Patent Description:
Dielectric coatings are commonly used in many optical systems and optical instruments. These dielectric coatings have a significant wavelength dependent contribution to the overall image quality budget. Although it has been known since the <NUM>'s that dielectric coatings, by design, will introduce wavelength dependent phase shifts to incoming wave fronts, this issue has to the inventor's knowledge not been addressed in the scientific literature.

In particular, the phase shift in reflection and transmission may depart from the as-designed phase shift due to manufacturing errors, and it may even be the case that structures are imprinted on the dielectric coating's reflected/transmitted wave front during the coating manufacturing process. The impact on the performance of the optical system including dielectric coatings may or may not be catastrophic depending on the robustness of the coatings design against manufacturing errors and on the design itself.

In several highly demanding applications (such as weak lensing astronomy, e.g., Euclid, gravitational wave detection, e.g., LISA, LIGO/VIRGO, polarization microscopy, earth observation missions, interferometry for spectrometry, etc.) an accurate knowledge of the phase of the light after reflection/transmission is necessary. However, the accurate phase is often impossible to assess unambiguously, thus leading to extremely stringent requirements on the manufacturing accuracy of the dielectric coatings, or to comparatively large error budgets.

Known techniques for determining the phase of light after reflection/transmission by a dielectric coating include ellipsometry and spectral reflectance/transmittance. The former technique necessitates a priori knowledge of the exact design of the dielectric coating and access to a wavelength scanning ellipsometer. The latter technique is based on Kramers-Kronig relations. It again requires a priori knowledge of the exact design of the dielectric coating and moreover provides non-unique solutions leading to ambiguities on the actual phase shift of the dielectric coating.

Details of the coating design are often retained by the coating manufacturer as part of his intellectual property. However, both the aforementioned techniques are unusable if the exact design of the dielectric coating is unknown. Moreover, both techniques are rather cumbersome to implement.

Thus, there is a need for methods and corresponding apparatus for determining a phase shift caused by reflection at, or transmission through, a dielectric coating as function of wavenumber. There is further need for such methods that require minimal knowledge of the design (e.g., layer design) of the dielectric coating. There is yet further need for methods for determining a layer design for a dielectric coating that improve robustness of the as-designed phase shift in the presence of deviations from the layer design during manufacturing.

Document <NPL> generally proposes a method to evaluate the coating wavefront distortion due to nonuniform thickness errors by using a reflectometer.

In view of this need, the present document proposes a method of determining a phase shift caused by reflection at, or transmission through, a dielectric coating as function of wavenumber, corresponding apparatus, corresponding computer-executable program, and corresponding computer-readable storage media, having the features of the respective independent claims. The dependent claims relate to preferred embodiments.

An aspect of the disclosure relates to a method of determining a phase shift caused by reflection at, or transmission through, a dielectric coating as function of wavenumber. The method includes obtaining a nominal phase shift (φc(k)) for the dielectric coating as function of wavenumber. The nominal phase shift is a known phase shift that is applied to a wave front in reflection or transmission by the dielectric coating as designed. The nominal phase shift may be provided by a manufacturer of the dielectric coating, for example. The nominal phase shift may be given for normal incidence or may be given for plurality of directions of incidence. The method further includes determining a first wavenumber and a second wavenumber for performing measurements of phase shift at these wavenumbers, based on the nominal phase shift. The method further includes determining a wavenumber shift based on a first measurement of phase shift at the first wavenumber, a second measurement of phase shift at the second wavenumber, and the nominal phase shift as function of wavenumber. The wavenumber shift may emulate a phase shift resulting from thickness variations in the dielectric coating. The method yet further includes determining the phase shift (φr(k) or φt(k)) as function of wavenumber based on the wavenumber shift and the nominal phase shift. The phase shift may relate to a wave front error (WFE). The determined phase shift may be an actual, or as-manufactured, phase shift of the dielectric coating. The phase shift as function of wavenumber may include a first portion ( <MAT>) resulting from thickness variations in the dielectric coating and a second portion (φs(k)) resulting from local deformation of the dielectric coating due to substrate deformation.

Configured as such, the proposed method allows to infer the phase shift that is applied by an as-manufactured dielectric coating (e.g., coating stack) to a wave front in reflection or transmission. Only minimal knowledge of the coating design and wave front measurements at two suitably chosen wavelengths are necessary for achieving this aim. Thus, the as-manufactured phase shift can be determined even if details of the coating stack are unknown (e.g., since they are retained by the manufacturer of the coating as part of his intellectual property). Thereby, the accuracy of wave front measurements can be improved by taking into account the actual phase shift that is induced by the as-manufactured dielectric coating, instead of only considering the as-designed phase shift. Notably, the proposed method can be applied to any optical instrument or application that employs dielectric coatings.

In some embodiments, the method may further include determining an estimate of the wavenumber shift based on a simulation of a deviation of the dielectric coating from an as-designed configuration thereof. The simulation may involve perturbing the as-designed configuration of the dielectric coating. Then, determining the first wavenumber and the second wavenumber may be further based on the estimate of the wavenumber shift. In some embodiments, the wavenumber shift may be determined iteratively.

The determining the phase shift as function of wavenumber based on the wavenumber shift and the nominal phase shift includes determining a first contribution to the phase shift as function of wavenumber, the first contribution depending on thickness variations of stacked layers of the dielectric coating from their respective nominal thickness. The determining the phase shift as function of wavenumber based on the wavenumber shift and the nominal phase shift yet further includes determining a second contribution to the phase shift as function of wavenumber, the second contribution depending on local deformation of the dielectric coating due to substrate deformation. Then, the phase shift as function of wavenumber may be determined based on the first contribution and the second contribution. The phase shift as function of wavenumber is based on a sum of the first contribution and the second contribution.

In some embodiments, the method may further include determining the first contribution by shifting the nominal phase shift as function of wavenumber by the wavenumber shift, such that the first contribution, at a given wavenumber, is given by the nominal phase shift at a shifted wavenumber that is obtained by shifting the given wavenumber by the wavenumber shift. In other words, the first contribution <MAT> may be given by <MAT>, where ϕc(k) is the nominal phase shift and δk is the wavenumber shift.

In some embodiments, the method may further include determining a total optical path difference that is caused by local deformation of the dielectric coating based on the first measurement of phase shift at the first wavenumber, the second measurement of phase shift at the second wavenumber, and the nominal phase shift. Then, the phase shift as function of wavenumber may be determined further based on the total optical path difference. Determining the second contribution may involve obtaining a product of the total optical path difference and the wavenumber. Thereby, the proposed method can also account for irregularities of the substrate on which the dielectric coating is provided.

In some embodiments, the first and second wavenumbers may be determined such that they satisfy <MAT> <MAT>, where k<NUM> and k<NUM> are the first and second wavenumbers, respectively, and R(k) includes higher order terms of second order or higher of the Taylor expansion of the nominal phase shift at wavenumber k. Using such wavenumbers ensures high accuracy of the determined phase shift.

