its convergence range needs to accommodate the initial uncertainties of only the (fiber-length related) interferometer excess delay and not those of the laser pseudo delay.

[0118] The proposed signal processing method (e.g., an algorithm embodying the method and associated look-up tables) may be implemented in software or in firmware (e.g., in a programmable firmware component such as a Field Programmable Gate Array FPGA).

[0119] Also disclosed is a phase generated carrier interferometer interrogator comprising at least one interferometer, being operable to perform any of the methods disclosed herein to process (e.g., demodulate) the detected interferometer signal. The phase generated carrier interferometer interrogator may comprise, for example, a multi-interferometer or multi-axis phase generated carrier interferometer interrogator which uses a common laser (e.g., laser diode) for each interferometer / axis.

[0120] Also disclosed is a position measurement system for measuring the position of a component of a machine, such as a lithographic apparatus (e.g., a scanner) comprising at least one interferometer, and being operable to perform any of the methods disclosed herein to process (e.g., demodulate) the detected interferometer signal. The position measurement system may comprise a (e.g., multi-axis) phase generated carrier interferometer interrogator as described. The component may comprise, for example, a stage (e.g., wafer stage or reticle stage) of the lithographic apparatus, or any other component such as a projection component (e.g., steering or 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 relating to a respective degree of freedom of the component being measured. In a specific example, position measurement system may be operable to measure six axes, the three spatial axes x, y, z and rotations Rx, Ry, Rz around each of these axes.

[0121] Also disclosed is a lithographic apparatus comprising at least one such position measurement system for measuring a position of a component of the lithographic apparatus.

[0122] Although specific reference may be made in this text to the use of a lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications. Possible other applications include the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat-panel displays, liquid- crystal displays (LCDs), thin-film magnetic heads, etc.

[0123] Although specific reference may be made in this text to embodiments of the invention in the context of a lithographic apparatus, embodiments of the invention may be used in other apparatus. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus that measures or processes an object such as a wafer (or other substrate) or mask (orother patterning device). These apparatus may be generally referred to as lithographic tools. Such a lithographic tool may use vacuum conditions or ambient (non-vacuum) conditions.

[0124] Although specific reference may have been made above to the use of embodiments of the invention in the context of optical lithography, it will be appreciated that the invention, where the context allows, is not limited to optical lithography and may be used in other applications, for example imprint lithography.

[0125] Where the context allows, embodiments of the invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the invention may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine-readable medium may include read only memory (ROM); random access memory (RAM); magnetic storage media; optical storage media; flash memory devices; electrical, optical, acoustical or other forms of propagated signals (e.g. carrier waves, infrared signals, digital signals, etc.), and others. Further, firmware, software, routines, instructions may be described herein as performing certain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc. and in doing that may cause actuators or other devices to interact with the physical world.