In some embodiments, the wavenumber shift may be determined as δk = <MAT>, where δk is the wavenumber shift, k<NUM> and k<NUM> are the first and second wavenumbers, respectively, OPD<NUM> and OPD<NUM> are optical path differences obtained in the first and second measurements, respectively, and ϕc(k) is the nominal phase shift as function of wavenumber.

In some embodiments, the steps of determining the wavenumber shift and determining the phase shift as function of wavenumber may be performed for each of a plurality of points on a surface of the dielectric coating. Thereby, a map of the phase shift across the surface or part of the surface of the dielectric coating may be obtained. Further, the steps of determining the wavenumber shift and determining the phase shift as function of wavenumber may be performed for each of a plurality of angles of incidence at the dielectric coating.

In some embodiments, the method may further include determining a phase shift at a third wavenumber that is different from both the first and second wavenumbers, based on the phase shift as function of wavenumber and the third wavenumber. Since the proposed method provides the phase shift as function of wavenumber, the phase shift (and thus the deformation of the wave front) can be calculated for any arbitrary wavenumber of interest.

In some embodiments, the method may further include obtaining a measured wave front after reflection by, or transmission through, the dielectric coating at the third wavenumber. Then, the method may yet further include determining a final output indicative of or depending on the measured wave front, based on the determined phase shift at the third wavenumber. This may involve calibrating the measurement based on the determined phase shift at the third wavenumber, correcting the measured wave front based on the determined phase shift at the third wavenumber, and/or modifying a quantity derived from the measured wave front based on the determined phase shift at the third wavenumber.

In some embodiments, the method may further include performing the first measurement of phase shift at the first wavenumber and the second measurement of phase shift at the second wavenumber. This may involve wave front measurements or wave front error measurements at the first and second wavenumbers.

In some embodiments, the method may further include obtaining design information relating to the dielectric coating. Then, the method may yet further include determining the nominal phase shift for the dielectric coating as function of wavenumber based on the design information.

In some embodiments, the dielectric coating may include a plurality of stacked layers. The plurality of stacked layers may be composed of different materials. Further, the plurality of layers may be supported by a substrate, e.g., may be stacked on the substrate.

In some embodiments, the design information may include a total optical thickness of the dielectric coating.

In some embodiments, the phase shift may further be a function of an angle of incidence at the dielectric coating.

In some embodiments, the method may further include correcting the phase shift as function of wavenumber based on a first spectral profile of a first light source that is used for the first measurement and/or a second spectral profile of a second light source that is used for the second measurement. Thereby, any deviation of the spectra of the first and second light sources from line spectra may be accounted for.

In some embodiments, the method may further include obtaining a result of a fourth measurement of a wave front after reflection by, or transmission through, the dielectric coating at a fourth wavenumber, the fourth wavenumber being different from the first and second wavenumbers. Then, the method may further include reconstructing the wave front after reflection by, or transmission through, the dielectric coating at the fourth wavenumber using the determined wavenumber shift and the nominal phase shift. The method may yet further include correcting the determined wavenumber shift based on a comparison of the measured and reconstructed wave fronts at the fourth wavenumber. Thereby, the determined wavenumber shift may be further refined and accuracy of the overall calculation of the phase shift may be increased.

Another aspect of the disclosure relates to an apparatus for determining a phase shift caused by reflection at, or transmission through, a dielectric coating as a function of wavenumber, the apparatus comprising one or more processors adapted to perform the method of the above aspect and its embodiments.

Another aspect of the disclosure relates to a computer-executable program that, when executed by a processor, causes the processor to perform the method of the above aspect and its embodiments.

Another aspect of the disclosure relates to a computer-readable storage medium storing the computer-executable program according to the preceding aspect.

An example suitable for understanding the invention of the disclosure relates to a method of determining a layer design for a dielectric coating. The dielectric coating may comprise a plurality of stacked layers. The method may include determining a plurality of layer designs that are in conformity with a desired optical property of the dielectric coating (e.g., a spectral reflectance R). The method may further include determining, for each of the layer designs, a metric Z that depends on the total optical thickness of the dielectric coating according to that layer design and a bandwidth of interest, in terms of wavenumbers, for which the dielectric coating is intended for use. The method may yet further include selecting, as a final layer design for the dielectric coating, that layer design among the plurality of layer designs that has the smallest value of the metric Z. The bandwidth of interest may be a pre-set parameter in accordance with a desired use of the dielectric coating.

By referring to the metric Z and measuring feasible layer designs against this metric, the proposed method aids the design of dielectric coatings with relaxed tolerance on thickness error.

In some possible implementations of the example, the metric Z may be given by or proportional to <MAT>, where x is the total optical thickness of the dielectric coating and Δk is the bandwidth of interest.

It will be appreciated that method steps and apparatus or system features may be interchanged in many ways. In particular, the details of the disclosed method can be implemented by an apparatus or system, and vice versa, as the skilled person will appreciate. Moreover, any of the above statements made with respect to methods are understood to likewise apply to apparatus and systems, and vice versa.

Exemplary embodiments of the disclosure are explained below with reference to the accompanying drawings, wherein.

In the following, example embodiments of the disclosure will be described with reference to the appended figures. Identical elements in the figures may be indicated by identical reference numbers, and repeated description thereof may be omitted.

In the following, the terms wavenumber k and wavelength λ may be used interchangeably, since these quantities are directly related to each other via k = 2π/λ. Thus, a function of wavenumber is implicitly also a function of wavelength, and vice versa.

The present disclosure addresses the problem in dielectric coatings that the phase shift departure of the as-manufactured dielectric coating departs from the as-designed phase shift. The phase shift is defined as the phase in reflection or in transmission added by a dielectric coating to an incoming wave front.

To address this problem, the present disclosure proposes a process enabling the design of dielectric coatings that are robust with respect to thickness errors from the phase shift point of view. The present disclosure further proposes a process for the reconstruction of the total phase of a wave front in reflection and transmission when reflected or transmitted by a dielectric coating for any arbitrary wavelength λ, with minimum prior knowledge on the details (e.g., layers definition) of the design of the dielectric coating. Notably, the proposed process only requires minimal knowledge on the as-designed dielectric coating and access to a wave front sensor with at least two wavelengths. Higher accuracy of the phase reconstruction of the reflected/transmitted wave front can be achieved if the wavelengths of the wave front sensor can be tuned.

Broadly speaking, the process for the reconstruction of the total phase of a wave front in reflection and transmission when reflected or transmitted by a dielectric coating for any arbitrary wavelength λ is based on the following basic property of the phase shift of a dielectric coating when submitted to small changes in the thickness of its layers. Namely, as the inventor has demonstrated, the phase shift of the dielectric coating as-manufactured can be approximated by a constant wavenumber shift δk applied to the phase shift of the dielectric coating as-designed. As a corollary, it has further been found that the robustness of a dielectric coating in terms of phase shift can be assessed by a simple unit-less metric <MAT> with x the total optical thickness seen by a ray going through the optical coating at normal incidence and Δk the bandwidth of interest in wavenumber.

Next, the theoretical framework for determining the phase shift of a dielectric coating will be described.

The phase shift φ with wavelength λ of multilayer coatings (e.g., dielectric coatings) is a well-known phenomenon [Baumeister, et al. The phase shift introduced by a simple reflection on a mirror is equal to π while the coating stack phase shift depends on its characteristic matrix [Born, et al. , <NUM>] [Abeles, <NUM>], the elements of which depend on the materials used and on the geometry of the layers of the coating stacks. In the absence of spatial non-uniformities, the phase φ is constant over the area covered by the beam footprint resulting in a piston term in the departing wave front. Thus, the point spread function of an imaging optical system at λ is not impacted by the coating phase shift. However, the coating non-uniformities introduced during manufacturing will have an impact on φ but a negligible one on the spectral reflectance [Strojnik, <NUM>]. Thus, the actual value of φ is dependent on λ, on the spatial coordinates on the coated surface and on the angle of incidence θ.

Theoretical work [Tikhonravov, et al. , <NUM>] has demonstrated that the phase shift of the reflected wave can be either an oscillating function of the wave number k = <NUM>π/λ or a decreasing function of k with the average slope being negative and twice the total thickness of the coating. The nature of the phase shift variation depends uniquely on the exact definition of the coating stack.

The problem of retrieving the phase shift introduced by the as-manufactured coating is equivalent to determining the characteristic matrix M of the as-manufactured coating. Since the coating is assumed to be non-uniform, the matrix M is dependent on the coordinates X = (x, y) on the coated surface. The aim is thus to determine a set of matrices, one per sampling point on the coated surface.

The elements of the matrix M(X) are dependent on the actual refractive index ni and thickness ti of each layer of the coating stack. The pair (ni, ti) can be determined experimentally using ellipsometry [Oreb, et al.

For the Laser Interferometer Gravitational-Wave Observatory (LIGO) project a characterization of the coating phase shift [Oreb, et al. , <NUM>] was performed by measuring the wave front error both in transmission and reflection, and ellipsometry measurements of the coatings to derive the thickness variation.

The approach used in [Oreb, et al. , <NUM>] requires the knowledge of the as-designed coating stack definition to retrieve the phase shift for each polarization. This information is usually not provided by coating manufacturers.

The characterization of φ is of primordial importance for the measurement of λ in spectroscopy using Fabry-Perot interferometers. For this application techniques were developed to retrieve φ from the measured spectral reflectance R(λ) using the Kramers-Kronig relations [Lichten, <NUM>] based on the definition of the reflectance <MAT>. This approach has two major drawbacks as far as, e.g., ESA's Euclid project is concerned. Namely, the resulting φ is not unique [Tikhonravov, et al. , <NUM>], and the measured R(λ) is averaged over an area larger than the sampling needed to retrieve the medium and high spatial frequencies.

A probability assessment method also has been proposed [Furman, et al. However, such method does not provide any knowledge on the spatial distribution of errors over the coated optical element.

The techniques discussed above are not compatible with, e.g., the Euclid project's needs and constraints. None provide easy access to the spatial distribution of phase error which will have an impact on the point spread function metrics. To obtain such knowledge large modifications in the current test plan would be needed (e.g., ellipsometry with high spatial sampling) and possibly the development of new optical test setups or modification of existing ones.

In the following, a process for deriving the wave front error (as example of phase shift phase shift) in reflection of a dielectric coating (e.g., coatings of a dichroic) is described. This process is based on the available information on the coatings. For the example of Euclid's dichroic, the available information includes: wave front error maps at two wavelengths in transmission and reflection of the dichroic assembly (with beam splitter coating and anti-reflection coating) at room temperature, at normal incidence with a <NUM>×<NUM> pixels resolution, the as-designed phase shift ϕc(λ) at normal incidence, the as-designed spectral reflectance R(λ) for different angles of incidence for the s and p polarizations, the as-designed total thickness of the coating stack, and the as-designed total thickness of each material in the coating stack. However, not all of the above information may be available in general.

An exact solution to calculate the derivatives δr(λ)/δdj and δt(λ)/δdj of the coating stack reflectance r(λ) and transmittance t(λ) versus the j-th layer's thickness dj has been derived in [Furman, et al. , <NUM>] and in [Mouchart, <NUM>]. However, the calculation requires to have a priori knowledge of the as-designed coating stack definition. An exact solution thus cannot be derived in the general case in which this information may not be available. The solution proposed by this disclosure will instead rely on measurements of the optical properties (e.g., wave front error) of the coating stack.

First, a description of the phases of interest will be given. The total phase λr(k,θ) in reflection/transmission is the combination of the phase shift <MAT> of the coating considered as a stack of perfectly parallel layers and the phase φs(k,θ) induced by local deformation of the coating layers due to substrate deformation. Thus, <MAT> with θ the angle of incidence on the first (e.g., uppermost) layer of the stack. If a given angle of incidence is considered, such as normal incidence, the argument θ may be omitted. For the purpose of this analysis, φs(k, θ) may be written as φs(k, θ) = k · l · cos(θ), with l the total optical path difference introduced by local surface defects. This term is independent of the wavelength λ. In some embodiments, the term φs(k, θ) may be neglected in Eq. (<NUM>).

Eq. (<NUM>) ignores the effect of the local slopes due to the coating substrate [Strojnik, <NUM>]. Typically, the impact is negligible.

As can be seen from Eq. (<NUM>), the total phase in reflection/transmission comprises (provided that the term φs(k, θ) is not neglected), a first contribution <MAT> that depends on thickness variations of stacked layers of the dielectric coating from their respective nominal thickness and a second contribution φs that depends on local deformation of the dielectric coating due to substrate deformation.

Next, phase variations with the thickness of the coating's layers will be described, following the approach described in [Tikhonravov, et al. , <NUM>] and considering the spectral coefficients of the multilayer coating in the complex wave-number plane v written as v = k + iσ with k = <NUM>π/λ. The spectral coefficients f<NUM> and f<NUM> are defined as <MAT> with t(v) the complex normalized amplitude transmittance and r(v) the complex normalized amplitude reflectance of the multilayer stack. It follows from Eq. (<NUM>) that the phase shift introduced by the coating stack in transmittance (respectively, in reflectance) is the argument of t(k) (respectively, r(k)) as defined in <MAT> with µj the zeros of f<NUM> and vj the zeros of f<NUM>. The geometrical meaning of Eq. (<NUM>) is explained in <FIG>.

The behavior of the phase shift with wavenumber k is dependent on the position of the zeros vj with respect to the k-axis. In case all the zeros are located in the upper half-plane, the phase shift is an oscillating function of k that does not change significantly in any wavenumber area. On the other hand, if the zeros are all located on the lower half-plane, the phase shift is a decreasing function of k with an average slope of <MAT>, with x the total optical thickness of the coating stack. The situation of all zeros vj being located in the lower half-plane is illustrated in <FIG>. In general, manufactured coatings have zeros located on either half-plane of the wavenumber complex plane.

The zeros of the functions f<NUM> and f<NUM> have the following properties: The average spacing between zeros is π/x with x the total optical path of the coating stack at normal incidence. Further, the range of values for σ is limited to a narrow band around the real wavenumber axis.