[0126] While specific embodiments of the invention have been described above, it will be appreciated that the invention may be practiced otherwise than as described. The descriptions above are intended to be illustrative, not limiting. Thus it will be apparent to one skilled in the art that modifications may be made to the invention as described without departing from the scope of the claims set out below. Other aspects of the invention are set-out as in the following numbered clauses. 1. A phase generated carrier interrogator, comprising: a complex multi-carrier synthesizer being operable to generate at least a laser modulation carrier signal for modulating a laser radiation source; a laser power monitor branch comprising a power monitor detector and being operable to measure a power metric of the laser radiation source to obtain a power monitor signal; a wavelength reference branch comprising a wavelength reference for wavelength stabilization and / or control of the laser radiation source, and a wavelength reference detector for measuring said wavelength reference to obtain a wavelength reference signal; and a signal processing module being operable to: estimate a monitor power-carrier delay describing a propagation delay of the power monitor signal in the laser power monitor signal path; and estimate, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrierdelay, the wavelength reference power-carrier delay describing a propagation delay of the wavelength reference signal in the wavelength reference branch. 2. A phase generated carrier interrogator as stated in clause 1, wherein said signal processing module is further operable to: complex demodulate a fundamental frequency component in the power monitor signal; determine a monitor power-carrier delay residual describing an error in the estimated monitor power- carrier delay, from the complex demodulated fundamental frequency component; and continually track said monitor power-carrier delay based on the monitor power-carrier delay residual. 3. A phase generated carrier interrogator as stated in clause 2, wherein said signal processing module is operable to determine the monitor power-carrier delay residual by rotating the complex demodulated fundamental-frequency component of the power monitor signal by an angle proportional to the monitor power-carrier delay estimate as determined either in a previous iteration or from an initial estimate. 4. A phase generated carrier interrogator as stated in any preceding clause, wherein said signal processing module is operable to estimate a wavelength reference excess delay in a first calibration, the first calibration comprising: obtaining a calibration wavelength reference signal while a wavelength of the laser radiation is maintained between two neighboring absorption lines of the wavelength reference; complex demodulating an odd-order harmonic frequency component of the calibration wavelength reference signal; and nullifying a wavelength reference excess delay residual determined from the complex demodulate of the odd-order harmonic frequency component of the calibration wavelength reference signal, the wavelength reference excess delay residual describing an error in the estimate of the wavelength reference excess delay. 5. A phase generated carrier interrogator as stated in clause 4, wherein the odd-order harmonic frequency component of the calibration wavelength reference signal comprises a fundamental frequency component of the calibration wavelength reference signal. 6. A phase generated carrier interrogator as stated in clause 4 or 5, wherein said signal processing module is operable to determine the wavelength reference excess delay residual by rotating the complex demodulate of the odd-order harmonic frequency component of the calibration wavelength reference signal by an angle proportional to the sum of the estimated monitor power- carrier delay and the estimated wavelength reference excess delay as determined either in a previous iteration or from an initial estimate. 7. A phase generated carrier interrogator as stated in any preceding clause, wherein said signal processing module is operable to: delay the power monitor signal or a reciprocal thereof by the determined wavelength reference excess carrier delay to obtain a first delayed power monitor signal; andpower-normalize the wavelength reference signal using the first delayed power monitor signal. 8. A phase generated carrier interrogator as stated in clause, wherein said signal processing module is operable to: estimate and / or continuously monitor a laser pseudo-delay while locked to an absorption line of the wavelength reference, the laser pseudo-delay describing a phase delay of the laser wavelength modulation carrier relative to the associated laser power modulation carrier and resulting from a drift of a thermal time constant of the laser. 9. A phase generated carrier interrogator as stated in clause 8, wherein said signal processing module is operable to perform said estimating and / or continuously monitoring a laser pseudo-delay by: complex demodulating an even-order harmonic frequency component of the wavelength reference signal; and nullifying a laser pseudo-delay residual determined from the even-order harmonic frequency component of the wavelength reference signal, the laser pseudo-delay residual describing an error in the estimate of the laser pseudo-delay. 