Next, the phase in transmission will be analyzed looking at the properties of the function f<NUM> at normal incidence. An error on the thickness of each individual layer will lead to a different structure in the coating stack and thus to a different definition of the zeros of the function f<NUM>. This is associated with a change in the total layer optical thickness denoted by x'. It will be assumed that the thickness variations are small enough to keep the sign of the of the coordinate σj of the zeros µj unchanged. In that case, for a given k, there is a wavenumber k'(j) such that arg <MAT> with <MAT> the zeros of the modified spectral coefficient. This is illustrated in <FIG>. The same holds for f<NUM>.

It will now be shown that, under certain conditions, the shift δkj = k'(j) - k is equal to δk = cste for all zeros and for both spectral coefficients. To this end, reference will be made to each term in Eq. (<NUM>) defining the phase shift in transmission for the as-manufactured coating stack.

The imaginary part σj of the complex wavenumber is limited to a narrow strip around the real wavenumber axis, thus the changes in σj can be neglected. Hence, <MAT>. Thus, under the condition δkj/kj « <NUM>, one obtains <MAT>.

Since <MAT>, it can be concluded that <MAT>. It is known that the zeros of the spectral coefficient f<NUM> are separated by π/x in average along the k-axis, and the same applies for the modified spectral coefficient <MAT> but with π/x'. Thus, one can write <MAT> with δj a second order term and <MAT>, with nj the refractive index and tj the thickness of the j-th layer of the coating stack. For the as-manufactured coating similar considerations hold. Thus, at first order: <MAT>.

Since the total optical paths x' and x are commensurate one can write <MAT>. With the condition that |ε| « <NUM>| it is straightforward to derive from Eq. (<NUM>) that ∀j, δkj ≈ δkj+<NUM> = δk. The above findings lead to the following conclusion: <MAT>.

Since the spectral dependence of f<NUM>(k) is analogous to that of f<NUM>(k), it follows that <MAT> arg[f<NUM>(k + δk)]. Consequently, the phase in reflection <MAT> of the as-manufactured coating stack can be derived from the as-designed phase arg[r(k)] = ϕc(k, <NUM>) as follows: <MAT>.

Eq. (<NUM>) describes an important relationship in the context of the present disclosure.

<FIG> is a flowchart schematically illustrating an example of a method <NUM> of determining a phase shift caused by reflection at a dielectric coating (φr(k)) or transmission through the dielectric coating (φt(k)) as function of wavenumber according to embodiments of the disclosure. This method is at least partly based on the fact that Eq. (<NUM>) does hold.

The dielectric coating comprises a plurality of stacked layers. By design, each layer has constant thickness. The plurality of stacked layers may be composed of different materials. In general, the plurality of stacked layers may have different indexes of refraction. The plurality of stacked layers may be provided on a substrate. The dielectric coating may be used in reflection mode and in transmission mode. In either case, a wave front incident on the dielectric coating will undergo a phase shift. This phase shift may be known for the as-designed (i.e., nominal) dielectric coating. However, the as-manufactured dielectric coating may be different from the as-designed dielectric coating due to manufacturing errors. Method <NUM> seeks to determine the actual phase shift of the as-manufactured dielectric coating, as function of wavenumber. The actual phase shift may have a portion resulting from thickness variations of the layers of the dielectric coating (assuming layers of constant thickness) and a portion resulting from local deformations of the dielectric coating due to substrate deformation.

Notably, the steps of method <NUM> may be performed for each of a plurality of points on the (surface of the) dielectric coating. Thereby, a map of the actual phase shift depending on position (x,y) on the dielectric coating can be obtained. Moreover, the steps of method <NUM> may be performed for each of a plurality of angles of incidence. Alternatively, the actual phase shift for a given angle of incidence may be analytically derived from the actual phase shift at another angle of incidence, e.g., normal incidence.

At step S110, a nominal phase shift ϕc(k) for the dielectric coating as function of wavenumber is obtained. The nominal phase shift is the (known) phase shift that would be applied to a wave front in reflection or transmission by the as-designed dielectric coating. The nominal phase shift may be given for normal incidence of light (e.g., of the wave front) on the dielectric coating. Alternatively, the nominal phase shift may be given for a plurality of distinct angles of incidence. In any case, either information is sufficient for determining the nominal phase shift at an arbitrary angle of incidence, e.g., via analytic relationships or interpolation.

The nominal phase shift φc(k) may be provided by the manufacturer of the dielectric coating, for example. Alternatively, the nominal phase shift may be determined from design information relating to the dielectric coating. In this case, step S110 would include the sub-steps of obtaining the design information of the dielectric coating (e.g., a configuration of the as-designed dielectric coating), and determining the nominal phase shift (as function of wavenumber) based on the design information. The design information may include a total optical thickness of the dielectric coating (as designed), for example.

At step S120, a first wavenumber k<NUM> and a second wavenumber k<NUM> for performing measurements of phase shift (e.g., wave front measurements or wave front error measurements) at these wavenumbers are determined based on the nominal phase shift.

Considering, in general, the measurement of the wave front of the coating stack in reflection at normal incidence at two different wavenumbers k<NUM> and k<NUM>, with k<NUM> < k<NUM>, the measured optical path difference OPDi is such that with the piston term removed <MAT> with OPDi the optical path difference dependent on the coordinates within the pupil measurement, l a constant optical path depending on the coating substrate surface irregularities, and i ∈ {<NUM>,<NUM>}. Notably, in some implementations of the present disclosure, coating substrate surface irregularities may be neglected (which would correspond to setting l = <NUM> in Eq. (<NUM>)).

The term ϕc(ki + δk) can be Taylor-expanded around ki: <MAT> with <MAT>. In some embodiments, N may be set to infinity, N = ∞. In other embodiments, a finite number of higher orders of the Taylor expansion of φc(ki + δk) may be retained. The set of equations in Eq. (<NUM>) have a unique solution for δk and l if and only if the wavenumbers k<NUM> and k<NUM> are chosen such that they satisfy (or substantially or approximately satisfy) <MAT>.

Thus, the first wavenumber k<NUM> and the second wavenumber k<NUM> may be determined such that they satisfy (or substantially or approximately satisfy) the relation <MAT>, where R(k) includes higher order terms (second order or higher) of the Taylor expansion of the nominal phase shift at wavenumber k. If a given set of fixed wavenumbers for performing measurements (e.g., wave front measurements or wave front error measurements) are available, those two wavenumbers may by chosen as the first and second wavenumbers that minimize χ. Accordingly, step S120 may also involve selecting, from a plurality of wavenumbers at which wave front measurements can be performed, that pair of wavenumbers that minimizes χ.

At step S130, a wavenumber shift δk is determined (e.g., calculated) based on a first measurement of phase shift at the first wavenumber (e.g., OPD<NUM>), a second measurement of phase shift at the second wavenumber (e.g., OPD<NUM>), and the nominal phase shift φc as function of wavenumber. This wavenumber shift δk emulates the phase shift resulting from thickness variations of the dielectric coating. That is, the as-manufactured phase shift at a given wavenumber k can be obtained by shifting the given wavenumber by the wavenumber shift δk and evaluating the nominal phase shift function ϕc at the shifted wavenumber k + δk.

For instance, with Eq. (<NUM>) satisfied, using Eq. (<NUM>) the set of equations in Eq. (<NUM>) can be solved for δk as <MAT>.

That is, the wavenumber shift δk can be determined using Eq. (<NUM>), for example.

In some embodiments, step S130 may further involve determining (e.g., calculating) a total optical path difference l that is caused by local deformation of the dielectric coating (e.g., due to substrate irregularities) based on the first measurement of phase shift at the first wavenumber (e.g., OPD<NUM>), the second measurement of phase shift at the second wavenumber (e.g., OPD<NUM>), and the nominal phase shift ϕc. The total optical path difference l may be further determined based on the first wavenumber k<NUM> and the second wavenumber k<NUM>.

For instance, the total optical path difference l can be derived from Eq. (<NUM>) and Eq. (<NUM>). In its simplest form the total optical path difference l can be expressed as: <MAT>.

In some embodiments, step S130 may include performing the first measurement of phase shift at the first wave number k<NUM> and performing the second measurement of phase shift at the second wave number k<NUM>.

At step S140, the phase shift as function of wavenumber is determined based on the wavenumber shift δk and the nominal phase shift ϕc. This phase shift is the actual phase shift that is applied to a wave front by the as-manufactured dielectric coating. The phase shift may relate to a wave front error, i.e., to a deviation of the actual wave front (the wave front resulting from reflection at, or transmission through, the as-manufactured dielectric coating) from the wave front that would be obtained for the as-designed (nominal) dielectric coating.

In some embodiments, the phase shift as function of wavenumber is determined further based on the total optical path difference l. For example, the second contribution of the phase shift may be determined based on the total optical path difference l.

Typically, the phase shift is further a function of angle of incidence at the dielectric coating, i.e., depends on the angle of incidence. To first approximation, the functional dependence on the angle of incidence may be approximated by a cosine function.

A more detailed derivation of phase variation with the angle of incidence θ will be provided below.

From [Baumeister, <NUM>] it is known that the phase shift ϕ<NUM> in reflection referenced at the top layer of the coating and the phase shift ϕe in reflection referenced at the last layer (laying on the substrate) are related by <MAT> with t the mechanical thickness of the coating stack. From Eq. (<NUM>) it is straightforward to show that for the as-manufactured coating stack one has <MAT>.

Then from Eq. (<NUM>), <MAT>. It is thus sufficient to know the phase shift as-designed, φc(k, θ).

The determination of the phase shift as-designed, ϕc(k, θ), requires to know the characteristic matrix of the coating accounting for the multi-reflections within each layer. The matrix in [Born, et al. , <NUM>] cannot be used as it is valid only for a single reflection for each layer. However, by calculating ϕc(k, θ<NUM>) for an angle of incidence θ<NUM> it is possible to derive the phase shift at any wavenumber by applying a wavenumber shift δkθ to the phase shift at normal incidence such that ϕc(k, θ) = φc(k + δkθ, <NUM>). Notably, this wavenumber shift δkθ is independent of the wavenumber shift δk: While the former implements (takes into account, or emulates) a difference in angle of incidence from an angle of incidence θ<NUM>, the latter implements (takes into account, or emulates) deviations of the as-manufactured coating from the as-designed coating. Thus, the wavenumber shift δkθ may be referred to as angular wavenumber shift, or the like. Finally, one obtains: <MAT> with δkθ<NUM> the wavenumber shift calculated for the angle of incidence θ<NUM>.

Eq. (<NUM>) is justified by the fact that the zeros spacing π/x of the coating stack will vary as cos(θ) since the total optical path will vary approximately as <NUM>/cos(θ).

To verify Eq. (<NUM>) an analysis of the phase shift for the coating of Euclid's folding mirror <NUM> (FOM3) has been performed based on the data available. The phase shift is calculated for this mirror for three different incidences for both polarizations TE and TM considering only the spectral range [<NUM>; <NUM>]. The wavenumber shift is derived by cross-correlation once the piston between the phase shift curves has been removed. The resulting (angular) wavenumber shift δkθ for TE and TM polarizations as function of angle of incidence θ is illustrated in <FIG>, in which curve <NUM> relates to TM polarization and curve <NUM> relates to TE polarization.

As expected, the wavenumber shift δkθ increases as cos(θ) increases and the variation is approximately linear. The correlation is lower for large angles of incidence and so is the resulting optical path error (OPE), as shown in <FIG> (OPE for TE polarization, θ<NUM> = <NUM>°, and θ = <NUM>°). The OPE can be significantly reduced if the phase at θ = <NUM>° is chosen as reference instead of θ<NUM> = <NUM>°. This is shown in <FIG> (OPE for TE polarization, θ<NUM> = <NUM>°, and θ = <NUM>°).

<FIG> is a flowchart schematically illustrating an example of method <NUM>, which is an implementation of step S120 in method <NUM> of <FIG>.

At step S210, an estimate of the wavenumber shift is determined based on a simulation of a deviation of the dielectric coating from an as-designed configuration thereof. The simulation may involve (analytically) perturbing the as-designed configuration of the dielectric coating.

At step S220, the first wavenumber and the second wavenumber are determined based on the nominal phase shift and the estimate of the wavenumber shift. That is, the first and second wavenumbers are determined further based on the estimate of the wavenumber shift.

For example, the first wavenumber k<NUM> and the second wavenumber k<NUM> may be determined such that they satisfy (or substantially or approximately satisfy) the relation <MAT>, where R(k) includes higher order terms (second order or higher) of the Taylor expansion of the nominal phase shift at wavenumber k, as defined in Eq. (<NUM>). At this point, the estimate of the wavenumber shift δk(e) may be used to evaluate R(k), e.g., via <MAT>, as described above. If a given set of fixed wavenumbers for performing measurements (e.g., wave front measurements or wave front error measurements) are available, those two wavenumbers may be chosen as the first and second wavenumbers that minimize χ. Accordingly, step S220 may also involve selecting, from a plurality of wavenumbers at which wave front measurements can be performed, that pair of wavenumbers that minimizes χ.

<FIG> is a flowchart schematically illustrating a method <NUM>, which is an implementation of step S140 in method <NUM> of <FIG>.

At step S310, a first contribution to the phase shift as function of wavenumber is determined. The first contribution depends on thickness variations of stacked layers of the dielectric coating from their respective nominal thickness. That is, the first contribution is that contribution to the total phase shift that depends on thickness variations (variations of constant-thickness layers from their nominal thickness) of the stacked layers of the dielectric coating.

The first contribution can be determined by shifting the nominal phase shift as function of wavenumber by the wavenumber shift. Accordingly, the first contribution, at a given wavenumber, would be given by the nominal phase shift evaluated at a shifted wavenumber, wherein the shifted wavenumber is obtained by shifting the given wavenumber by the wavenumber shift. In other words, the wavenumber shift indicates by how much the actual wavenumber would have to be shifted so that evaluating the nominal phase shift function at the shifted wavenumber yields the actual phase shift for the as-manufactured dielectric coating. If ϕc(k) indicates the nominal phase shift and <MAT> indicates the first contribution, the first contribution would be given by <MAT>, with δk the wavenumber shift.