10. A phase generated carrier interrogator as stated in clause 9, wherein said signal processing module is operable to determine the laser pseudo-delay residual by rotating the complex demodulate of the even-order harmonic frequency component of the wavelength reference signal by an angle proportional to the sum of the estimated monitor power-carrier delay, the estimated wavelength reference excess delay and the estimated laser pseudo-delay determined either in a previous iteration or from an initial estimate. 11. A phase generated carrier interrogator as stated in clause 9 or 10, wherein said even-order harmonic frequency component of the wavelength reference signal comprises the second harmonic frequency component of the wavelength reference signal. 12. A phase generated carrier interrogator as stated in any of clauses 8 to 11, wherein said signal processing module is operable to: estimate a wavelength reference wavelength-carrier delay as a sum of the estimated laser pseudo- delay, the estimated monitor power-carrier delay and the estimated wavelength reference excess delay, while locked to an absorption line of the wavelength reference. 13. A phase generated carrier interrogator as stated in clause 12, wherein said signal processing module is operable to: complex demodulate an odd-order harmonic frequency component of the wavelength reference signal; rotating the complex demodulate of the odd-order harmonic frequency component by said wavelength reference wavelength-carrier delay estimate; and using a real part of the rotated complex demodulate of the odd-order harmonic frequency component to control a mean wavelength of the laser radiation source.14. A phase generated carrier interrogator as stated in clause 13, wherein said using the real part of the rotated complex demodulate of the odd-order harmonic frequency component to control the mean wavelength of the laser radiation source comprises using the real part of the rotated complex demodulate of the odd-order harmonic frequency component to control a bias current of said laser radiation source. 15. A phase generated carrier interrogator as stated in clause 13 or 14, wherein said using the real part of the rotated complex demodulate of the odd-order harmonic frequency component to control the mean wavelength of the laser radiation source comprises controlling the laser radiation source so as to nullify said real part of the rotated complex demodulate of the odd-order harmonic frequency component. 16. A phase generated carrier interrogator as stated in clause 13, 14 or 15, wherein the odd- order harmonic frequency component comprises a fundamental frequency component. 17. A phase generated carrier interrogator as stated in any of clauses 8 to 16, further comprising: one or more interferometers, each said one or more interferometers being operable to receive modulated radiation from said laser radiation source and to generate a respective interferometer signal comprising a phase of interest induced by a measurand, each said interferometer signal also comprising a periodic phase modulation induced by said modulated radiation; and at least one detector operable to detect each said interferometer signal; wherein said complex multi-carrier synthesizer is operable to generate complex demodulation phasors for each said interferometer signal; and said signal processing module is operable, respectively for each said one or more interferometers, to: determine at least two partial complex demodulates from the complex demodulation phasors and each said interferometer signal; and estimate an interferometer excess delay from said at least two partial complex demodulates, the interferometer excess delay describing a difference of an interferometer power-carrier delay and the monitor power-carrier delay. 18. A phase generated carrier interrogator as stated in clause 17, wherein said signal processing module is operable to: delay the power monitor signal or reciprocal thereof by the determined interferometer excess delay to obtain a second delayed power monitor signal; and power-normalize each said interferometer signal using the second delayed power monitor signal prior to said demodulation step. 19. A phase generated carrier interrogator as stated in clause 17 or 18, wherein said signal processing module is operable to: predict an interferometer wavelength-carrier delay from a sum of the estimated interferometer excess delay, the estimated monitor power-carrier delay and the estimated laser pseudo-delay; andcorrect, using the predicted interferometer wavelength-carrier delay, the at least two partial complex demodulates corresponding to the respective interferometer signal to obtain corrected partial complex demodulates. 20. A phase generated carrier interrogator as stated in clause 19, wherein said correcting step comprises rotating each of said at least two partial complex demodulates by an angle proportional to said interferometer wavelength-carrier delay prediction. 21. A phase generated carrier interrogator as stated in any of clause 19 to 20, wherein said signal processing module is operable to estimate said interferometer excess delay based on an interferometer excess delay residual determined from a first weighted combination of imaginary parts of said corrected partial complex demodulates, the first weighted combination comprising weighting by a set of first weighting coefficients, each of said first weighting coefficients being dependent on a respective modulation index estimate for each said one or more interferometers, the modulation index describing an amplitude of the periodic phase modulation. 22. A phase generated carrier interrogator as stated in clause 21, wherein said interferometer excess delay is estimated in a second calibration. 23. A phase generated carrier interrogator as stated in clause 22, wherein said first weighted combination of said imaginary parts of said corrected partial complex demodulates comprises a first weighted sum of said imaginary parts of said corrected partial complex demodulates. 24. A phase generated carrier interrogator as stated in any of clause 21 to 23, wherein said signal processing module is operable, for each of said one or more interferometers, to: determine full demodulates from real parts of said corrected partial complex demodulates; and estimate said phase of interest from said full demodulates., 25. A phase generated carrier interrogator as stated in clause 24, wherein the signal processing module is further operable to: estimate the modulation index respectively for each said one or more interferometers to obtain the respective estimated modulation index for each said one or more interferometers; determine said first weighting coefficients from the respective estimated modulation index for each said one or more interferometers. 26. A phase generated carrier interrogator as stated in clause 25, wherein said signal processing module is operable, for each of said one or more interferometers, to use the respective estimated modulation index for that interferometer to define a set of second weighting coefficients for said full demodulates, said second weighting coefficients being defined such that a second weighted combination of said full demodulates, comprising weighting by said second weighting coefficients, yields a phase residual phasor comprising in its argument, a phase residual representing an error in the estimated phase of interest value. 27. A phase generated carrier interrogator as stated in clause 25 or 26, wherein said signal processing module is operable to estimate the modulation index respectively for each said one or moreinterferometers by iteratively updating the estimated modulation index based on a modulation index residual representing an error in said estimated modulation index. 28. A phase generated carrier interrogator as stated in any of clauses 19 to 27, wherein said signal processing module is operable to determine a phase residual representing an error in the estimated phase of interest value; and said signal processing module further comprises a dynamic state estimator being operable to determine a new state comprising said estimated phase of interest value from a previous state and from said phase residual. 29. A phase generated carrier interrogator as stated in any preceding clause, comprising a programmable firmware component programmed to implement said signal processing module 30. A phase generated carrier interrogator as stated in any preceding clause, further comprising a plurality of said interferometers. 31. A phase generated carrier interrogator as stated in clause 30, further comprising a common radiation source for generating radiation to be modulated to obtain said modulated radiation for each of said plurality of said interferometers. 32. A multi-axis position measuring system comprising the phase generated carrier interrogator of clause 30 or 31, wherein each of said plurality of interferometers is used to measure a respective axis of said multi-axis position measuring system. 33. A multi-axis position measuring system as stated in clause 32, wherein said multi-axis position measuring system comprises at least three axes for measuring three rigid-body position coordinates related to the measurand. 34. A multi-axis position measuring system as stated in clause 32, wherein said multi-axis position measuring system comprises at least six axes for measuring three rigid-body position coordinates and three rigid-body attitude coordinates related to the measurand. 35. A lithographic apparatus comprising the multi-axis position measuring system of any of clauses 32 to 34, wherein said multi-axis position measuring system is operable to measure the position coordinates and / or attitude coordinates of a component of said lithographic apparatus. 36. A metrology apparatus comprising the multi-axis position measuring system of any of clauses 32 to 34, wherein said multi-axis position measuring system is operable to measure the position coordinates and / or attitude coordinates of a component of said metrology apparatus. 37. A method of estimating at least a wavelength reference excess delay relating to a wavelength reference, the method comprising: measuring a power metric of a laser radiation source to obtain a power monitor signal; stabilizing and / or controlling the wavelength of the laser radiation source using a wavelength reference signal from a wavelength reference; estimating a monitor power-carrier delay describing a propagation delay of the power monitor signal in a laser power monitor signal path; andestimating, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrier delay, the wavelength reference power-carrier delay describing a propagation delay of the wavelength reference signal in a wavelength reference branch. 