At step S320, a second contribution to the phase shift as function of wavenumber is determined. The second contribution depends on local deformation of the dielectric coating due to substrate deformation. That is, the second contribution is that contribution to the total phase shift that depends on local deformation of the dielectric coating due to substrate deformation.

In some embodiments, determining the second contribution φs(k) may involve obtaining a product of the total optical path difference l and the wavenumber k. In particular, the second contribution φs(k,<NUM>) may be given by φs(k) = k · l · cosθ, as noted above.

At step S330, the phase shift as function of wavenumber is determined based on the first contribution and the second contribution. For example, the phase shift may be obtained by summing the first and second contributions, e.g., via <MAT>. In general, the phase shift may depend on a sum of the first and second contributions.

Using the method set out in <FIG> with optional details as set out in <FIG> and/or <FIG>, the phase shift φr can be calculated for any angle of incidence θ for both polarizations. A schematic summary of the determination of the phase shift is illustrated in <FIG>.

<FIG> is a flowchart schematically illustrating an example of method <NUM> implementing steps according to embodiments of the disclosure that may be performed in conjunction with (e.g., subsequent to) the steps in the flowchart of <FIG>.

At step S410, a result of a fourth measurement of a wave front after reflection by, or transmission through, the dielectric coating at a fourth wavenumber k<NUM> is obtained. The fourth wavenumber k<NUM> is different from the first and second wavenumbers k<NUM>, k<NUM>.

At step S420, a wave front after reflection by, or transmission through, the dielectric coating at the fourth wavenumber is reconstructed using the determined wavenumber shift δk and the nominal phase shift φc(k). This may involve determining (e.g., calculating) the phase shift φr(k<NUM>) at the fourth wavenumber and determining an outgoing wave front using the (known) incoming wave front and the phase shift φr(k<NUM>) at the fourth wavenumber, e.g., by adding the phase shift to the incoming wave front for each location on the coating.

At step S430, the determined wavenumber shift δk is corrected based on a comparison of the measured and reconstructed wave fronts at the fourth wavenumber k<NUM>. The corrected wavenumber shift may then be used instead of the wavenumber shift determined at step S130 of method <NUM>, e.g., in step S140 of method <NUM> and/or in the steps of method <NUM> described below.

<FIG> is a flowchart schematically illustrating an example of method <NUM> implementing further steps according to embodiments of the disclosure that may be performed in conjunction with (e.g., subsequent to) the steps in the flowchart of <FIG>.

At step S510, a phase shift at a third wavenumber k<NUM> that is different from both the first and second wavenumbers k<NUM>, k<NUM> is determined, based on the phase shift φr(k) as function of wavenumber and the third wavenumber k<NUM>, e.g., by evaluating the phase shift φr(k) as function of wavenumber at the third wavenumber k<NUM>.

At step S520, a measured wave front after reflection by, or transmission through, the dielectric coating at the third wavenumber k<NUM> is obtained.

At step S530, a final output indicative of or depending on the measured wave front is determined, based on the determined phase shift φr(k<NUM>) at the third wavenumber k<NUM>. This may involve calibrating the measurement of the wave front at the third wavenumber k<NUM> based on the determined phase shift ϕr(k<NUM>) at the third wavenumber k<NUM>, correcting the measured wave front at the third wavenumber k<NUM> based on the determined phase shift φr(k<NUM>) at the third wavenumber k<NUM>, and/or modifying (e.g., correcting) a quantity derived from the measured wave front based on the determined phase shift φr(k<NUM>) at the third wavenumber k<NUM>.

<FIG> is a flowchart schematically illustrating an example of a method <NUM> of determining a layer design for a dielectric coating according to embodiments of the disclosure. As noted above, the dielectric coating comprises a plurality of stacked layers, e.g., stacked on a substrate.

At step S610, a plurality of layer designs are determined that are in conformity with a desired optical property of the dielectric coating. The desired optical property may be a certain spectral reflectance, for example.

At step S620, a metric Z is determined for each of the layer designs. The metric Z depends on the total optical thickness x of the dielectric coating according to the respective layer design and a bandwidth Δk of interest, in terms of wavenumbers, for which the dielectric coating is intended for use. For example, the metric Z may be given by or proportional to <MAT>, where x is the total optical thickness of the dielectric coating and Δk is the bandwidth of interest.

At step S630, that layer design among the plurality of layer designs is selected as a final layer design for the dielectric coating, that has the smallest value of the metric Z.

In the above, a method of determining a phase shift after reflection/transmission by a dielectric coating as well as a method of determining a suitable layer design for a dielectric coating have been described. Using these methods, an example process according to embodiments of the disclosure may unfold as follows:.

The reconstructed wave front maps can be used to retrieve the final performance of an optical system including a dielectric coating. For example, the optical system can be an imaging system, a spectrometer, an interferometer etc. However, any optical application using dielectric coatings can profit from the methods and procedures of the present disclosure.

The accuracy of the reconstruction depends on the accuracy of the WFE measurement, the value of χ (the lower the better), and the intrinsic error of the approximation of the as-manufactured coating phase shift by a shift of the as-designed phase shift.

The only inputs needed from the coating manufacturer are the as-designed coating's phase shift ϕc, for example at normal incidence θ = <NUM>. For other angles of incidence θ there are two possibilities: Either to reconstruct the characteristic matrix of the as-designed coating, or to obtain the as-designed coating phase shift for discrete values of the angle of incidence θ (as many as necessary). For reconstructing the characteristic matrix, the following information is needed: the total mechanical thickness of the coating, the materials used in the coating (i.e., their optical properties, e.g., indexes of refraction), and the total mechanical thickness of each material in the as-designed coating.

In the remainder of this disclosure the phase shift in reflection induced by the dichroic coating of Euclid's Bridge Dichroic Model (BDM) will be described as an example.

In the context of this example, first, it will be checked that the conditions of applicability and accuracy of the model described above are met.

As discussed above, the spectral behavior of the phase changes dramatically when the zeros of the functions f<NUM> and f<NUM> move from the upper half-plane to the lower half-plane in the complex (σ; k) plane. Thus any sign changes in any zeros will have a detectable impact on the curve φ(k). The simplest tool available to check whether any changes occur is the normalized cross-correlation factor C(δk) = (φperturbed ★ φdesign)(δk) between the phase φperturbed of the coating stack with thickness errors and the as-designed phase curve φdesign. At the same time the maximum value of C(δkmax) = Cmax provides an estimate δkmax of the wavenumber shift for a given thickness error of the layers.

A sensitivity analysis of the phase versus the layers' thickness error has been performed. The analysis is divided in two steps: First, a constant change of thickness is applied to each layer of the coating stack. The values selected for the change are, e.g., <NUM>%,<NUM>%, <NUM>% and <NUM>%. Second, a Monte-Carlo analysis is performed in which the thickness of each individual layer departs randomly from its as-designed value with a standard deviation of ±<NUM>%. This procedure may be used for estimating the wavenumber shift in step S210 of method <NUM>, for example.

The outcome of the sensitivity analysis is illustrated in <FIG>, which shows the wrapped phase for different thickness errors, for λ ∈ [<NUM>, <NUM>]. From this figure, it can be concluded that a positive relative increase of the coating stack thickness induces a positive wavenumber shift δkmax > <NUM>. By symmetry it is assumed that for thinner coating stacks one would obtain δkmax < <NUM>.

Next, the first step of the sensitivity analysis will be described. For each case the factor C(δk) is calculated for different values of δk. The result is shown in <FIG>, which shows the normalized cross-correlation factor C(δk) at different thickness errors, for λ ∈ [<NUM>, <NUM>].

For a change of thickness of <NUM>%, Cmax ≈ <NUM> is found. Cmax drops quickly with the thickness error. Without applying the method described in the present disclosure, the difference between the as-designed phase shift and the phase shift of the perturbed coating stack can be tremendous, depending of course on the thickness error. This is illustrated in <FIG>, which shows the optical path error (OPE) calculated as OPE = (φas designed (k) - φas perturbed (k))/k as function of wavelength. For instance, for an error of <NUM>% the mean difference of OPE between the as-designed and as-perturbed coating is <NUM>, the absolute values of the OPE can be higher. Applying the wavenumber shift δkmax the OPE difference is significantly reduced. For <NUM>% the mean OPE is <NUM>. This is illustrated in <FIG> which shows the OPE calculated as OPE = (φas designed (k + δkmax) - φas perturbed (k))/k as function of wavelength.

Next, the second step of the sensitivity analysis will be described. The OPE for <NUM> runs of the Monte-Carlo analysis is shown in <FIG>, which shows the OPE as function of wavelength. The absolute OPE can reach very high values (e.g., > <NUM>) in some cases. The model accuracy is thus highly dependent on which layers are the most perturbed. However, the averaged (or mean) OPE over the band λ ∈ [<NUM>; <NUM>] amounts to <NUM> ± <NUM> and OPE ≤ <NUM>.

It is worth noticing that OPE is correlated to Cmax. This is illustrated in <FIG>, which shows the mean OPE, OPE, for different values of Cmax with <MAT>. The correlation with δkmax is less obvious. This can be seen from <FIG>, which shows the mean OPE, OPE for different values of δkmax.

Next, a summary of results of the sensitivity analysis will be provided. A constant change in the layer thickness is not necessarily the worst case scenario. For instance, for ±<NUM>% the error is about the same size as for +<NUM>%. Therefore the sensitivity analysis shall only be performed with a Monte-Carlo analysis with a large number of runs (e.g., ><NUM>). The results are summarized in Table <NUM> below.

Next, conditions on bias sources in the phase shift calculation will be described. The above condition |ε| « <NUM> can be rewritten as <MAT>. In the worst case all the layers of the as-manufactured coating stack would have their thickness changed by <NUM>% (as typically guaranteed by dichroic manufacturers). Thus, |ε| = <NUM>/<NUM> in this case.

The above condition δkmax/kj « <NUM> is aimed to eliminate a source of bias in the phase calculation. The estimated value δkmax = <NUM> × <NUM>-<NUM> rad. nm-<NUM> is used to assess the ratio δkmax/kj which is linear with λ.

Thus, one finds <MAT> for <NUM> ≤ λ ≤ <NUM>. The limitation to the spectral band of interest can be made since the contribution of the zeros µj or vj to the total phase shift decreases with increasing kj.

Next, conditions on the choice of wavelengths λ<NUM> and λ<NUM> (corresponding to wavenumbers k<NUM> and k<NUM>, respectively) will be described. The phase residual χ defined in Eq. (<NUM>) is assessed in <FIG> for different pairs of wavelengths (λ<NUM>; λ<NUM>) for δk = <NUM> · <NUM>-<NUM> rad. This value of δk corresponds to Cmax for a thickness error of <NUM>%. The smallest absolute value of the phase residual χ is achieved for (λ<NUM> = <NUM>; λ<NUM> = <NUM>). However, when applying all possible pairs of wavelengths to the reconstruction process it is consistently found that δk ≈ <NUM>-<NUM> rad. nm-<NUM> on average. <FIG> shows the phase residual χ for different pairs of wavelengths (λ<NUM>;λ<NUM>) for δk = <NUM>-<NUM> rad. As noted above, there is no correlation between the reconstruction error and the value of δk. It seems that the actual thickness errors are distributed such that δk is much larger than calculated in the sensitivity analysis. From <FIG> it can be seen that the wavelength pair (λ<NUM> = <NUM>; λ<NUM> = <NUM>) provides systematically low values of χ when compared to the others possibilities. However due to measurements biases for λ = <NUM> (see below) this wavelength pair may be disfavored.

When using the value δk ≈ <NUM>-<NUM> rad. nm-<NUM> the optimum wavelength pair is another one and the results of the reconstruction are much better. In the following, λ<NUM> = <NUM>, λ<NUM> = <NUM> will be used for the present example.

Next, a simplified model will be described. Eq. (<NUM>) can be applied to both polarizations. The wavenumber shift δkθ to be applied to account for the angle of incidence θ will be different depending on the polarization. Thus the phase shift must be determined for the two polarizations independently for a given angle of incidence θ<NUM> so that the terms δkTM(θ<NUM>) and δkTE(θ<NUM>) can be determined.

The phase shift of the coating as manufactured is thus provided by <MAT> with P the polarization (TM or TE), δkM the wavenumber shift caused by manufacturing errors (as determined, e.g., in step S130 of method <NUM>) and δkθ(P) the wavenumber shift caused by the angle of incidence for the polarization P.

Next, results of measurements and WFE reconstruction in the context of the present example will be described. Several measurements of WFEs in reflection at normal incidence (θ = <NUM> rad) were performed at the following wavelengths: <NUM>, <NUM>, <NUM>, and <NUM>. The measured WFE maps in reflection are shown in <FIG>.

A Zernike analysis of the WFE maps shows that there is a significant remnant focus for λ = <NUM>. Results of this analysis are summarized in Table <NUM>, which shows focus terms (Zernike fringe definition) in the measured WFE maps.

It can be concluded that the remnant defocus term added in the l map is about <MAT> nm RMS.

Also an analysis on the maps difference WFEwith focus - WFEno focus for each wavelength λ shows that the difference of focus content in each WFE map is not necessarily due to the coating. Results of this analysis are illustrated in <FIG>. Indeed, for λ = <NUM> there is a strong tilt remaining in the map being caused by the test set-up itself. The other WFE maps also have a remnant tilt term but to a lower extent.

In the WFE maps for λ = <NUM> and λ = <NUM>, some outliers data points are present resulting from the incomplete removal of the ghost signal created by the test set-up. Large WFEs are present locally around the center of the aperture. <FIG> shows the WFE map at <NUM> without tilt. The scale is in waves.