38. A method as stated in clause 37, comprising: Complex demodulating a fundamental frequency component in the power monitor signal; determining a monitor power-carrier delay residual describing an error in the estimated monitor power-carrier delay, from the demodulated fundamental frequency component; and continually tracking said monitor power-carrier delay based on the monitor power-carrier delay residual. 39. A method as stated in clause 38, comprising determining the monitor power-carrier delay residual by rotating the complex demodulated fundamental frequency component by an angle proportional to the monitor power-carrier delay as determined either in a previous iteration or from an initial estimate. 40. A method as stated in any of clauses 37 to 39, comprising: estimating a wavelength reference excess delay in a first calibration, the first calibration comprising: obtaining a calibration wavelength reference signal while a wavelength of the laser radiation is maintained between two neighboring absorption lines of the wavelength reference; complex demodulating an odd-order harmonic frequency component of the calibration wavelength reference signal; and nullifying a wavelength reference excess delay residual determined from the complex demodulate of the odd-order harmonic frequency component of the calibration wavelength reference signal, the wavelength reference excess delay residual describing an error in the estimate of the wavelength reference excess delay. 41. A method as stated in clause 40, wherein the odd-order harmonic frequency component comprises a fundamental frequency component. 42. A method as stated in clause 40 or 41, comprising determining the wavelength reference excess delay residual by rotating the complex demodulate of the odd-order harmonic frequency component by an angle proportional to the sum of the estimated monitor power-carrier delay and the estimated wavelength reference excess delay as determined in either a previous iteration or from an initial estimate. 43. A method as stated in any of clauses 37 to 42, comprising delaying the power monitor signal or a reciprocal thereof by the determined wavelength reference excess carrier delay to obtain a first delayed power monitor signal; and power-normalizing the wavelength reference signal using the first delayed power monitor signal. 44. A method as stated in clause 43, comprising:estimating and / or continuously tracking a laser pseudo-delay while locked to an absorption line of the wavelength reference, the laser pseudo-delay describing a phase delay of the laser wavelength modulation carrier relative to the associated laser power modulation carrier and resulting from drift of a thermal time constant of the laser. 45. A method as stated in clause 44, comprising performing said estimating and / or continuously tracking a laser pseudo-delay by: complex demodulating an even-order harmonic frequency component of the power-normalized wavelength reference signal; and nullifying a laser pseudo-delay residual determined from the complex demodulate of the even-order harmonic frequency component, the laser pseudo-delay residual describing an error in the estimate of the laser pseudo-delay. 46. A method as stated in clause 45, comprising determining the laser pseudo-delay residual by rotating the complex demodulate of the even-order harmonic frequency component by an angle proportional to the sum of the estimated monitor power-carrier delay, the estimated wavelength reference excess delay and the estimated laser pseudo-delay determined either in a previous iteration or from an initial estimate. 47. A method as stated in clause 45 or 46, wherein said even-order harmonic frequency component comprises the second harmonic frequency component. 48. A method as stated in any of clauses 44 to 47, comprising: determining an wavelength reference wavelength-carrier delay as a sum of the estimated laser pseudo- delay, the estimated monitor power-carrier delay and the estimated wavelength reference excess delaying, while locked to an absorption line of the wavelength reference. 49. A method as stated in clause 48, comprising: complex demodulating an odd-order harmonic frequency component of the power-normalized wavelength reference signal; rotating the complex demodulate of the odd-order harmonic frequency component by said wavelength reference wavelength-carrier delay; and using a real part of the rotated complex demodulate to control the wavelength of the laser radiation source. 50. A method as stated in clause 49, wherein said using the real part of the rotated complex demodulate to control the wavelength of the laser radiation source comprises using the real part of the rotated complex demodulate to control a bias current of said laser radiation source. 51. A method as stated in clause 49 or 50, wherein said using the real part of the rotated complex demodulate to control the wavelength of the laser radiation source comprises controlling the laser radiation source so as to nullify said real part of the rotated complex demodulate. 52. A method as stated in clause 49, 50 or 51, wherein the odd-order harmonic frequency component comprises a fundamental frequency component.53. A method as stated in any of clauses 44 to 52, further comprising: generating a carrier signal; generating complex demodulation phasors; modulating radiation of said laser radiation source with said carrier signal to apply a periodic phase modulation to each of one or more interferometers; detecting a respective interferometer signal for each said one or more interferometers, each interferometer signal comprising a phase of interest induced by the measurand in addition to said periodic phase modulation; determining at least two partial complex demodulates from the complex demodulation phasors and said interferometer signal; and estimating an interferometer excess delay from said at least two partial demodulation components, the interferometer excess delay describing a difference of an interferometer power-carrier delay describing a propagation delay of the laser modulation power-carrier signal and the monitor power- carrier delay. 54. A method as stated in clause 53, comprising: delaying the power monitor signal or a reciprocal thereto by the determined interferometer excess delay to obtain a second delayed power monitor signal; and power-normalizing each said interferometer signal using the second delayed power monitor signal prior to said demodulation step. 55. A method as stated in clause 53 or 54, comprising: predicting an interferometer wavelength-carrier delay from a sum of the estimated interferometer excess delay, the estimated monitor power-carrier delay and the estimated laser pseudo-delay; and correcting, using the predicted interferometer wavelength-carrier delay, the at least two partial complex demodulates corresponding to the respective interferometer signal to obtain corrected partial complex demodulates. 56. A method as stated in clause 55, wherein said correcting step comprises rotating each of said at least two partial complex demodulates by an angle proportional to said interferometer wavelength- carrier delay prediction. 57. A method as stated in any of clauses 55 to 56, comprising estimating said interferometer excess delay based on an interferometer excess delay residual determined from a first weighted combination of imaginary parts of said corrected partial complex demodulates, the first weighted combination comprising weighting by a set of first weighting coefficients, each of said first weighting coefficients being dependent on a respective modulation index estimate for each said one or more interferometers, the modulation index describing an amplitude of the periodic phase modulation. 58. A method as stated in clause 57, wherein said interferometer excess delay is estimated in a second calibration.59. A method as stated in clause 58, wherein said first weighted combination of said imaginary parts of said corrected partial complex demodulates comprises a first weighted sum of said imaginary parts of said corrected partial complex demodulates. 60. A method as stated in any of clauses 57 to 59, comprising, for each of said one or more interferometers: determining full demodulates from the real parts of said corrected partial complex demodulates; and determine said estimated phase of interest from said full demodulates. 61. A method as stated in clause 60, comprising: estimating the modulation index respectively for each said one or more interferometers to obtain the respective estimated modulation index for each said one or more interferometers; determining said first weighting coefficients from the respective estimated modulation index for each said one or more interferometers. 62. A method as stated in clause 61, comprising, for each of said one or more interferometers, to use the respective estimated modulation index for that interferometer to define a set of second weighting coefficients for said full demodulates, said second weighting coefficients being defined such that a second weighted combination of said full demodulates, comprising weighting by said second weighting coefficients, yields a phase residual phasor comprising in its argument, a phase residual representing an error in the estimated phase of interest value. 63. A method as stated in clause 61 or 62, comprising estimating the modulation index respectively for each said one or more interferometers by iteratively updating the estimated modulation index based on a modulation index residual representing an error in said estimated modulation index. 64. A method as stated in any of clauses 55 to 63, comprising determining a phase residual representing an error in the estimated phase of interest value; and determining a new state comprising said estimated phase of interest value from a previous state and from said phase residual. 65. A method as stated in any of clauses 53 to 64, wherein the method comprises processing a respective interferometer signal from each of a plurality of said interferometers, the plurality of said interferometers using a common laser radiation source. 66. A method as stated in any of clauses 53 to 65, comprising determining a respective position value describing a position coordinate of a component of a machine for each estimated phase of interest value. 67. A method as stated in clause 66, wherein said machine comprises a lithographic apparatus. 68. A method as stated in clause 66, wherein said machine comprises a metrology apparatus. 69. A computer program comprising computer readable instructions operable to perform the method of any of clauses 38 to 66.70. A programmable firmware component programmed to implement the method of any of clauses 38 to 66.