Next, the calculation of δk and l will be described. To take into account the spectral width of the laser diodes used in the measurements at λ<NUM> and λ<NUM>, the wavelengths listed in Table <NUM> may be used in the calculations.

The actual wavelengths are derived from the spectral profile of the laser diodes. The actual wavelengths may be the centroid wavelengths for each source. In the remainder of the present disclosure, reference will be made to the theoretical wavelengths for simplicity. For the calculations the actual values will be used.

Applying Eq. (<NUM>) for λ<NUM> = <NUM> and λ<NUM> = <NUM> the maps for δk and l shown in <FIG> can be derived. The left panel shows the δk map (units in rad. The right panel shows the l map (units in nm).

The statistics of the surface defects map l are <NUM> RMS and <NUM> Peak-to-valley (PtV). It is worth noting that the WFE map l is comparable both in statistics and features to the Surface Figure Error (SFE) map of the bare surface (without coating) measured over a diameter of <NUM> while the l map covers an area of Ø100 mm, giving confidence in the correctness of the calculations. The SFE map of the bare surface is illustrated in <FIG>. Before polishing the equivalent WFE is <NUM> RMS and <NUM> PtV in line with the values for the l map.

Next, WFE reconstruction will be described. The RMS values of the WFE maps calculated according to methods of the present disclosure are illustrated in <FIG> and <FIG>. Of these, <FIG> shows the RMS WFE as function of wavelength. <FIG> shows the residual maps obtained by subtracting the reconstructed WFE maps from the measured WFE maps. Residual statistics are also summarized in Table <NUM>. These residuals are considered to assess the validity of the WFE map reconstruction according to the present disclosure. Further, Table <NUM> shows a comparison of RMS and PtV between the reconstructed and the measured WFE maps.

Next, fine tuning of the wavenumber shift δk will be described. Both in general and in the context of the present example, the condition of Eq. (<NUM>) may not be fully met by the wavelengths λ<NUM> and λ<NUM> that are available for the WFE measurements, thus introducing a bias on δk. This bias may be compensated by an additive correction term δkc and the map l may be recalculated accordingly. The results for δkc = <NUM> × <NUM>-<NUM> nm-<NUM> are illustrated in <FIG>, which shows the RMS of reconstructed WFE maps as function of wavelength for δkc = <NUM> × <NUM>-<NUM> nm-<NUM>. The errors bars along the wavelength axis and RMS WFE axis are respectively given by the full width at half maximum of the diodes used in the measurement and the error budget on the WFE measurement (<NUM> RMS). The value of δkc = <NUM> × <NUM>-<NUM> nm-<NUM> is a compromise allowing the lowest value for the RMS residuals and the best fit for the RMS of each measured WFE map. In general, the determined wavenumber shift (e.g., determined at step S130 of method <NUM>) may be corrected using a correction term that depends on a deviation of χ from zero for the wavenumbers k<NUM> and k<NUM> actually used in the measurements.

Next, WFE map reconstruction residuals will be described. Using the fine-tuned δk and l maps the WFE maps for all the wavelengths in Table <NUM> are calculated and compared to the measured WFE maps. The statistics of the residuals calculated with the corrected δk are given in Table <NUM> and Table <NUM>. <FIG> shows the residual maps obtained by subtracting the reconstructed WFE maps (using the fine-tuned δk) from the measured WFE maps. <FIG> shows the corrected maps for δk (left panel) and l (right panel) without piston. <FIG> shows the reconstructed WFE maps (using the fine-tuned δk).

Next, an appraisal of the model error will be performed. Table <NUM> provides a first assessment of the WFE reconstruction error. It seems that the residuals are higher than the WFE measurement error for λ = <NUM> and λ = <NUM>. The explanations for these discrepancies are as follows.

For λ = <NUM>, the deviation between the reconstructed WFE map and the measured WFE map is driven by a few data points which can be discarded in a Zernike fit of the WFE map. Those deviating points are present in the measured WFE map for λ = <NUM>. However by removing the outliers in the reconstructed map at λ = <NUM> the RMS of residuals drops to <NUM> RMS and only data points close to the central obscuration are removed. The outliers points are due to an incorrect masking of the ghost signal present in the test setup for λ < <NUM>. <FIG> shows the filtered residual maps with outlier points removed.

For λ = <NUM>, the residual has a remnant focus (see <FIG>) and a strong remnant tilt (<FIG>) in addition to the outliers points. After removing these points, the RMS of the residual drops to <NUM> RMS. As can be seen from <FIG>, the focus introduced in the map l is about <NUM> RMS. Removing this focus bias from the residual map and the focus from the measured map will lead to a residual of <NUM> RMS. Taking into account the <NUM> RMS measurement accuracy, the final residual amounts to <NUM> RMS.

In summary, the present disclosure provides a method for reconstructing the phase of a wave front in reflection or transmission at a dielectric coating for arbitrary wavelength λ. From this method, a simplified model is derived by combining Eq. (<NUM>) and Eq. (<NUM>), yielding <MAT>.

The proposed model is valid for any design of the dielectric coating (e.g., dichroic). Only the parameter values will change with the design.

As an example, this simplified model is applied to the dichroic coating as-designed for Euclid's BDM. The phase in reflection in the visible channel is derived. Parameter values for the BDM dichroic are given in Table <NUM>.

δkM and l are determined from WFE measurements performed at λ<NUM> = <NUM> and λ<NUM> = <NUM> and with an additive correction δkc. The term δkθ(P) is calculated using Eq. (<NUM>).

For further and more accurate reconstruction it is recommended to first "clean" the measured WFE maps before calculating the δkM and l maps. The residual tilt and outliers data points should be removed from the WFE maps.

In addition, the as-designed phase shift at an angle θ<NUM> > <NUM>° should be used to derive δkθ<NUM> so as to reduce the error on the calculated phase shift.

Claim 1:
A method (<NUM>) of determining a phase shift caused by reflection at, or transmission through, a dielectric coating as function of wavenumber, the method (<NUM>) comprising:
obtaining (S110) a nominal phase shift for the dielectric coating as function of wavenumber, wherein the nominal phase shift is a known phase shift that is applied to a wave front in reflection or transmission by the dielectric coating as designed;
determining (S120) a first wavenumber and a second wavenumber for performing measurements of phase shift at these wavenumbers, based on the nominal phase shift;
determining (S130) a wavenumber shift based on a first measurement of phase shift at the first wavenumber, a second measurement of phase shift at the second wavenumber, and the nominal phase shift as function of wavenumber; and
determining (S140) the phase shift as function of wavenumber based on the wavenumber shift and the nominal phase shift,
wherein the determining (S140) the phase shift as function of wavenumber based on the wavenumber shift and the nominal phase shift comprises:
determining (S310) a first contribution to the phase shift as function of wavenumber, the first contribution depending on thickness variations of stacked layers of the dielectric coating from their respective nominal thickness; and
determining (S320) a second contribution to the phase shift as function of wavenumber, the second contribution depending on local deformation of the dielectric coating due to substrate deformation, and
wherein the phase shift as function of wavenumber is based on a sum of the first contribution and the second contribution.