Claims

CLAIMS 1. A phase generated carrier interrogator, comprising: a complex multi-carrier synthesizer being operable to generate at least a laser modulation carrier signal for modulating a laser radiation source; a laser power monitor branch comprising a power monitor detector and being operable to measure a power metric of the laser radiation source to obtain a power monitor signal; a wavelength reference branch comprising a wavelength reference for wavelength stabilization and / or control of the laser radiation source, and a wavelength reference detector for measuring said wavelength reference to obtain a wavelength reference signal; and a signal processing module being operable to: estimate a monitor power-carrier delay describing a propagation delay of the power monitor signal in the laser power monitor signal path; and estimate, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrier delay, the wavelength reference power-carrier delay describing a propagation delay of the wavelength reference signal in the wavelength reference branch.

2. A phase generated carrier interrogator as claimed in claim 1, wherein said signal processing module is further operable to: complex demodulate a fundamental frequency component in the power monitor signal; determine a monitor power-carrier delay residual describing an error in the estimated monitor power- carrier delay, from the complex demodulated fundamental frequency component; and continually track said monitor power-carrier delay based on the monitor power-carrier delay residual.

3. A phase generated carrier interrogator as claimed in claim 2, wherein said signal processing module is operable to determine the monitor power-carrier delay residual by rotating the complex demodulated fundamental-frequency component of the power monitor signal by an angle proportional to the monitor power-carrier delay estimate as determined either in a previous iteration or from an initial estimate.

4. A phase generated carrier interrogator as claimed in any preceding claim, wherein said signal processing module is operable to estimate a wavelength reference excess delay in a first calibration, the first calibration comprising: obtaining a calibration wavelength reference signal while a wavelength of the laser radiation is maintained between two neighboring absorption lines of the wavelength reference;complex demodulating an odd-order harmonic frequency component of the calibration wavelength reference signal; and nullifying a wavelength reference excess delay residual determined from the complex demodulate of the odd-order harmonic frequency component of the calibration wavelength reference signal, the wavelength reference excess delay residual describing an error in the estimate of the wavelength reference excess delay.

5. A phase generated carrier interrogator as claimed in claim 4, wherein the odd-order harmonic frequency component of the calibration wavelength reference signal comprises a fundamental frequency component of the calibration wavelength reference signal.

6. A phase generated carrier interrogator as claimed in claim 4 or 5, wherein said signal processing module is operable to determine the wavelength reference excess delay residual by rotating the complex demodulate of the odd-order harmonic frequency component of the calibration wavelength reference signal by an angle proportional to the sum of the estimated monitor power- carrier delay and the estimated wavelength reference excess delay as determined either in a previous iteration or from an initial estimate.

7. A phase generated carrier interrogator as claimed in any preceding claim, wherein said signal processing module is operable to: delay the power monitor signal or a reciprocal thereof by the determined wavelength reference excess carrier delay to obtain a first delayed power monitor signal; and power-normalize the wavelength reference signal using the first delayed power monitor signal.

8. A method of estimating at least a wavelength reference excess delay relating to a wavelength reference, the method comprising: measuring a power metric of a laser radiation source to obtain a power monitor signal; stabilizing and / or controlling the wavelength of the laser radiation source using a wavelength reference signal from a wavelength reference; estimating a monitor power-carrier delay describing a propagation delay of the power monitor signal in a laser power monitor signal path; and estimating, using the estimated monitor power-carrier delay, a wavelength reference excess delay describing a difference of a wavelength reference power-carrier delay and the monitor power-carrier delay, the wavelength reference power-carrier delay describing a propagation delay of the wavelength reference signal in a wavelength reference branch.

9. A method as claimed in claim 8, comprising:Complex demodulating a fundamental frequency component in the power monitor signal; determining a monitor power-carrier delay residual describing an error in the estimated monitor power-carrier delay, from the demodulated fundamental frequency component; and continually tracking said monitor power-carrier delay based on the monitor power-carrier delay residual.

10. A method as claimed in claim 9, comprising determining the monitor power-carrier delay residual by rotating the complex demodulated fundamental frequency component by an angle proportional to the monitor power-carrier delay as determined either in a previous iteration or from an initial estimate.

11. A method as claimed in any of claims 8 to 10, comprising: estimating a wavelength reference excess delay in a first calibration, the first calibration comprising: obtaining a calibration wavelength reference signal while a wavelength of the laser radiation is maintained between two neighboring absorption lines of the wavelength reference; complex demodulating an odd-order harmonic frequency component of the calibration wavelength reference signal; and nullifying a wavelength reference excess delay residual determined from the complex demodulate of the odd-order harmonic frequency component of the calibration wavelength reference signal, the wavelength reference excess delay residual describing an error in the estimate of the wavelength reference excess delay.

12. A method as claimed in claim 11, wherein the odd-order harmonic frequency component comprises a fundamental frequency component.

13. A method as claimed in claim 11 or 12, comprising determining the wavelength reference excess delay residual by rotating the complex demodulate of the odd-order harmonic frequency component by an angle proportional to the sum of the estimated monitor power-carrier delay and the estimated wavelength reference excess delay as determined in either a previous iteration or from an initial estimate.

14. A computer program comprising computer readable instructions operable to perform the method of any of claims 8 to 13.

15. A programmable firmware component programmed to implement the method of any of claims 8 to 13.

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